diff --git a/工具/关键字筛选题号.py b/工具/关键字筛选题号.py index d5b078a3..5908a3eb 100644 --- a/工具/关键字筛选题号.py +++ b/工具/关键字筛选题号.py @@ -2,7 +2,7 @@ import os,re,json """---设置关键字, 同一field下不同选项为or关系, 同一字典中不同字段间为and关系, 不同字典间为or关系, _not表示列表中的关键字都不含, 同一字典中的数字用来供应同一字段不同的条件之间的and---""" keywords_dict_table = [ - {"content":["离子"]} + {"origin":["风暴"],"origin2":["40","45","57","67","78","81"]} ] """---关键字设置完毕---""" # 示例： keywords_dict_table = [ diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index 68b80a2f..1cc2a385 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,1471 +1,5624 @@ -ans +tags +015352 +第一单元 -021441 -错误, 正确, 错误, 错误 +015353 +第一单元 +015354 +第一单元 -021442 -D +015355 +第一单元 +015356 +第一单元 -021443 -C +015357 +第一单元 +015358 +第一单元 -021444 -A +015359 +第一单元 +015360 +第一单元 -021445 -C +015361 +第一单元 +015362 +第一单元 -021446 -D +015363 +第一单元 +015364 +第一单元 -021447 -$-390^\circ$ +015365 +第一单元 +015366 +第一单元 -021448 -$304^\circ$, $-56^\circ$ +015367 +第一单元 +015368 +第一单元 -021449 -$-144^\circ$ +015369 +第一单元 +015370 +第一单元 -021450 -二, 四 +015371 +第一单元 +015372 +第一单元 -021451 -(1) $\{\alpha|\alpha=60^\circ+k\cdot 360^\circ, \ k\in \mathbf{Z}\}$, $-300^\circ$, $60^\circ$, $420^\circ$; (2) $\{\alpha|\alpha = -21^\circ+k\cdot 360^\circ, \ k \in \mathbf{Z}\}$, $-21^\circ$, $339^\circ$, $699^\circ$ +015373 +第一单元 +015374 +第一单元 -021452 -\begin{tikzpicture}[>=latex] -\fill [pattern = north east lines] (30:2) arc (30:60:2) -- (0,0) -- cycle; -\draw (30:2) -- (0,0) -- (60:2); -\draw [->] (-2,0) -- (2,0) node [below] {$x$}; -\draw [->] (0,-2) -- (0,2) node [left] {$y$}; -\draw (0,0) node [below left] {$O$}; -\end{tikzpicture} +015375 +第一单元 +015376 +第一单元 -021453 -$-1290^{\circ}$;第二象限 +015377 +第一单元 +015378 +第一单元 -021454 -(1) $ \{\alpha|\alpha=45^{\circ}+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$;\\ -(2) $\{\alpha|\alpha=135^{\circ}+k\cdot 180^{\circ}, \ k \in \mathbf{Z}\}$;\\ -(3) $\{\alpha|\alpha=45^{\circ}+k\cdot 90^{\circ}, \ k \in \mathbf{Z}\}$;\\ -(4) $\{\alpha|180^{\circ}+k\cdot 360^{\circ}<\alpha<270^{\circ}+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$. +015379 +第一单元 +015380 +第一单元 -021455 -(1) $ \{\beta|\beta=\alpha+180^{\circ}+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$;\\ -(2) $\{\beta|\beta=\alpha+90^{\circ}+k\cdot 180^{\circ}, \ k \in \mathbf{Z}\}$;\\ -(3) $\{\beta|\beta=-\alpha+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$;\\ -(4) $\{\beta|\beta=90^{\circ}-\alpha+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$. +015381 +第一单元 +015382 +第一单元 -021456 -C +015383 +第一单元 +015384 +第一单元 -021457 -B +015385 +第一单元 +015386 +第一单元 -021458 -$\dfrac{\pi}{12}$; $\dfrac{7\pi}{12}$; $\dfrac{5\pi}{4}$; $300^{\circ}$; $324^{\circ}$; $315^{\circ}$; $(\dfrac{270}{\pi})^{\circ}$ +015387 +第一单元 +015388 +第一单元 -021459 -(1)$\frac{50\pi+180}{9}$;(2)$\frac{250\pi}{9}$ +015389 +第一单元 +015390 +第一单元 -021460 -$\sqrt{3}$ +015391 +第一单元 +015392 +第一单元 -021461 -(1)$\frac{\pi}{3}$;(2)$\frac{2\pi}{3}$ +015393 +第一单元 +015394 +第一单元 -021462 -(1)$16\pi+\frac{2\pi}{3}$,二;\\ -(2)$-18\pi+\frac{4\pi}{3}$,三;\\ -(3)$-2\pi+\frac{7\pi}{5}$,三;\\ -(4)$-2\pi+\frac{3\pi}{4}$,二. +015395 +第一单元 +015396 +第一单元 -021463 -$\frac{1}{2}$ +015397 +第一单元 +015398 +第一单元 -021464 -(1) $\{\alpha|-\frac{\pi}{2}+2k\pi<\alpha<2k\pi,\ k \in \mathbf{Z}\}$;\\ -(2) $\{\alpha|\alpha=\frac{k\pi}{2},\ k \in \mathbf{Z}\}$. +015399 +第一单元 +015400 +第一单元 -021465 -(1) $\beta=\alpha+2k\pi,\ k \in \mathbf{Z}$;\\ -(2) $\beta=-\alpha+2k\pi,\ k \in \mathbf{Z}$;\\ -(3) $\beta=-\alpha+\pi+2k\pi,\ k \in \mathbf{Z}$;\\ -(4) $\beta=\alpha+\pi+2k\pi,\ k \in \mathbf{Z}$. +015401 +第一单元 +015402 +第一单元 -021466 -(1) $\{\alpha|-\frac{\pi}{4}+2k\pi \le \alpha \le \frac{\pi}{2}+2k\pi,\ k \in \mathbf{Z}\}$;\\ -(2) $\{\alpha|\frac{\pi}{6}+k\pi \le \alpha \le \frac{5\pi}{6}+k\pi,\ k \in \mathbf{Z}\}$. +015403 +第一单元 +015404 +第一单元 -021467 -(1) 第四象限；第四象限；\\ -(2) 第二象限或者第四象限；第一象限或第二象限或者$y$轴正半轴. +015405 +第一单元 +015406 +第一单元 -021468 -$A\cap B=\{\alpha | 2k \pi+\dfrac{5\pi}{6}<\alpha<2k \pi+\dfrac{7\pi}{6},\ k \in \mathbf{Z} \}$ +015407 +第一单元 +015408 +第一单元 -021469 -\begin{tabular}{|c|c|c|c|c|c|} -\hline &$P(-5,12)$&$P(0,-6)$&$P(6,0)$&$P(-9,-12)$&$P(1,-\sqrt{3})$\\ -\hline$\sin \alpha$&$\dfrac{12}{13}$ &$-1$ & $0$&$-\dfrac{4}{5}$ &$-\dfrac{\sqrt{3}}2$ \\ -\hline$\cos \alpha$&$-\dfrac{5}{13}$ &$0$ & $1$&$-\dfrac{3}{5}$ &$\dfrac 12$ \\ -\hline$\tan \alpha$&$-\dfrac{12}{5}$ &不存在 & $0$&$\dfrac{4}{3}$ &$-\sqrt{3}$ \\ -\hline$\cot \alpha$&$-\dfrac{5}{12}$ &$0$ & 不存在 &$\dfrac {3}{4}$ &$-\dfrac{\sqrt{3}}3$ \\ -\hline -\end{tabular} +015409 +第一单元 +015410 +第一单元 -021470 -$2\sqrt{5}$ +015411 +第一单元 +015412 +第一单元 -021471 -$\frac{2\sqrt{13}}{13}$;$-\frac{2}{3}$ +015413 +第一单元 +015414 +第一单元 -021472 -$ \left( -2,\frac{2}{3} \right)$ +015415 +第一单元 +015416 +第一单元 -021473 -$<$ +015417 +第一单元 +015418 +第一单元 -021474 -5 +015419 +第一单元 +015420 +第一单元 -021475 -2 +015421 +第一单元 +015422 +第一单元 -021476 -当$t=\sqrt{5}$时, $\cos \alpha=- \frac{\sqrt{6}}{4}$, $\tan \alpha =- \frac{\sqrt{15}}{3}$;\\ -当$t=-\sqrt{5}$时, $\cos \alpha=- \frac{\sqrt{6}}{4}$, $\tan \alpha = \frac{\sqrt{15}}{3}$;\\ -当$t=0$时, $\cos \alpha=-1$, $\tan \alpha = 0$. +015423 +第一单元 +015424 +第一单元 -021477 -当$\alpha$在第二象限时，$ \sin \alpha =\frac{4}{5}$, $\tan \alpha=-\frac{4}{3}$;\\ -当$\alpha$在第三象限时，$ \sin \alpha =-\frac{4}{5}$, $\tan \alpha=\frac{4}{3}$. +015425 +第一单元 +015426 +第一单元 -021478 -$-\frac{\sqrt{3}}{4}$ +015427 +第一单元 +015428 +第一单元 -021479 -(1) 第四象限； (2) 第一、四象限；(3)第一、三象限；(4)第一、三象限. +015429 +第一单元 +015430 +第一单元 -021480 -$A=\left\{ -2,-0,4 \right\}$ +015431 +第一单元 +015432 +第一单元 -021481 -(1) $\{\alpha|2k\pi \le \alpha \le \frac{\pi}{2}+2k\pi,\ k \in \mathbf{Z}\}$;\\ -(2) $[0,3)$ +015433 +第一单元 +015434 +第一单元 -021482 -\begin{center} -\begin{tabular}{|c|c|c|c|c|c|} -\hline$\alpha$&$\dfrac{\pi}{3}$&$\dfrac{7 \pi}{4}$&$\dfrac{2021 \pi}{2}$&$-\dfrac{\pi}{6}$&$-\dfrac{22 \pi}{3}$\\ -\hline$\sin \alpha$& $\frac{\sqrt{3}}{2}$ &$-\frac{\sqrt{2}}{2}$ & $1$&$-\frac{1}{2}$ &$\frac{\sqrt{3}}{2}$ \\ -\hline$\cos \alpha$&$\frac{1}{2}$ &$\frac{\sqrt{2}}{2}$ & $0$&$\frac{\sqrt{3}}{2}$ &$-\frac{1}{2}$ \\ -\hline$\tan \alpha$&$\sqrt{3}$ &$-1$ & 不存在 &$-\frac{\sqrt{3}}{3}$ &$-\sqrt{3}$\\ -\hline$\cot \alpha$&$\frac{\sqrt{3}}{3}$ &$-1$ & $ 0$&$-\sqrt{3}$ &$-\frac{\sqrt{3}}{3}$ \\ -\hline -\end{tabular} -\end{center} +015435 +第一单元 +015436 +第一单元 -021483 -(1) $\{x|x=\frac{4\pi}{3}+2k \pi$或$ x=\frac{5\pi}{3}+2k \pi,\ k \in \mathbf{Z} \}$;\\ -(2) $\{-\frac{2\pi}{3},-\frac{\pi}{3},\frac{4\pi}{3} ,\frac{5\pi}{3},\frac{10\pi}{3},\frac{11\pi}{3} \}$ +015437 +第一单元 +015438 +第一单元 -021484 -$-\frac{2\sqrt{5}}{5}$;$2$ +015439 +第一单元 +015440 +第一单元 -021485 -\textcircled{2} \textcircled{4} +015441 +第一单元 +015442 +第一单元 -021486 -当$\alpha$在第一象限时，$ \sin \alpha =\frac{3\sqrt{10}}{10}$, $\cos \alpha =\frac{\sqrt{10}}{10}$,$\tan \alpha=3$;\\ -当$\alpha$在第三象限时，$ \sin \alpha =-\frac{3\sqrt{10}}{10}$,$\cos \alpha =-\frac{\sqrt{10}}{10}$, $\tan \alpha=3$. +015443 +第一单元 +015444 +第一单元 -021487 -$\sin k\pi =0$;\\$\cos k\pi=\left\{ - \begin{array}{lc} - $1$, & k=2n \\ - $ -1$ , &k=2n-1\\ - \end{array} -\right.$ ($n \in \mathbf{Z}$). +015445 +第一单元 +015446 +第一单元 -021488 -(1) $\{\theta | 2k \pi+\dfrac{\pi}{3}<\theta<2k \pi+\dfrac{2\pi}{3},\ k \in \mathbf{Z} \}$;\\ -(2) $\{\theta | k \pi-\dfrac{\pi}{2}<\theta \le k \pi-\dfrac{\pi}{6},\ k \in \mathbf{Z} \}$;\\ -(3) $\{\theta | k \pi+\dfrac{\pi}{3} \le \theta \le k \pi+\dfrac{2\pi}{3},\ k \in \mathbf{Z} \}$. +015447 +第一单元 +015448 +第一单元 -021489 -第二象限 +015449 +第一单元 +015450 +第一单元 -021490 -(1) 当$\dfrac{\alpha}{2}$在第二象限时，点$P$在第四象限；\\ -当$\dfrac{\alpha}{2}$在第四象限时，点$P$在第二象限.\\ -(2) $\sin (\cos \alpha) \cdot \cos (\sin \alpha)<0$ +015451 +第一单元 +015452 +第一单元 -021491 -当$m=0$时，$ \cos (\alpha+1905^{\circ})=-1$,$\tan (\alpha-615^{\circ})=0$;\\ -当$m=\sqrt{5}$时，$ \cos (\alpha+1905^{\circ}) =-\frac{\sqrt{6}}{4}$,$\tan (\alpha-615^{\circ})=-\frac{\sqrt{15}}{3}$;\\ -当$m=-\sqrt{5}$时，$ \cos (\alpha+1905^{\circ}) =-\frac{\sqrt{6}}{4}$,$\tan (\alpha-615^{\circ})=\frac{\sqrt{15}}{3}$. +015453 +第一单元 +015454 +第一单元 -021492 -$-\dfrac{3}{8}$ +015455 +第一单元 +015456 +第一单元 -021493 -$-\dfrac{1}{20}$ +015457 +第一单元 +015458 +第一单元 -021494 -$\dfrac{7\sqrt{2}}{4}$ +015459 +第一单元 +015460 +第一单元 -021495 -$\dfrac{3\sqrt{5}}{5}$ +015461 +第一单元 +015462 +第一单元 -021496 -$11$ +015463 +第一单元 +015464 +第一单元 -021497 -$5$;$-\dfrac{12}{5}$;$\dfrac{4}{9}$ +015465 +第一单元 +015466 +第一单元 -021498 -$\sin ^2 \alpha$ +015467 +第一单元 +015468 +第一单元 -021499 -$1$ +015469 +第一单元 +015470 +第一单元 -021502 -$-\dfrac{12}{5}$ +015471 +第一单元 +015472 +第一单元 -021503 -$-\dfrac{\sqrt{3}}{2}$ +015473 +第一单元 +015474 +第一单元 -021504 -$\dfrac{\sqrt{7}}{2}$;$\dfrac{\sqrt{7}}{4}$ +015475 +第一单元 +015476 +第一单元 -021505 -$-\dfrac{\sqrt{11}}{3}$ +015477 +第一单元 +015478 +第一单元 -021506 -$\dfrac{\pi}{3}$ +015479 +第一单元 +015480 +第一单元 -021507 -$\left[ 0,\pi \right )$ +015481 +第一单元 +015482 +第一单元 -021508 -$-\dfrac{\sqrt{3}}{2}$;$-\dfrac{\sqrt{2}}{2}$;$-\sqrt{3}$;$-\sqrt{3}$ +015483 +第一单元 +015484 +第一单元 -021509 -$69^{\circ}$;$72^{\circ}$;$\dfrac{\pi}{9}$;$\dfrac{7 \pi}{15}$ +015485 +第一单元 +015486 +第一单元 -021510 -$\cot \alpha$ +015487 +第一单元 +015488 +第一单元 -021511 -$-1$ +015489 +第一单元 +015490 +第一单元 -021512 -$-1$ +015491 +第一单元 +015492 +第一单元 -021513 -$ \sin 2-\cos 2$ +015493 +第一单元 +015494 +第一单元 -021514 -$0$ +015495 +第一单元 +015496 +第一单元 -021515 -$0$ +015497 +第一单元 +015498 +第一单元 -021516 -$-\dfrac{\sqrt{1-a^2}}{a}$ +015499 +第一单元 +015500 +第一单元 -021517 -$-\dfrac{2+\sqrt{3}}{3}$ +015501 +第一单元 +015502 +第一单元 -021518 -(1) $\dfrac{\sqrt{3}}{2}$;(2) $\dfrac{1}{4}$. +015503 +第一单元 +015504 +第一单元 -021519 -(1) $-\dfrac{2}{3}$; \\ -(2) $\dfrac{2}{3}$; \\ -(3) $-\dfrac{\sqrt{5}}{3}$;\\ -(4) $\dfrac{\sqrt{5}}{2}$. +015505 +第一单元 +015506 +第一单元 -021520 -(1) $\sin 69^{\circ}$ ; (2) $-\cos 8^{\circ}$ ; -(3) $-\tan \dfrac{\pi}{9}$; (4) $\cot \dfrac{7\pi}{15}$. +015507 +第一单元 +015508 +第一单元 -021521 -$\dfrac{2}{5}$ +015509 +第一单元 +015510 +第一单元 -021522 -$(3,4)$ +015511 +第一单元 +015512 +第一单元 -021523 -$0$ +015513 +第一单元 +015514 +第一单元 -021524 -$\sin \alpha$ +015515 +第一单元 +015516 +第一单元 -021525 -$-\dfrac{1}{5}$ +015517 +第一单元 +015518 +第一单元 -021526 -(1) $\dfrac{\sqrt{6}}{6}-\sqrt{3}$;\\ -(2) $-\dfrac{\sqrt{6}}{3}$;\\ -(3) $1$ +015519 +第一单元 +015520 +第一单元 -021527 -(1) $\dfrac{6 \pi}{5}$; (2) $\dfrac{4 \pi}{5}$; (3) $\dfrac{13 \pi}{10}$; (4) $\dfrac{17 \pi}{10}$. +015521 +第一单元 +015522 +第一单元 -021528 -(1) 当$\alpha$在第一象限时, $\sin (2 \pi-\alpha)=-\dfrac{\sqrt{3}}{2}$; -当$\alpha$在第三象限时, $\sin (2 \pi-\alpha)=\dfrac{\sqrt{3}}{2}$.\\ -(2) 当$\alpha$在第一象限时, $\dfrac{1}{\tan [\dfrac{(2 k+1) \pi}{2}+\alpha]}=-\sqrt{3}$; -当$\alpha$在第四象限时, $\dfrac{1}{\tan [\dfrac{(2 k+1) \pi}{2}+\alpha]}=\sqrt{3}$. +015523 +第一单元 +015524 +第一单元 -021529 -(1) $\{x | x=k \pi+ (-1)^k \cdot \dfrac{\pi}{4},\ k \in \mathbf{Z}\}$;\\ -(2) $\{x | x=2k \pi \pm \dfrac{2\pi}{3},\ k \in \mathbf{Z}\}$;\\ -(3) $\{x | x=k \pi + \dfrac{5\pi}{6},\ k \in \mathbf{Z}\}$;\\ -(4) $\{x | x=2k \pi + \dfrac{5\pi}{6}$ 或$x=2k \pi + \dfrac{3\pi}{2} ,\ k \in \mathbf{Z}\}$;\\ -第二种写法： $\{x | x=k \pi+ (-1)^k \cdot \dfrac{\pi}{6}+\dfrac{2\pi}{3},\ k \in \mathbf{Z}\}$;\\ -(5) $\{x | x=k \pi - \arctan \dfrac{\sqrt{3}}{2}+ \dfrac{\pi}{4},\ k \in \mathbf{Z}\}$;\\ -(6) $\{x | x=\dfrac{2k \pi}{5} + \dfrac{7\pi}{60}$ 或$ x=\dfrac{2k \pi}{5} - \dfrac{13\pi}{60} ,\ k \in \mathbf{Z}\}$;\\ -(7) $\{x | x=k \pi - \dfrac{5\pi}{8}$ 或$x=k \pi - \dfrac{3\pi}{8} ,\ k \in \mathbf{Z}\}$; +015525 +第一单元 +015526 +第一单元 -021530 -(1) $\{ \dfrac{\pi}{12},\dfrac{17\pi}{12} \}$;\\ -(2) $\{ \dfrac{5\pi}{6} \}$;\\ -(3) $\{ \dfrac{\pi}{12},\dfrac{5\pi}{12} \}$;\\ -(4) $\{ \dfrac{5\pi}{6} \}$. +015527 +第一单元 +015528 +第一单元 -021531 -(1) $\{x | x= \dfrac{2k \pi}{5} ,\ k \in \mathbf{Z}\}$;\\ -(2) $\{x | x= \dfrac{2k \pi}{3} +\dfrac{ \pi}{6},\ k \in \mathbf{Z}\}$;\\ -(3) $\{x | x= 2k \pi$ 或$x=k \pi +(-1)^k \cdot \dfrac{ \pi}{6},\ k \in \mathbf{Z}\}$;\\ -(4) $\{x | x= k \pi+\dfrac{ \pi}{3}$ 或$x=k \pi -\dfrac{ \pi}{6},\ k \in \mathbf{Z}\}$. +015529 +第一单元 +015530 +第一单元 -021532 -$\dfrac{3+4\sqrt{3}}{10}$ +015531 +第一单元 +015532 +第一单元 -021533 -$-1$ +015533 +第一单元 +015534 +第一单元 -021534 -$-\dfrac{33}{50}$ +015535 +第一单元 +015536 +第一单元 -021535 -(1) $\dfrac{\sqrt{6}-\sqrt{2}}{4}$; -(2) $\dfrac{\sqrt{6}+\sqrt{2}}{4}$; -(3) $0$. +015537 +第一单元 +015538 +第一单元 -021536 -(1) $\sqrt{3} \sin \alpha$; -(2) $\cos(\alpha-2\beta)$. +015539 +第一单元 +015540 +第一单元 -021537 -$\dfrac{140}{221}$ +015541 +第一单元 +015542 +第一单元 -021538 -$\dfrac{2\sqrt{6}-1}{6}$ +015543 +第一单元 +015544 +第一单元 -021540 -C +015545 +第一单元 +015546 +第一单元 -021541 -A +015547 +第一单元 +015548 +第一单元 -021542 -$\dfrac{3\sqrt{10}+6\sqrt{2}+2\sqrt{14}-\sqrt{70}}{24}$ +015549 +第一单元 +015550 +第一单元 -021543 -$\dfrac{8\sqrt{3}-21}{20}$ +015551 +第一单元 +015552 +第一单元 -021544 -$\dfrac{\pi}{2}$ +015553 +第一单元 +015554 +第一单元 -040018 -(1) $\dfrac{\pi}{4}$; (2) $\dfrac{\pi}{6}$; (3) $\dfrac{\pi}{10}$; (4) $\dfrac{\pi}{3}$; (5) $\dfrac{5\pi}{12}$; (6) $\dfrac{\pi}{15}$ +015555 +第一单元 +015556 +第一单元 -040019 -(1) $60^{\circ}$; (2) $36^{\circ}$; (3) $45^{\circ}$; (4) $75^{\circ}$; (5) $40^{\circ}$; (6) $54^{\circ}$ +015557 +第一单元 +015558 +第一单元 -040020 -(1) $2k\pi+\dfrac{\pi}{2}$; (2) $2k\pi+\dfrac{3\pi}{2}$; (3) $2k\pi+\dfrac{7\pi}{6}$; (4) $k\pi+\dfrac{\pi}{4}$; (5) $\dfrac{k\pi}{2}+\dfrac{\pi}{6}$ +015559 +第一单元 +015560 +第一单元 -040021 -(1) $k \times 360^{\circ}+60^{\circ}$;\\ -(2) $k \times 360^{\circ}+330^{\circ}$; \\ -(3) $k \times 360^{\circ}-210^{\circ}$; \\ -(4) $k \times 180^{\circ}-45^{\circ}$; \\ -(5) $k \times 90^{\circ}+50^{\circ}$ +015561 +第一单元 +015562 +第一单元 -040022 -(1) $330^{\circ}$; (2) $240^{\circ}$; (3) $210^{\circ}$; (4) $300^{\circ}$ +015563 +第一单元 +015564 +第一单元 -040023 -(1) $\dfrac{4\pi}{3}$; (2) $\dfrac{11\pi}{6}$; (3) $10-2\pi$; (4) $-10+4\pi$ +015565 +第一单元 +015566 +第一单元 -040024 -$18$ +015567 +第一单元 +015568 +第一单元 -040025 -$3$,$-2$ +015569 +第一单元 +015570 +第一单元 -040026 -(1) $1037$; (2) $-4k+53$; (3) $500$ +015571 +第一单元 +015572 +第一单元 -040027 -$-2n+10$ +015573 +第一单元 +015574 +第一单元 -040028 -15 +015575 +第一单元 +015576 +第一单元 -040029 -$7$ +015577 +第一单元 +015578 +第一单元 -040030 -$(4,\dfrac{14}{3}]$ +015579 +第一单元 +015580 +第一单元 -040031 -$2n-1$ +015581 +第一单元 +015582 +第一单元 -040032 -$(3,\dfrac{35}{9})$或$(\dfrac{35}{9},3)$ +015583 +第一单元 +015584 +第一单元 -040033 -$200$ +015585 +第一单元 +015586 +第一单元 -040034 -略 +015587 +第一单元 +015588 +第一单元 -040035 -$a_n=\begin{cases}1, & n=1,\\ 2n, & n=2k, \\ 2n-2, & n=2k+1\end{cases}$($k\in \mathbf{N}$, $k\ge 1$) +015589 +第一单元 +015590 +第一单元 -040036 -$6n-3$ +015591 +第一单元 +015592 +第一单元 -040057 -$\dfrac{19}{28}\sqrt{7}$ +015593 +第一单元 +015594 +第一单元 -040058 -$\dfrac{79}{156}$ +015595 +第一单元 +015596 +第一单元 -040059 -$2$ +015597 +第一单元 +015598 +第一单元 -040060 -$-\dfrac{\sqrt{1-m^2}}{m}$ +015599 +第一单元 +015600 +第一单元 -040061 -$-\dfrac{1}{5}, \dfrac{1}{5}$ +015601 +第一单元 +015602 +第一单元 -040062 -$-\dfrac{1}{3}, 3$ +015603 +第一单元 +015604 +第一单元 -040063 -$\dfrac{1}{2}, -2$ +015605 +第一单元 +015606 +第一单元 -040064 -$\dfrac{\sqrt{6}}{3}$ +015607 +第一单元 +015608 +第一单元 -040065 -$\dfrac{1}{3}, -\dfrac{9}{4}$ +015609 +第一单元 +015610 +第一单元 -040066 -$\dfrac{1}{3}, \dfrac{7}{9}$ +015611 +第一单元 +015612 +第一单元 -040067 -$\pm\dfrac{\sqrt{2}}{3}$ +015613 +第一单元 +015614 +第一单元 -040068 -$\dfrac{1}{4}, \dfrac{2}{5}$ +015615 +第一单元 -040069 -$\dfrac{1-\sqrt{17}}{4}$ +015616 +第二单元 +015617 +第二单元 -040070 -(1) 三； (2) 三 +015618 +第二单元 +015619 +第二单元 -040071 -(1) $[-\dfrac{1}{2},\dfrac{1}{2})\cup\{1\}$; (2) $[-\dfrac{\pi}{3},\dfrac{\pi}{3})$; (3) $\{-\dfrac{1}{2}\}$ +015620 +第二单元 +015621 +第二单元 -040072 -(1) $-\tan \alpha-\cot \alpha$; (2) $-\dfrac{\sqrt{2}}{\sin \alpha}$; (3) $-1$; (4) $0$ +015622 +第二单元 +015623 +第二单元 -040073 -略 +015624 +第二单元 +015625 +第二单元 -040074 -$-\dfrac{10}{9}$ +015626 +第二单元 +015627 +第二单元 -040075 -$a_n=\dfrac{1}{3n-2}$ +015628 +第二单元 +015629 +第二单元 -040076 -$a_n=\dfrac{1}{n}$ +015630 +第二单元 +015631 +第二单元 -040077 -$(n-\dfrac{4}{5})5^n$ +015632 +第二单元 +015633 +第二单元 -040078 -$2^{n+1}-3$ +015634 +第二单元 +015635 +第二单元 -040079 -$1078$ +015636 +第二单元 +015637 +第二单元 -040080 -$S_n=\begin{cases}\dfrac{n^2}{2}+n-\dfrac 23+\dfrac 23\cdot 2^n, & n\text{为偶数},\\ \dfrac{n^2}{2}-\dfrac 76+\dfrac 23\cdot 2^{n+1}, & n\text{为奇数} \end{cases}$ +015638 +第二单元 +015639 +第二单元 -040081 -(1) 略； (2) $n^2$ +015640 +第二单元 +015641 +第二单元 -040082 -(1) 不存在； (2) 存在, 如$c_n=2^{n-1}$ +015642 +第二单元 +015643 +第二单元 -040083 -$\dfrac{\sqrt{3}}{2}$ +015644 +第二单元 +015645 +第二单元 -040084 -$0$ +015646 +第二单元 +015647 +第二单元 -040085 -$\{0,-2\pi\}$ +015648 +第二单元 +015649 +第二单元 -040086 -$-\dfrac{\pi}6,\dfrac 56\pi$ +015650 +第二单元 +015651 +第二单元 -040087 -$\cot \alpha$ +015652 +第二单元 +015653 +第二单元 -040088 -$7+4\sqrt{3}$ +015654 +第二单元 +015655 +第二单元 -040089 -$\dfrac{\sqrt{2}-\sqrt{6}}{4}$ +015656 +第二单元 +015657 +第二单元 -040090 -$\dfrac{\sqrt{3}+\sqrt{35}}{12}$ +015658 +第二单元 +015659 +第二单元 -040091 -$\dfrac 12$ +015660 +第二单元 +015661 +第二单元 -040092 -$5$ +015662 +第二单元 +015663 +第二单元 -040093 -$-\dfrac 12$ +015664 +第二单元 +015665 +第二单元 -040094 -$\dfrac{\pi}{12}$ +015666 +第二单元 +015667 +第二单元 -040095 -$\{x|x=\pm\frac 23 \pi+2k\pi,k \in \mathbf{Z}\}$ +015668 +第二单元 +015669 +第二单元 -040096 -$\dfrac 43 \pi$ +015670 +第二单元 +015671 +第二单元 -040097 -\textcircled{4} +015672 +第二单元 +015673 +第二单元 -040098 -C +015674 +第二单元 +015675 +第二单元 -040099 -$\dfrac{-2\sqrt{2}-\sqrt{3}}6$ +015676 +第二单元 +015677 +第二单元 -040100 -$-\dfrac 7{25}$ +015678 +第二单元 +015679 +第二单元 -040101 -$-\dfrac {\pi}3$ +015680 +第二单元 +015681 +第二单元 -040102 -$(-\dfrac {12}{13}, \dfrac{5}{13})$ +015682 +第二单元 +015683 +第二单元 -040103 -$(\dfrac {5-12\sqrt{3}}{2}, \dfrac{12-5\sqrt{3}}{2})$ +015684 +第二单元 +015685 +第二单元 -040104 -略 +015686 +第二单元 +015687 +第二单元 -040105 -$\dfrac {171} {221}, -\dfrac {21} {221}$ +015688 +第二单元 +015689 +第二单元 -040106 -$\{-\pi\}$ +015690 +第二单元 +015691 +第二单元 -040107 -$\dfrac{8\sqrt{2}-3}{15}$ +015692 +第二单元 +015693 +第二单元 -040108 -$\sin \theta$ +015694 +第二单元 +015695 +第二单元 -040109 -$-\dfrac{56}{65}$ +015696 +第二单元 +015697 +第二单元 -040110 -$\dfrac {\pi}4$ +015698 +第二单元 +015699 +第二单元 -040111 -略 +015700 +第二单元 +015701 +第二单元 -040112 -略 +015702 +第二单元 +015703 +第二单元 -040131 -$-\dfrac{25}{12}$ +015704 +第二单元 +015705 +第二单元 -040132 -$\dfrac 52$ +015706 +第二单元 +015707 +第二单元 -040133 -$-\dfrac{\pi}4$ +015708 +第二单元 +015709 +第二单元 -040134 -$-\dfrac 12$ +015710 +第二单元 +015711 +第二单元 -040135 -$\dfrac 6{19}$ +015712 +第二单元 +015713 +第二单元 -040136 -$-\dfrac {\sqrt{3}}3$ +015714 +第二单元 +015715 +第二单元 -040137 -$\dfrac 3{22}$ +015716 +第二单元 +015717 +第二单元 -040138 -$4$ +015718 +第二单元 +015719 +第二单元 -040139 -$-\dfrac{63}{65}$ +015720 +第二单元 +015721 +第二单元 -031288 -$[7,10]$ +015722 +第二单元 +015723 +第二单元 -031289 -$(-\infty,-2)\cup(-2,3]$ +015724 +第二单元 +015725 +第二单元 -031290 -$2$ +015726 +第二单元 +015727 +第二单元 -031291 -$7$ +015728 +第二单元 +015729 +第二单元 -031292 -$a\ge3$ +015730 +第二单元 +015731 +第二单元 -031293 -$-9$或$3$ +015732 +第二单元 +015733 +第二单元 -031294 -$\dfrac{1}{27}$ +015734 +第二单元 +015735 +第二单元 -031295 -$[-3,3]$ +015736 +第二单元 +015737 +第二单元 -031296 -$45$ +015738 +第二单元 +015739 +第二单元 -031297 -$(1,\dfrac 32]$ +015740 +第二单元 +015741 +第二单元 -031298 -$[0,1]$ +015742 +第二单元 +015743 +第二单元 -031299 -$\dfrac{\sqrt{6}}{4}$ +015744 +第二单元 +015745 +第二单元 -031300 -D +015746 +第二单元 +015747 +第二单元 -031301 -B +015748 +第二单元 +015749 +第二单元 -031302 -A +015750 +第二单元 +015751 +第二单元 -031303 -A +015752 +第二单元 +015753 +第二单元 -031304 -$(1)a_n=-3n+19,b_n=4^{3-n}\\ -(2)1\le n \le 28,S_n>T_n;n=29,S_n=T_n;n \ge 30,S_n1\\}$, $N=\\{x | \\log _2(x-a)<1\\}$, $M \\subseteq N$, 用区间表示$a$的取值组成的集合.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -398452,7 +398470,9 @@ "id": "015361", "content": "设集合$M=\\{(x,y) | x^2+y^2 \\geq 1\\}$, $N=\\{(x,y) | y>x+\\sqrt{2}\\}$, 求证: $N \\subset M$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -398472,7 +398492,9 @@ "id": "015362", "content": "记关于$x$的不等式$\\dfrac{3}{x}>1(x \\in \\mathbf{Z})$的解集为$A$, 关于$x$的方程$x^2-m x+2=0$的解集为$B$, 且$B \\subseteq A$.\\\\\n(1) 求集合$A$;\\\\\n(2) 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -398492,7 +398514,9 @@ "id": "015363", "content": "设集合$A=\\{x | \\dfrac{3}{x}>1\\}$, 集合$B=\\{x | x^2-m x+2<0\\}$且$B \\subseteq A$, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -398512,7 +398536,9 @@ "id": "015364", "content": "设集合$A=\\{x | \\dfrac{3}{x}>1\\}$, 集合$B=\\{x | x^2-m x+2<0\\}$, 是否存在实数$m$, 使得$A \\subseteq B$, 如果存在, 求出实数$m$的取值范围, 如果不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -398532,7 +398558,9 @@ "id": "015365", "content": "已知集合$A=\\{x| x=a^2-b^2,\\ a,b \\in \\mathbf{Z}\\}$, 则下列说法中不正确的是\\bracket{20}.\n\\twoch{$2021 \\in A$}{所有质数都在集合$\\mathrm{A}$中}{所有奇数都在集合$A$中}{所有$4$的倍数都在集合$A$中}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -398552,7 +398580,9 @@ "id": "015366", "content": "集合$A=\\{y | y=-x^2+4,\\ x \\in \\mathbf{N},\\ y \\in \\mathbf{N}\\}$的真子集的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -398572,7 +398602,9 @@ "id": "015367", "content": "下列命题: \\textcircled{1} 很小的实数可以构成集合; \\textcircled{2} 集合$\\{y | y=x^2-1\\}$与集合$\\{(x,y) | y=x^2-1\\}$相等; \\textcircled{3}$1,\\dfrac{3}{2},\\dfrac{6}{4},|-\\dfrac{1}{2}|,0.5$这些数组成的集合有$5$个元素; \\textcircled{4} 集合$\\{(x,y) | x y \\leq 0,\\ x,y \\in \\mathbf{R}\\}$是指第二和第四象限内的点集. 其中假命题的序号为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -398592,7 +398624,9 @@ "id": "015368", "content": "已知集合$A=\\{x | a x^2-x+1=0\\}$只有一个元素, 则实数$a$所构成的集合$B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -398612,7 +398646,9 @@ "id": "015369", "content": "设集合$M=\\{1,3,5,7,9\\}$, $N=\\{x | x>a\\}$, 且$M \\subset N$, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -398632,7 +398668,9 @@ "id": "015370", "content": "若集合$A=\\{x | x^2-2 x-3 \\leq 0\\}$, $B=\\{x | x>a\\}$, 且$A \\cap B=\\varnothing$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -398652,7 +398690,9 @@ "id": "015371", "content": "集合$A=\\{y | y=x^2+2 x+4,\\ x \\in \\mathbf{R}\\}$, $B=\\{z | z=a x^2-2 x+4 a,\\ x \\in \\mathbf{R}\\}$, 若$A \\subseteq B$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -398672,7 +398712,9 @@ "id": "015372", "content": "设$x=\\dfrac{1}{3-5 \\sqrt{2}}$, $y=3+\\sqrt{2} \\pi$, 集合$M=\\{m | m=a+b \\sqrt{2},\\ a \\in \\mathbf{Q},\\ b \\in \\mathbf{Q}\\}$, 那么$x, y$与集合$M$的关系是\\bracket{20}.\n\\fourch{$x \\in M$, $y \\in M$}{$x \\in M$, $y \\notin M$}{$x \\notin M$, $y \\in M$}{$x \\notin M$, $y \\notin M$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -398692,7 +398734,9 @@ "id": "015373", "content": "设集合$A=\\{x | 1 \\leq x \\leq 2\\}$, $B=\\{x | x \\geq a\\}$, 若$A \\subseteq B$, 则$a$的范围是\\bracket{20}.\n\\fourch{$a<1$}{$a \\leq 1$}{$a<2$}{$a \\leq 2$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -398712,7 +398756,9 @@ "id": "015374", "content": "已知集合$M=\\{a | \\dfrac{6}{5-a} \\in \\mathbf{N},\\ a \\in \\mathbf{Z}\\}$, 则$M$等于\\bracket{20}.\n\\fourch{$\\{2,3\\}$}{$\\{1,2,3,4\\}$}{$\\{1,2,3,6\\}$}{$\\{-1,2,3,4\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -398732,7 +398778,9 @@ "id": "015375", "content": "已知含有三个元素的集合$\\{a,\\dfrac{b}{a},1\\}=\\{a^2,a+b,0\\}$, 求$a^{2021}+b^{2022}$的值.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -398752,7 +398800,9 @@ "id": "015376", "content": "若集合$A=\\{x | \\log _a (x^2-x-2)>2,\\ a>0$且$a \\neq 1\\}$.\\\\\n(1) 若$a=2$, 求集合$A$;\\\\\n(2) 若$\\dfrac{9}{4} \\in A$, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -398772,7 +398822,9 @@ "id": "015377", "content": "已知集合$A=\\{x | x=2k+1,\\ k \\in \\mathbf{Z}\\}$, $B=\\{x | x=4 k+1,\\ k \\in \\mathbf{Z}\\}$, $C=\\{x | x=4 k \\pm 1,\\ k \\in \\mathbf{Z}\\}$. 证明:\\\\\n(1) $B \\subset A$;\\\\\n(2) $A=C$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -398792,7 +398844,9 @@ "id": "015378", "content": "设集合$M=\\{x | 00\\}$, 求$A \\cap B$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -398972,7 +399042,9 @@ "id": "015387", "content": "设全集$U=\\{x | x \\geq 2,\\ x \\in \\mathbf{N}\\}$, 集合$A=\\{x | x^2 \\geq 5,\\ x \\in \\mathbf{N}\\}$, 则$\\overline{A}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -398992,7 +399064,9 @@ "id": "015388", "content": "设集合$A=\\{x || x-1 |<2\\}$, $B=\\{y | y=2^x,\\ x \\in[0,2]\\}$, 则$A \\cup B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399012,7 +399086,9 @@ "id": "015389", "content": "设全集$U=\\mathbf{R}$, 集合$A=\\{x | x<-3$或$x>3\\}$, $B=(-\\infty,1) \\cup(4,+\\infty)$, 则$\\overline{A} \\cup B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399032,7 +399108,9 @@ "id": "015390", "content": "若$A=\\{1,4,x\\}$, $B=\\{1,x^2\\}$且$A \\cap B=B$, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399052,7 +399130,9 @@ "id": "015391", "content": "$50$名学生做物理、化学两种实验, 每人两种实验各做一次. 已知物理实验做得正确的有$40$人, 化学实验做得正确的有$31$人, 两种实验都做错的有$5$人, 则这两种实验都做对的有\\blank{50}人.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399072,7 +399152,9 @@ "id": "015392", "content": "定义$A\\backslash B=\\{x | x \\in A$且$x \\notin B\\}$, 若$A=\\{1,2,3,4,5\\}$, $B=\\{2,3,6\\}$, 则$A\\backslash B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399092,7 +399174,9 @@ "id": "015393", "content": "已知集合$A=\\{0,1,2,3,4,5\\}$, $B=\\{x | x^2-2 x-8<0\\}$, 则$A \\cap B$的一个真子集为\\bracket{20}.\n\\fourch{$\\{5\\}$}{$\\{3,4\\}$}{$\\{1,2,3\\}$}{$\\{0,1,2,3\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -399112,7 +399196,9 @@ "id": "015394", "content": "已知全集$U=\\{1,2,3,4\\}$, 集合$A=\\{1,2\\}$, $B=\\{2,3\\}$, 则$\\overline{A \\cup B}$等于\\bracket{20}.\n\\fourch{$\\{1,3,4\\}$}{$\\{3,4\\}$}{$\\{3\\}$}{$\\{4\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -399132,7 +399218,9 @@ "id": "015395", "content": "已知集合$S=\\{s | s=2 n+1,\\ n \\in \\mathbf{Z}\\}$, $T=\\{t|t=4 n+1,\\ n \\in \\mathbf{Z}\\}$, 则$S \\cap T$\\bracket{20}.\n\\fourch{$\\varnothing$}{$S$}{$T$}{$\\mathbf{Z}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -399152,7 +399240,9 @@ "id": "015396", "content": "设常数$a \\in \\mathbf{R}$, 集合$A=\\{x|(x-1)(x-a) \\geq 0\\}$, $B=\\{x | x \\geq a-1\\}$, 若$A \\cup B=\\mathbf{R}$, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -399172,7 +399262,9 @@ "id": "015397", "content": "全集$U=\\{1,3,x^3+3 x^2+2 x\\}$, $A=\\{1,|2 x-1|\\}$, 使得$\\overline{A}=\\{0\\}$的实数$x$是否存在? 若存在, 求出$x$; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -399192,7 +399284,9 @@ "id": "015398", "content": "已知集合$A=\\{x | \\dfrac{6}{x+1} \\geq 1,\\ x \\in \\mathbf{R}\\}$, $B=\\{x | x^2-2 x-m<0\\}$.\\\\\n(1) 当$m=3$时, 求$A \\cap \\overline{B}$;\\\\\n(2) 若$A \\cap B=\\{x |-11$''是条件乙: $a>\\sqrt{a}$''的\\bracket{20}.\n\\twoch{既不充分也不必要条件}{充要条件}{充分不必要条件}{必要不充分条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -399352,7 +399460,9 @@ "id": "015406", "content": "设$p: |4 x-3| \\leq 1$, $q: x^2-(2 a+1) x+a(a+1) \\leq 0$, 若$p$是$q$的充分不必要条件, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399372,7 +399482,9 @@ "id": "015407", "content": "等比数列$\\{a_n\\}$的公比为$q$, 前$n$项和为$S_n$, 设甲: $q>0$, 乙: $\\{S_n\\}$是严格递增数列, 则\\bracket{20}.\n\\twoch{甲是乙的充分条件但不是必要条件}{甲是乙的必要条件但不是充分条件}{甲是乙的充要条件}{甲既不是乙的充分条件也不是乙的必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -399392,7 +399504,9 @@ "id": "015408", "content": "已知函数$y=f(x)$的定义域为$\\mathbf{R}$, 下列是$f(x)$无最大值的充分条件是\\bracket{20}.\n\\twoch{$f(x)$是偶函数且关于点$(1,1)$对称}{$f(x)$是偶函数且关于直线$x=1$对称}{$f(x)$是奇函数且关于点$(1,1)$对称}{$f(x)$是奇函数且关于直线$x=1$对称}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -399412,7 +399526,9 @@ "id": "015409", "content": "在平面直角坐标系$x O y$中, 直线$l$与抛物线$y^2=2 x$相交于$A$、$B$两点.\\\\\n(1) 求证: ``如果直线$l$过点$T(3,0)$, 那么$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=3$''是真命题;\\\\\n(2) ``如果$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=3$, 那么直线$l$过点$T(3,0)$''是真命题还是假命题, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -399432,7 +399548,9 @@ "id": "015410", "content": "已知$f(x)=x^2+p x+q$, 求证: $|f(1)|,|f(2)|,|f(3)|$中至少有一个不小于$\\dfrac{1}{2}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -399452,7 +399570,9 @@ "id": "015411", "content": "若实数$a$、$b$满足$a \\geq 0$, $b \\geq 0$且$a b=0$, 则称$a$与$b$互补, 记$\\varphi(a,b)=\\sqrt{a^2+b^2}-a-b$, 那么``$\\varphi(a,b)=0$''是``$a$与$b$互补''的\\bracket{20}.\n\\twoch{必要而不充分的条件}{充分而不必要的条件}{充要条件}{即不充分也不必要的条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -399472,7 +399592,9 @@ "id": "015412", "content": "命题``存在实数$x$, 使得$x^2+2 x+2 \\leq 0$成立''的否定是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399492,7 +399614,9 @@ "id": "015413", "content": "设有两个命题: \\textcircled{1} 不等式$|x|+|x-1|>m$的解集为$\\mathbf{R}$; \\textcircled{2} 函数$f(x)=-(7-3 m)^x$是严格减函数. 如果这两个命题中有且仅有一个命题是真命题, 则$m$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399512,7 +399636,9 @@ "id": "015414", "content": "已知$p: (x-3)(x+1)>0$, $q: x^2-2 x+1-m^2>0$($m>0$), 若$p$是$q$的充分不必要条件, 则实数$m$的范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399532,7 +399658,9 @@ "id": "015415", "content": "若由$A$: ``$2(1-x^2)-3 x>0$''能推出$B$: ``$x>a$'', 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399552,7 +399680,9 @@ "id": "015416", "content": "``直线$x+y-a=0$与圆$C: (x-1)^2+(y-2)^2=2$相交''的一个充要条件为$啊\\in$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399572,7 +399702,9 @@ "id": "015417", "content": "已知集合$A=\\{x |-\\dfrac{1}{2} \\leq x<2\\}$, 集合$B=\\{x | x^2-(a+2) x+2 a<0\\}$若``$x \\in A$''是``$x \\in B$''的充分不必要条件, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399592,7 +399724,9 @@ "id": "015418", "content": "``$x<0$''是``$\\ln (x+1)<0$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -399612,7 +399746,9 @@ "id": "015419", "content": "``$\\alpha=\\beta$''是``$\\sin ^2 \\alpha+\\cos ^2 \\beta=1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -399632,7 +399768,9 @@ "id": "015420", "content": "用反证法证明命题``设$a$、$b$为实数, 则方程$x^2+a x+b=0$至少有一个实根''时, 要作出的假设是\\bracket{20}.\n\\twoch{方程$x^2+a x+b=0$没有实根}{方程$x^2+a x+b=0$至多有一个实根}{方程$x^2+a x+b=0$至多有两个实根}{方程$x^2+a x+b=0$恰好有两个实根}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -399652,7 +399790,9 @@ "id": "015421", "content": "已知陈述句$p$: 方程$a^2 x^2+a x-2=0$在$[-1,1]$上有解; 陈述句$q$: 只有一个实数$x$满足不等式$x^2+2 a x+2 a \\leq 0$, 若$p$与$q$至少有一个是错的, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -399672,7 +399812,9 @@ "id": "015422", "content": "已知集合$A=\\{y | y=x^2-\\dfrac{3}{2} x+1,\\ x \\in[\\dfrac{3}{4},2]\\}$, $B=\\{x | x+m^2 \\geq 1\\}$, $p: x \\in A$, $q: x \\in B$, 且$p$是$q$的充分条件, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -399692,7 +399834,9 @@ "id": "015423", "content": "陈述句$P$: 存在实数$a \\neq 0$, 对于任意的$x \\in \\mathbf{R}$, 都有$f(x+a)0$; 陈述句$q_2: f(x)$是$\\mathbf{R}$上的严格增函数且存在$x_0<0$使得$f(x_0)=0$. 证明: $q_1,q_2$都是$P$的充分条件.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -399712,7 +399856,9 @@ "id": "015424", "content": "设集合$A=\\{x |-21\\}$, $B=\\{-2,-1,1,2\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399852,7 +400010,9 @@ "id": "015431", "content": "$50$名学生参加跳远和铅球两项测试, 跳远、铅球测试及格的分别有$40$人和$31$人, 两项测试均不及格的有$4$人, 两项测试全都及格的人数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399872,7 +400032,9 @@ "id": "015432", "content": "已知全集$U=\\mathbf{R}$, 集合$M=\\{x |-2 \\leq x-1 \\leq 2\\}$和$N=\\{x | x=2 k-1,\\ k \\in\\mathbf{N},\\ k \\ge 1\\}$的关系的韦恩图如图所示, 则阴影部分所示的集合为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,-1) rectangle (3,1);\n\\begin{scope}\n\\clip (1,0) circle (0.8);\n\\clip (2,0) circle (0.9);\n\\filldraw [pattern = north east lines] (1,0) circle (0.8);\n\\filldraw [pattern = north east lines] (2,0) circle (0.9);\n\\end{scope}\n\\draw (1,0) circle (0.8);\n\\draw (2,0) circle (0.9);\n\\draw (0.4,0) node {$N$};\n\\draw (2.6,0) node {$M$};\n\\draw (0,1) node [below right] {$U$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399892,7 +400054,9 @@ "id": "015433", "content": "集合$A=\\{0,2,a\\}$, $B=\\{1,a^2\\}$, 若$A \\cup B=\\{0,1,2,4,16\\}$, 则$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399912,7 +400076,9 @@ "id": "015434", "content": "已知集合$A=\\{x | \\log _2 x \\leq 2\\}$, $B=(-\\infty,a)$, 若$A \\subseteq B$, 则实数$a$的取值范围是$(c,+\\infty)$, 其中$c=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399932,7 +400098,9 @@ "id": "015435", "content": "若实数集合$\\{1,2,3,x\\}$的最大元素与最小元素之差等于该集合的所有元素之和, 则$x$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -399952,7 +400120,9 @@ "id": "015436", "content": "已知集合$M=\\{-1,0,1\\}$, $N=\\{x | x=a b,\\ a,b \\in M$且$a \\neq b\\}$, 则集合$M$与集合$N$的关系是\\bracket{20}.\n\\fourch{$M=N$}{$M \\subset N$}{$M \\supset N$}{$M \\cap N=\\varnothing$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -399972,7 +400142,9 @@ "id": "015437", "content": "已知集合$A$是由满足$x=3 k-1$($k \\in \\mathbf{Z}$)的$x$的值组成, 则集合$A$中的元素\\bracket{20}.\n\\fourch{除以$3$余数为$-1$}{除以$3$余数为$1$}{除以$3$余数为$2$}{能被$3$整除}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -399992,7 +400164,9 @@ "id": "015438", "content": "若$a,b \\in \\mathbf{R}$, 则``$a+b>4$''是``$a>2$且$b>2$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -400012,7 +400186,9 @@ "id": "015439", "content": "已知集合$A=\\{x | x=a^2-b^2,\\ a,b \\in \\mathbf{Z}\\}$, 则下列说法中不正确的是\\bracket{20}.\n\\twoch{$2023 \\in A$}{所有奇数都在集合$A$中}{所有质数都在集合$A$中}{所有$4$的倍数都在集合$A$中}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -400032,7 +400208,9 @@ "id": "015440", "content": "已知集合$A=\\{x | 0b$, 使得$a(c+1)^2>b(c+1)^2$成立的条件为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -400692,7 +400934,9 @@ "id": "015473", "content": "设$a$、$b$、$c$、$d \\in \\mathbf{R}$, 且$a>b$, $c>d$, 下列结论: \\textcircled{1} $a+c>b+d$; \\textcircled{2} $ab-d$; \\textcircled{3} $a c>b d$; \\textcircled{4} $\\dfrac{a}{d}>\\dfrac{b}{c}$, 其中正确的序号为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -400712,7 +400956,9 @@ "id": "015474", "content": "如果$ab>c>d$, 下列选项中正确的是\\bracket{20}.\n\\fourch{$a+d>b+c$}{$a+c>b+d$}{$ad>bc$}{$a cb$, (乙) $\\dfrac{1}{a}<\\dfrac{1}{b}$, (丙) $\\dfrac{1}{a^2}>\\dfrac{1}{b^2}$.\\\\\n(1) 若$a>0$, 则甲是乙的必要非充分条件, 试说明理由;\\\\\n(2) 若$a<0$, $b<0$, 判断丙是甲的什么条件, 并加以说明.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -400812,7 +401066,9 @@ "id": "015479", "content": "利用不等式的基本性质证明: 若$a>b>0$, $d\\dfrac{c}{d}>0$和$a db$、$a-\\dfrac{1}{a}>b-\\dfrac{1}{b}$同时成立, 则$a$、$b$应满足的条件是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -400912,7 +401176,9 @@ "id": "015484", "content": "若$x \\in\\{y | y=x^2-2 x+2,\\ x \\in \\mathbf{R}\\}$, 则$\\dfrac{1}{x+1}$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -400932,7 +401198,9 @@ "id": "015485", "content": "设$x<0$, $xy<0$, 则$|y-x+1|-|x-y-5|$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -400952,7 +401220,9 @@ "id": "015486", "content": "已知$a$、$b$、$c$满足$ca c$; \\textcircled{2} $c(b-a) \\leq 0$; \\textcircled{3} $c b^20$中, 正确的序号为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -400972,7 +401242,9 @@ "id": "015487", "content": "下列命题: \\textcircled{1} $a>b>0 \\Rightarrow a \\lg \\dfrac{2}{3}>b \\lg \\dfrac{2}{3}$; \\textcircled{2} $|a|>b \\Rightarrow a^2>b^2$; \\textcircled{3} $a>|b| \\Rightarrow a^2>b^2$; \\textcircled{4} $a^3>b^3 \\Rightarrow a>b$; \\textcircled{5} $a^2>b^2 \\Rightarrow a>b$. 其中真命题有\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -400992,7 +401264,9 @@ "id": "015488", "content": "在下列命题中, 不可能成立的是\\bracket{20}.\n\\twoch{$a>b$, $cb-d$}{$a>b$, $a b>0 \\Rightarrow \\dfrac{1}{a}<\\dfrac{1}{b}$}{$a>b \\Rightarrow|a|>b$}{$a>2$, $b>2 \\Rightarrow a b0$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既不充要又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -401032,7 +401308,9 @@ "id": "015490", "content": "记方程\\textcircled{1} $x^2+a_1 x+1=0$, 方程\\textcircled{2} $x^2+a_2 x+2=0$, 方程\\textcircled{3} $x^2+a_3 x+4=0$, 其中$a_1,a_2,a_3$是正实数. 当$a_2^2=a_1 a_3$时, 下列选项中, 能推出方程\\textcircled{3} 无实根的是\\bracket{20}.\n\\twoch{方程\\textcircled{1}有实根, 且\\textcircled{2}有实根}{方程\\textcircled{1}有实根, 且\\textcircled{2}无实根}{方程\\textcircled{1}无实根, 且\\textcircled{2}有实根}{方程\\textcircled{1}无实根, 且\\textcircled{2}无实根}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -401052,7 +401330,9 @@ "id": "015491", "content": "已知$a$、$b$是正实数, 证明: ``不等式组$\\begin{cases}x+y>a+b,\\\\ x y>a b\\end{cases}$成立''是``不等式组$\\begin{cases}x>a,\\\\ y>b\\end{cases}$成立''的必要不充分条件.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401072,7 +401352,9 @@ "id": "015492", "content": "已知三个不等式: \\textcircled{1} $a b>0$; \\textcircled{2} $\\dfrac{c}{a}>\\dfrac{d}{b}$; \\textcircled{3} $b c>a d$, 以其中两个作为条件, 余下一个作为结论, 试写出其中的正确命题 (要给出推理过程).", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401092,7 +401374,9 @@ "id": "015493", "content": "有三个房间需要粉刷, 粉刷方案要求: 每个房间只用一种颜色, 且三个房间颜色各不相同. 已知三个房间的粉刷面积(单位, $\\text{m}^2$)分别为$x, y, z$, 且$x0$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401152,7 +401440,9 @@ "id": "015496", "content": "不等式$\\dfrac{2 x+5}{x-2}<1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401172,7 +401462,9 @@ "id": "015497", "content": "下列不等式中, 与不等式$\\dfrac{x+8}{x^2+2 x+3}<2$解集相同的是\\bracket{20}.\n\\twoch{$(x+8)(x^2+2 x+3)<2$}{$x+8<2(x^2+2 x+3)$}{$\\dfrac{1}{x^2+2 x+3}<\\dfrac{2}{x+8}$}{$\\dfrac{x^2+2 x+3}{x+8}>\\dfrac{1}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -401192,7 +401484,9 @@ "id": "015498", "content": "已知$a \\in \\mathbf{R}$, 不等式$\\dfrac{x-3}{x+a} \\geq 1$的解集为$P$, 且$-2 \\notin P$, 则$a$的取值范围是\\bracket{20}.\n\\fourch{$a>-3$}{$-32$或$a<-3$}{$a \\geq 2$或$a<-3$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -401212,7 +401506,9 @@ "id": "015499", "content": "已知$k \\geq 0$, 解关于$x$的不等式$k x^2-(1+k) x+1>0$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401232,7 +401528,9 @@ "id": "015500", "content": "关于$x$的不等式$a x^2+b x+c<0$的解集为$\\{x | x<-2,\\ x>-\\dfrac{1}{2}\\}$, 求关于$x$的不等式$a x^2-b x+c>0$的解集.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401252,7 +401550,9 @@ "id": "015501", "content": "一般而言, 把$b-a$称为区间$(a,b)$的``长度''. 已知关于$x$的不等式$x^2-k x+2 k<0$有实数解, 且解集区间长度不超过$3$个单位, 求实数$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401272,7 +401572,9 @@ "id": "015502", "content": "设$a_1,b_1,c_1,a_2,b_2,c_2$均为非零实数; 不等式$a_1 x^2+b_1 x+c_1>0$和$a_2 x^2+b_2 x+c_2>0$的解集分别为$M$、$N$, 则``$\\dfrac{a_1}{a_2}=\\dfrac{b_1}{b_2}=\\dfrac{c_1}{c_2}$''是``$M=N$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要非充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -401292,7 +401594,9 @@ "id": "015503", "content": "解分式不等式$x+\\dfrac{2}{x+1}>2$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401312,7 +401616,9 @@ "id": "015504", "content": "不等式$\\dfrac{x(x+1)}{1-x} \\leq 0$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401332,7 +401638,9 @@ "id": "015505", "content": "不等式$\\dfrac{x(x-1)^2}{x+1} \\leq 0$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401352,7 +401660,9 @@ "id": "015506", "content": "解不等式$\\dfrac{2 x^2+3}{x-a}<2 x$($a \\in \\mathbf{R}$).", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401372,7 +401682,9 @@ "id": "015507", "content": "解关于$x$的不等式$\\dfrac{x^2-a x+a-2}{x-2} \\leq 1$($a \\neq 0$, $a \\in \\mathbf{R}$).", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401392,7 +401704,9 @@ "id": "015508", "content": "解关于$x$的不等式$\\dfrac{a x^2}{a x-1}>x$($a \\in \\mathbf{R}$).", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401412,7 +401726,9 @@ "id": "015509", "content": "在关于$x$的不等式$x^2-(a+1) x+a<0$的解集中恰有两个整数, 则$a$的取值范围是\\bracket{20}.\n\\fourch{$(3,4)$}{$(-2,-1) \\cup(3,4)$}{$(3,4]$}{$[-2,-1) \\cup(3,4]$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -401432,7 +401748,9 @@ "id": "015510", "content": "已知$A=\\{x|| 2 x+1 |>3\\}$, $B=\\{x | x^2+x \\leq 6\\}$, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401452,7 +401770,9 @@ "id": "015511", "content": "设不等式$\\dfrac{a(x-2)}{x+3}<2$的解集为$A$, 且$1 \\notin A$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401472,7 +401792,9 @@ "id": "015512", "content": "关于$x$的不等式$x^2-2 a x-8 a^2<0$($a>0$)的解集$(x_1,x_2)$, 且$x_2-x_1=15$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401492,7 +401814,9 @@ "id": "015513", "content": "不等式$x<\\dfrac{1}{x}0$的解集为$(\\dfrac{1}{a},1)$, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401552,7 +401880,9 @@ "id": "015516", "content": "已知不等式$\\dfrac{x-a}{2-x}>0$的解集为$\\{x|-20$的解集为\\bracket{20}.\n\\fourch{$\\varnothing$}{$(-2,-1)$}{$(-\\infty,-1) \\cup(2,+\\infty)$}{$(-\\infty,-2) \\cup(1,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -401572,7 +401902,9 @@ "id": "015517", "content": "集合$A=\\{x | 1<\\dfrac{(3-x)^2}{2} \\leq 2\\}$, $B=\\{1,5\\}$, 则$x \\in A$是$x \\in B$的\\bracket{20}.\n\\twoch{充分非必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -401592,7 +401924,9 @@ "id": "015518", "content": "若$a>0$, $b>0$, 则不等式$-b<\\dfrac{1}{x}\\dfrac{1}{a}$}{$x<-\\dfrac{1}{a}$或$x>\\dfrac{1}{b}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -401612,7 +401946,9 @@ "id": "015519", "content": "设函数$f(x)=a x^2+(b-2) x+3$($a \\neq 0$), 若不等式$f(x)>0$的解集为$(-1,3)$.\\\\\n(1) 求$a$、$b$的值;\\\\\n(2) 若函数$f(x)$在$x \\in[m,1]$上的最小值为$1$, 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401632,7 +401968,9 @@ "id": "015520", "content": "已知函数$f(x)=\\begin{cases}a-x,& x \\leq 0,\\\\ a,& x>0\\end{cases}$($a>0$), 解关于$x$的不等式$\\dfrac{f(x)}{x-2}<1$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401652,7 +401990,9 @@ "id": "015521", "content": "设函数$f(x)=-4 x+b$, 不等式$|f(x)|0$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401672,7 +402012,9 @@ "id": "015522", "content": "在实数范围内, 不等式$||x-2|-1| \\leq 1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401692,7 +402034,9 @@ "id": "015523", "content": "不等式$2^{x^2-x}<4$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401712,7 +402056,9 @@ "id": "015524", "content": "若关于$x$的不等式$|a x-2|<3$的解集为$(-\\dfrac{5}{3},\\dfrac{1}{3})$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401732,7 +402078,9 @@ "id": "015525", "content": "不等式组$\\begin{cases}x(x+2)>0,\\\\ |x|<1\\end{cases}$的解集为\\bracket{20}.\n\\fourch{$\\{x |-21\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -401752,7 +402100,9 @@ "id": "015526", "content": "设集合$A=\\{x | x^2-1>0\\}$, $B=\\{x | \\log _2 x>0\\}$, 则$A \\cap B$等于\\bracket{20}.\n\\fourch{$\\{x | x>1\\}$}{$\\{x | x>0\\}$}{$\\{x | x<-1\\}$}{$\\{x | x>1$或$x<-1\\}$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -401772,7 +402122,9 @@ "id": "015527", "content": "解不等式组$\\begin{cases}\\dfrac{(x-1)(2 x-1)}{x^2+2 x+2}<0,\\\\|x-3|-|x+1|>0.\\end{cases}$", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401792,7 +402144,9 @@ "id": "015528", "content": "解不等式$|x-3|-|x+1|<1$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401812,7 +402166,9 @@ "id": "015529", "content": "如果不等式$|x+2|+|x|>m$的解集为$\\mathbf{R}$, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401832,7 +402188,9 @@ "id": "015530", "content": "设$f(x)=\\begin{cases}2\\mathrm{e}^{x-1},& x<2,\\\\ \\log _3(x^2-1),& x \\geq 2,\\end{cases}$解不等式$f(x)>2$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401852,7 +402210,9 @@ "id": "015531", "content": "已知函数$f(x)=\\begin{cases}\\dfrac{1}{x},& x<0,\\\\ (\\dfrac{1}{3})^{x},& x \\geq 0,\\end{cases}$解不等式$|f(x)| \\geq \\dfrac{1}{3}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401872,7 +402232,9 @@ "id": "015532", "content": "证明: 对任意的实数$a$、$b$, 有$|a+b| \\leq |a|+|b|$, 当且仅当$a b \\geq 0$时等号成立(三角不等式).", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401892,7 +402254,9 @@ "id": "015533", "content": "已知函数$f(x)=|2 x-a|+a$.\\\\\n(1) 当$a=2$时, 求不等式$f(x) \\leq 6$的解集;\\\\\n(2) 设函数$g(x)=|2 x-1|$. 当$x \\in \\mathbf{R}$时, $f(x)+g(x) \\geq 3$, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401912,7 +402276,9 @@ "id": "015534", "content": "已知函数$f(x)=|2 x-a|+a$.\\\\\n(1) 若不等式$f(x) \\leq 6$的解集为$\\{x |-2 \\leq x \\leq 3\\}$, 求实数$a$的值;\\\\\n(2) 在 (1) 的条件下, 若存在实数$n$使$f(n) \\leq m-f(-n)$成立, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -401932,7 +402298,9 @@ "id": "015535", "content": "设$x \\in \\mathbf{R}$, 则不等式$|x-3|<1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401952,7 +402320,9 @@ "id": "015536", "content": "不等式$3^{x^{2}-3 x+1}>(\\dfrac{1}{3})^{x}$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401972,7 +402342,9 @@ "id": "015537", "content": "不等式$\\log _{2}(2-x)<\\log _{4} x$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -401992,7 +402364,9 @@ "id": "015538", "content": "若不等式$|x-1|0$, 不等式$|a x+b|0$的解集.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -402132,7 +402518,9 @@ "id": "015545", "content": "已知函数$f(x)=|x=2 a|=|x+1|$.\\\\\n(1) 当$a=1$时, 求不等式$f(x) \\geq 1$的解集;\\\\\n(2) 若$f(x)-a-2 \\leq 0$恒成立, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -402152,7 +402540,9 @@ "id": "015546", "content": "已知函数$f(x)=2^{x}-\\dfrac{1}{2^{|x|}}$.\\\\\n(1) 若$f(x)>2$, 求$x$的取值范围;\\\\\n(2) 若$2^{t} f(2 t)+m f(t) \\geq 0$对于$t \\in[1,2]$恒成立, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -402172,7 +402562,9 @@ "id": "015547", "content": "已知$a$、$b \\in (0,+\\infty)$. 若$a+b=1$, 则$a b$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402192,7 +402584,9 @@ "id": "015548", "content": "函数$y=|x+\\dfrac{4}{x}|$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402212,7 +402606,9 @@ "id": "015549", "content": "已知函数$f(x)=3^x+\\dfrac{a}{3^x+1}$($a>0$)的最小值为$5$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402232,7 +402628,9 @@ "id": "015550", "content": "下列不等式中恒成立的是\\bracket{20}.\n\\fourch{$a+b \\leq 2 \\sqrt{|a b|}$}{$a^2+b^2 \\geq -2 a b$}{$a+b \\geq -2 \\sqrt{|a b|}$}{$a^2+b^2 \\leq 2 a b$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -402252,7 +402650,9 @@ "id": "015551", "content": "要制作一个容积为$4 \\text{m}^3$, 高为$1 \\text{m}$的无盖长方体容器, 已知该容器的底面造价是每平方米$20$元, 侧面造价是每平方米$10$元, 则该容器的最低总造价是\\bracket{20}.\n\\fourch{$80$元}{$120$元}{$160$元}{$240$元}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -402272,7 +402672,9 @@ "id": "015552", "content": "若$x, y \\in (0,+\\infty)$, 且$\\dfrac{1}{x}+2 y=3$, 则$\\dfrac{y}{x}$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402292,7 +402694,9 @@ "id": "015553", "content": "已知$|x|>4$, 则$x+\\dfrac{25}{x-1}$的值不可能为\\bracket{20}.\n\\fourch{$-9$}{$-10$}{$11$}{$12$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -402312,7 +402716,9 @@ "id": "015554", "content": "下列说法:\\\\\n\\textcircled{1} 不等式$a+b \\geq 2 \\sqrt{a b}$恒成立;\\\\\n\\textcircled{2} 存在$a$, 使得不等式$a+\\dfrac{1}{a} \\leq 2$成立;\\\\\n\\textcircled{3} 若$a$、$b \\in(0,+\\infty)$, 则$\\dfrac{b}{a}+\\dfrac{a}{b} \\geq 2$;\\\\\n\\textcircled{4} 若正实数$x$、$y$满足$x+2 y=1$, 则$\\dfrac{2}{x}+\\dfrac{1}{y} \\geq 8$, 其中正确的序号为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402332,7 +402738,9 @@ "id": "015555", "content": "若$a>0$, $b>0$, 则$\\dfrac{1}{a}+\\dfrac{a}{b^2}+b$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402352,7 +402760,9 @@ "id": "015556", "content": "已知$x, y$为正数, 且$x+\\dfrac{y}{2}=1$,求:\\\\\n(1) $\\dfrac{1}{x}+\\dfrac{2}{y}$的最小值;\n(2) $\\sqrt{x(1+y)}$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -402372,7 +402782,9 @@ "id": "015557", "content": "某快递公司在某市的货物转运中心, 拟引进智能机器人分栋系统, 以提高分栋效率和降低物流成本, 已知购买$x$台机器人的总成本$p(x)=\\dfrac{1}{600} x^2+x+150$万元.\n(1)若使每台机器人的平均成本最低, 问应买多少台?\n(2) 现按 (1) 中的数量购买机器人, 需要安排$m$人将邮件放在机器人上, 机器人将邮件送达指定落袋格口完成分拣, 经实验知, 每台机器人的日平均分拣量$q(m)=$$\\begin{cases}\\dfrac{8}{15} m(60-m),& 1 \\leq m \\leq 30,\\\\ 480,& m>30\\end{cases}$(单位: 件), 已知传统人工分拣每人每日的平均分拣量为$1200$件, 问引进机器人后, 日平均分拣量达最大值时, 用人数量比引进机器人前的用人数量最多可减少多少?", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -402392,7 +402804,9 @@ "id": "015558", "content": "对于$c>0$, 当非零实数$a$、$b$满足$4 a^2-2 a b+b^2-c=0$且使$|2 a+b|$最大时, $\\dfrac{1}{a}+\\dfrac{2}{b}+\\dfrac{4}{c}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402412,7 +402826,9 @@ "id": "015559", "content": "对任意的实数$x>\\dfrac{1}{2}$, $y>1$, 不等式$\\dfrac{4 x^2}{a^2(y-1)}+\\dfrac{y^2}{a^2(2 x-1)} \\geq 1$恒成立, 则实数$a$的最大值是\\bracket{20}.\n\\fourch{$2 \\sqrt{2}$}{$4$}{$\\dfrac{\\sqrt{14}}{2}$}{$2$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -402432,7 +402848,9 @@ "id": "015560", "content": "若$a>1$, 则$4 a+\\dfrac{1}{a-1}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402452,7 +402870,9 @@ "id": "015561", "content": "当$x \\geq 2$时, 函数$y=x+\\dfrac{1}{x}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402472,7 +402892,9 @@ "id": "015562", "content": "已知正数$x,y$满足$x+y=x y$, 则$x+y$的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402492,7 +402914,9 @@ "id": "015563", "content": "已知$a+b=t$($a>0$, $b>0$), $t$为常数, 且$a b$的最大值为$2$, 则$t=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402512,7 +402936,9 @@ "id": "015564", "content": "已知$\\dfrac{5}{x}+\\dfrac{3}{y}=2$($x>0$, $y>0$), 则$x y$的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402532,7 +402958,9 @@ "id": "015565", "content": "若实数$a$使得不等式$|x-2 a|+|2 x-a| \\geq a^2$对任意实数$x$恒成立, 则实数$a$的取值范围\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402552,7 +402980,9 @@ "id": "015566", "content": "如果正数$a$、$b$、$c$、$d$满足$a+b=c d=4$, 则下列各式中恒成立的是\\bracket{20}.\n\\fourch{$a bc+d$}{$a b \\geq_c+d$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -402572,7 +403002,9 @@ "id": "015567", "content": "若$a$、$b \\in \\mathbf{R}$, 且$a b>0$, 则下列不等式中, 恒成立的是\\bracket{20}.\n\\fourch{$a^2+b^2>2 a b$}{$a+b \\geq 2 \\sqrt{a b}$}{$\\dfrac{1}{a}+\\dfrac{1}{b}>\\dfrac{2}{\\sqrt{a b}}$}{$\\dfrac{b}{a}+\\dfrac{a}{b} \\geq 2$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -402592,7 +403024,9 @@ "id": "015568", "content": "下列不等式中一定成立的是\\bracket{20}.\n\\twoch{$\\lg (x^2+\\dfrac{1}{4})>\\lg x$($x>0$)}{$\\sin x+\\dfrac{1}{\\sin x} \\geq 2$($x \\neq k \\pi$, $k \\in \\mathbf{Z}$)}{$x^2+1 \\geq 2|x|$($x \\in \\mathbf{R}$)}{$\\dfrac{1}{x^2+1}>1$($x \\in \\mathbf{R}$)}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -402612,7 +403046,9 @@ "id": "015569", "content": "已知不等式$(1+k^2) x \\leq k^4+4$对任意实常数$k$均成立, 试求实数$x$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -402632,7 +403068,9 @@ "id": "015570", "content": "已知函数$f(x)=\\dfrac{a+x^2}{x}$($a>0$, $b>0$, $x \\in(0,b]$), 求$f(x)$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -402652,7 +403090,9 @@ "id": "015571", "content": "某镇努力打造``生态水果特色小镇''. 经调研发现: 某珍稀水果树的单株产量$W$(单位: 干克)与施用肥料$x$(单位: 千克)满足如下关系: $W(x)=\\begin{cases}5(x^2+3),& 0 \\leq x \\leq 2,\\\\ \\dfrac{50 x}{1+x},& 2b>0$时, 下列不等式中成立的是\\bracket{20}.\n\\fourch{$a>b>\\dfrac{a+b}{2}>\\sqrt{a b}$}{$a>\\dfrac{a+b}{2}>\\sqrt{a b}>b$}{$a>\\dfrac{a+b}{2}>b>\\sqrt{a b}$}{$a>\\sqrt{a b}>\\dfrac{a+b}{2}>b$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -402692,7 +403134,9 @@ "id": "015573", "content": "若$a \\geq 0$, $b \\geq 0$, 且$a+b=2$, 则\\bracket{20}.\n\\fourch{$a b \\leq \\dfrac{1}{2}$}{$a b \\geq \\dfrac{1}{2}$}{$a^2+b^2 \\geq 2$}{$a^2+b^2 \\leq 3$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -402712,7 +403156,9 @@ "id": "015574", "content": "若$a>0$, $b>0$, 则``$a+b \\leq 4$''是``$a b \\leq 4$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -402732,7 +403178,9 @@ "id": "015575", "content": "已知$a$、$b$是两个不相等的正数, $M=\\dfrac{\\sqrt{a}+\\sqrt{b}}{2}$, $N=\\sqrt{\\dfrac{a+b}{2}}$, 则$M$与$N$的大小关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402752,7 +403200,9 @@ "id": "015576", "content": "若$x \\geq 1$, 则$\\sqrt{x+1}-\\sqrt{x}$与$\\sqrt{x}-\\sqrt{x-1}$的大小关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402772,7 +403222,9 @@ "id": "015577", "content": "设$a$、$b$、$c$是互不相等的正数, 判断下列等式是否恒成立, 说明理由.\\\\\n(1) $a^2+\\dfrac{1}{a^2} \\geq a+\\dfrac{1}{a}$;\\\\\n(2) $|a-b| \\mp \\dfrac{1}{a-b} \\geq 2$;\\\\\n(3) $|a-b| \\leq|a-c|+|b-c|$;\\\\\n(4) $\\sqrt{a+3}-\\sqrt{a+1} \\leq \\sqrt{a+2}-\\sqrt{a}$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -402792,7 +403244,9 @@ "id": "015578", "content": "设$x \\geq 1$, $y \\geq 1$; 证明: $x+y+\\dfrac{1}{x y} \\leq \\dfrac{1}{x}+\\dfrac{1}{y}+x y$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -402812,7 +403266,9 @@ "id": "015579", "content": "设$10$, $b>0$, 则$a^2+b^2+2$\\blank{50}$2 a+2 b$(比较大小).", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402912,7 +403376,9 @@ "id": "015584", "content": "若$a$、$b \\in \\mathbf{R}$, 则$|a|+\\dfrac{1}{2}|b|$\\blank{50}$\\sqrt{2} \\cdot \\sqrt{|a b|}$(比较大小).", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402932,7 +403398,9 @@ "id": "015585", "content": "当$x \\in(0,\\dfrac{\\pi}{2})$时, $1-\\cos x$\\blank{50}$\\sin x$(比较大小).", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402952,7 +403420,9 @@ "id": "015586", "content": "若$a$、$b \\in \\mathbf{R}$, 则$a^2+3+\\dfrac{4}{a^2+3}$\\blank{50}4(比较大小)", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402972,7 +403442,9 @@ "id": "015587", "content": "设$x>0$, $P=2^x+2^{-x}$, $Q=(\\sin x+\\cos x)^2$, 则$P$、$Q$之间的大小关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -402992,7 +403464,9 @@ "id": "015588", "content": "设$a>0$, $b>0$, 则以下不等式: \\textcircled{1} $(a+b)(\\dfrac{1}{a}+\\dfrac{1}{b}) \\geq 4$; \\textcircled{2} $a^3+b^3 \\geq 2 a b^2$; \\textcircled{3} $\\sqrt{|a-b|} \\geq\\sqrt{a}-\\sqrt{b}$. 其中恒成立的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403012,7 +403486,9 @@ "id": "015589", "content": "如果正数$a$、$b$、$c$、$d$满足$a+b=c d=4$, 那么\\bracket{20}.\n\\onech{$a b \\leq c+d$, 且等号成立时$a \\vee b, c$、$d$的取值唯一}{$a b \\geq c+d$, 且等号成立时$a$、$b$、$c$、$d$的取值唯一}{$a b \\leq c+d$, 且等号成立时$a$、$b$、$c$、$d$的取值不唯一}{$a b \\geq_c+d$, 且等号成立时$a$、$b$、$c$、$d$的取值不唯一}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -403032,7 +403508,9 @@ "id": "015590", "content": "设$a$、$b \\in \\mathbf{R}$, 则``$a^2+b^2<1$''是``$a b+1>a+b$''的\\bracket{20}.\n\\twoch{充要条件}{必要不充分条件}{充分不必要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -403052,7 +403530,9 @@ "id": "015591", "content": "设$z_1,z_2\\in\\mathbf{C}$, $M=z_1 \\overline{z_2}+\\overline{z_1} z_2$, $N=z_1 \\overline{z_1}+z_2 \\overline{z_2}$, 则$M$、$N$的大小关系为\\bracket{20}.\n\\fourch{$M \\geq N$}{$M>N$}{$M \\leq N$}{不能比较大小}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -403072,7 +403552,9 @@ "id": "015592", "content": "已知$a>0$, $b>0$且$a^2+b^2=1$, 证明下列不等式:\\\\\n(1) $a b+1>a+b$;\\\\\n(2) $(\\dfrac{1}{a}+\\dfrac{1}{b})(a^5+b^5) \\geq 1$;\\\\\n(3) $\\log _2 a+\\log _2 b \\leq -1$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -403092,7 +403574,9 @@ "id": "015593", "content": "设函数$f(x)=|\\lg x|$, 若$0f(b)$, 证明: $a b<1$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -403112,7 +403596,9 @@ "id": "015594", "content": "已知集合$M=\\{x |-10$的解集为$(1,2)$, 解关于$x$的不等式$c x^2-$$b x+a>0$'', 有如下解决方案:\\\\\n解: 由$c x^2-b x+a>0 \\Rightarrow c-b(\\dfrac{1}{x})+a(\\dfrac{1}{x})^2>0$, 令$y=\\dfrac{1}{x}$, 则$c-b y+a y^2>0$,\n所以$y \\in(1,2)$, 所以不等式$c x^2-b x+a>0$的解集为$(\\dfrac{1}{2},1)$.\\\\\n参考上述解法, 已知关于$x$的不等式$\\dfrac{k}{x+a}+\\dfrac{x+b}{x+c}<0$的解集为$(-2,-1) \\cup(2,3)$,\n则关于$x$的不等式$\\dfrac{k x}{a x+1}+\\dfrac{b x+1}{c x+1}<0$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403292,7 +403794,9 @@ "id": "015603", "content": "设常数$a>0$, 若$9 x+\\dfrac{a^2}{x} \\geq a+1$对一切正实数$x$成立, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403312,7 +403816,9 @@ "id": "015604", "content": "某公司一年购买某种货物$400$吨, 每次都购买$x$吨, 运费为$4$万元/次, 一年的总存储费用为$4 x$万元, 要使一年的总运费与总存储费用之和最小, 则$x=$\\blank{50}吨.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403332,7 +403838,9 @@ "id": "015605", "content": "设$x, y$是不全为零的实数, 则$2 x^2+y^2$与$x^2+x y$的大小关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403352,7 +403860,9 @@ "id": "015606", "content": "对于正数$a$、$b$, 称$\\dfrac{a+b}{2}$是$a$、$b$的算术平均值, 并称$\\sqrt{a b}$是$a$、$b$的几何平均值. 设$x>1$, $y>1$, 若$\\ln x,\\ln y$的算术平均值是 1 , 则$\\mathrm{e}^x,\\mathrm{e}^y$的几何平均值($\\mathrm{e}$是自然对数的底) 的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403372,7 +403882,9 @@ "id": "015607", "content": "若$a<0\\dfrac{1}{b}$}{$-a>b$}{$a^2>b^2$}{$a^30$, 则下列不等式中正确的是\\bracket{20}.\n\\fourch{$b-a>0$}{$a^3+b^3<0$}{$b+a<0$}{$a^2-b^2>0$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -403432,7 +403948,9 @@ "id": "015610", "content": "若不等式$|x-a|-|x|<2-a^2$当$x \\in \\mathbf{R}$时总成立, 则实数$a$的取值范围是\\bracket{20}.\n\\fourch{$(-2,2)$}{$(-2,1)$}{$(-1,1)$}{$(-\\infty,-1) \\cup(1,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -403452,7 +403970,9 @@ "id": "015611", "content": "已知函数$f(x)=\\dfrac{1}{3}|x-a|$($a \\in \\mathbf{R}$).\\\\\n(1) 当$a=2$时, 解不等式$|x-\\dfrac{1}{3}|+f(x) \\geq 1$;\\\\\n(2) 设不等式$|x-\\dfrac{1}{3}|+f(x) \\leq x$的解集为$M$, 若$[\\dfrac{1}{3},\\dfrac{1}{2}] \\subseteq M$, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -403472,7 +403992,9 @@ "id": "015612", "content": "已知实数$x$、$y$满足$|x+y| \\leq \\dfrac{1}{6}$, $|x-y| \\leq \\dfrac{1}{4}$, 求证: $|x+5 y| \\leq 1$.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -403492,7 +404014,9 @@ "id": "015613", "content": "已知函数$f(x)=\\begin{cases}2^x,& x \\leq 0,\\\\ \\log _2 x,& x>0.\\end{cases}$\\\\\n(1) 解不等式$x \\cdot f(x) \\leq 0$;\\\\\n(2) 设$k, m$均为实数, 当$x \\in(-\\infty,m]$时, $f(x)$的最大值为 1 , 且满足此条件的任意实数$x$及$m$的值, 使得关于$x$的不等式$f(x) \\leq m^2-(k-2) m+3 k-10$恒成立, 求$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第一单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -403512,7 +404036,9 @@ "id": "015614", "content": "若$a$、$b \\in(0,+\\infty)$, $2 c>a+b$, 求证:\\\\\n(1) $c^2>a b$;\\\\\n(2) $c-\\sqrt{c^2-a b}0$).", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403592,7 +404124,9 @@ "id": "015618", "content": "若$a^2=\\dfrac{16}{81}$($a>0$), 则$\\log _{\\frac{2}{3}} a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403612,7 +404146,9 @@ "id": "015619", "content": "若$\\dfrac{m}{n}=\\mathrm{e}^3$, 则$\\ln m-\\ln n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403632,7 +404168,9 @@ "id": "015620", "content": "已知$4^a=2$, $\\lg x=a$, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403652,7 +404190,9 @@ "id": "015621", "content": "化简下列各式:\\\\\n(1) $(x^{\\frac{\\sqrt{3}}{2}})^{\\sqrt{3}} \\cdot \\sqrt{x}=$\\blank{50}($x>0$);\\\\\n(2) $\\dfrac{(a^{\\frac{2}{3}} b^{\\frac{1}{2}})(-3 a^{\\frac{1}{2}} b^{\\frac{1}{3}})}{\\dfrac{1}{3} a^{\\frac{1}{6}} b^{\\frac{5}{6}}}=$\\blank{50}($a>0$, $b>0$);\\\\\n(3) $27^{-\\frac{1}{3}}-(-\\dfrac{1}{7})^{-2}+256^{\\frac{3}{4}}-3^{-1}+(\\sqrt{2}-1)^0=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403672,7 +404212,9 @@ "id": "015622", "content": "下列各式中正确的是\\bracket{20}.\\\\\n\\textcircled{1} $\\log _2(8-2)=\\log _28-\\log _22=2$;\n\\textcircled{2} $\\log _2(8-2)=\\dfrac{\\log _28}{\\log _22}=3$;\n\\textcircled{3} $\\log _2 \\dfrac{8}{4}=\\log _28-\\log _24=1$;\n\\textcircled{4} $\\dfrac{\\log _28}{\\log _22}=\\log _28-\\log _22=2$;\n\\textcircled{5} $\\log _2[(-2)(-8)]=\\log _2(-2)+\\log _2(-8)=-4$.\n\\fourch{\\textcircled{1}\\textcircled{4}\\textcircled{5}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{3}}{全正确}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -403692,7 +404234,9 @@ "id": "015623", "content": "青少年视力是社会普遍关注的问题, 视力情况可借助视力表测量. 通常用五分记录法和小数记录法记录视力数据, 五分记录法的数据$L$和小数记录法的数据$V$满足$L=5+\\lg V$. 已知某同学视力的五分记录法的数据为$4.9$, 则其视力的小数记录法的数据为\\bracket{20}.\n\\fourch{$1.5$}{$1.2$}{$0.8$}{$0.6$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -403712,7 +404256,9 @@ "id": "015624", "content": "已知$b>0$, $\\log _5 b=a$, $\\lg b=c$, $5^d=10$, 求证: $a=c d$;", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -403732,7 +404278,9 @@ "id": "015625", "content": "若$2^{6 a}=3^{3 b}=6^{2 c}$, 求证: $3 a b-2 a c=b c$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -403752,7 +404300,9 @@ "id": "015626", "content": "已知$\\log _{18} 9=a$, $18^b=5$, 用$a$、$b$表示$\\log _{30} 36$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -403772,7 +404322,9 @@ "id": "015627", "content": "方程$\\lg x=8-2 x$的根$x \\in(k,k+1)$, $k \\in \\mathbf{Z}$, 则$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403792,7 +404344,9 @@ "id": "015628", "content": "对于任意实数$x$, 符号$[x]$表示不超过$x$的最大整数. 计算:\n$[\\log _21]+[\\log _22]+[\\log _23]+[\\log _24]+\\cdots+[\\log _21024]$的值$=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403812,7 +404366,9 @@ "id": "015629", "content": "若$\\lg x=a$, 则$\\lg (1000 x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403832,7 +404388,9 @@ "id": "015630", "content": "若$\\log _3 x=\\dfrac{1}{4}$, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403852,7 +404410,9 @@ "id": "015631", "content": "$\\dfrac{\\log _89}{\\log _23}$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403872,7 +404432,9 @@ "id": "015632", "content": "$\\lg 5^2+\\dfrac{2}{3} \\lg 8+\\lg 5 \\lg 20+(\\lg 2)^2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403892,7 +404454,9 @@ "id": "015633", "content": "已知$f(x)=a^{x-\\frac{1}{2}}$, 且$f(\\lg a)=\\sqrt{10}$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403912,7 +404476,9 @@ "id": "015634", "content": "若$3^a=0.618$, $a \\in[k,k+1)$, $k \\in \\mathbf{Z}$, 则$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -403932,7 +404498,9 @@ "id": "015635", "content": "下列关系式中正确的是\\bracket{20}.\n\\fourch{$2^{\\log _43} \\in \\mathbf{N}$}{$-2^{\\log _3 \\frac{1}{3}} \\in \\mathbf{N}$}{$2^{\\log _13} \\in \\mathbf{Q}$}{$-2^{\\log _3 \\frac{1}{3}} \\in \\mathbf{Q}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -403952,7 +404520,9 @@ "id": "015636", "content": "若$\\log _{10} m=b-\\log _{10} n$, 则$m$为\\bracket{20}.\n\\fourch{$\\dfrac{b}{n}$}{$10^b \\cdot n$}{$b-10^n$}{$\\dfrac{10^b}{n}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -403972,7 +404542,9 @@ "id": "015637", "content": "根据有关资料, 围棋状态空间复杂度的上限$M$约为$3^{361}$, 而可观测宇宙中普通物质的原子总数$N$约为$10^{80}$. 则下列各数中与$\\lg\\dfrac{M}{N}$最接近的是\\bracket{20}.\n\\fourch{$33$}{$53$}{$73$}{$93$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -403992,7 +404564,9 @@ "id": "015638", "content": "若$\\log _4(3 a+4 b)=\\log _2 \\sqrt{a b}$, 求$a+b$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404012,7 +404586,9 @@ "id": "015639", "content": "已知$\\log _{14} 7=a$, $\\log _{14} 5=b$, 用$a$、$b$表示$\\log _{35} 28$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404032,7 +404608,9 @@ "id": "015640", "content": "设$a$、$b$、$c$都是正数, 且$3^a=4^b=6^c$, 求证: $\\dfrac{2}{c}=\\dfrac{2}{a}+\\dfrac{1}{b}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404052,7 +404630,9 @@ "id": "015641", "content": "下列各选项给出的两个函数中, 表示相同函数的是\\bracket{20}.\n\\twoch{$f(x)=x$与$g(x)=\\sqrt{x^2}$}{$f(t)=|t-1|$与$g(x)=|x-1|$}{$f(x)=\\lg x$与$g(x)=\\dfrac{1}{2} \\lg x^2$}{$f(x)=\\dfrac{x^2-1}{x+1}$与$g(x)=x-1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404072,7 +404652,9 @@ "id": "015642", "content": "如图, 其中表示函数图像的是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-2,2)--(0,0)--(2,2);\n\\end{tikzpicture}\n}{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-2,1)--(0.5,1)--(0.5,-1)--(2,-1);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) circle (1.5);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$}; \n\\draw (-2,-2)--(2,2) (-2,2)--(2,-2);\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404092,7 +404674,9 @@ "id": "015643", "content": "已知$a \\in \\mathbf{R}$, 函数$f(x)=\\begin{cases}x^2-4,& x>2,\\\\ |x-3|+a,& x \\leq 2,\\end{cases}$若$f[f(\\sqrt{6})]=3$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404112,7 +404696,9 @@ "id": "015644", "content": "若函数$f(x)=\\sqrt{a-x}$的定义域为$A$, 函数$g(x)=\\lg (x-1)$, $x \\in[2,11]$的值域为$B$, 若$A \\cap B=B$, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404132,7 +404718,9 @@ "id": "015645", "content": "函数$f(x)=2^x-\\dfrac{1}{x}$的零点个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404152,7 +404740,9 @@ "id": "015646", "content": "如图为某加油站地下圆柱体储油罐示意图, 已知储油罐长度为$d$, 轴截面半径为$r$($d, r$为常量), 油面高度为$h$, 油面宽度为$w$, 储油量为$v$($h,w,v$为变量), 则下列说法: \\textcircled{1} $w$是$v$的函数; \\textcircled{2} $v$是$w$的函数; \\textcircled{3} $h$是$w$的函数; \\textcircled{4} $w$是$h$的函数. 其中正确的个数是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-sqrt(0.91)},-0.3) coordinate (A);\n\\draw ({sqrt(0.91)},-0.3) coordinate (B);\n\\draw (B)++(0,0,-5) coordinate (C);\n\\draw (A)++(0,0,-5) coordinate (D);\n\\draw (A)--(B)--(C);\n\\draw [dashed] (C)--(D)--(A);\n\\draw (0,-0.3) -- (0,-1) node [midway, right] {$h$};\n\\fill [gray!30] (A)--(B)--(C)--(D)--cycle;\n\\draw [domain = 0:360] plot ({cos(\\x)},{sin(\\x)},0);\n\\draw [domain = -45:135] plot ({cos(\\x)},{sin(\\x)},-5);\n\\draw [domain = 135:315, dashed] plot ({cos(\\x)},{sin(\\x)},-5);\n\\draw (-45:1) --++ (0,0,-5) (135:1) --++ (0,0,-5);\n\\draw (-45:1) ++ (0,-0.1) --++ (0,-0.8) (-45:1) ++ (0,0,-5) ++ (0,-0.1) --++ (0,-0.8);\n\\draw [<->] (-45:1) ++ (0,-0.5) --++ (0,0,-5) node [midway, fill = white] {$d$};\n\\draw ($(C)!0.5!(D)$) node [above] {$w$};\n\\draw (0,0) --++ (135:1) node [midway, below left] {$r$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$1$个}{$2$个}{$3$个}{$4$个}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404172,7 +404762,9 @@ "id": "015647", "content": "设$D$是含数$1$的有限实数集, $f(x)$是定义在$D$上的函数, 若$f(x)$的图像绕原点逆时针旋转$\\dfrac{\\pi}{6}$后与原图像重合, 则在以下各项中, $f(1)$的取值只能是\\bracket{20}.\n\\fourch{$\\sqrt{3}$}{$\\dfrac{\\sqrt{3}}{2}$}{$\\dfrac{\\sqrt{3}}{3}$}{$0$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404192,7 +404784,9 @@ "id": "015648", "content": "在平面直角坐标系$xOy$中, 若直线$y=2 a$与函数$y=|x-a|-1$的图像只有一个交点, 则$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404212,7 +404806,9 @@ "id": "015649", "content": "求下列函数的定义域:\\\\\n(1) $y=\\sqrt{\\log _{\\frac{1}{2}}(4 x-3)}$;\\\\\n(2) $y=\\sqrt{\\sqrt[3]{9}-(\\dfrac{1}{3})^x}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404232,7 +404828,9 @@ "id": "015650", "content": "已知函数$f(x)$的定义域为$[0,1]$, 求函数$y=f(\\lg |x+1|)$的定义域.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404252,7 +404850,9 @@ "id": "015651", "content": "已知函数$f(x)=\\begin{cases}2^x,& x<1,\\\\ f(x-1),& x \\geq 1,\\end{cases}$求$f(2022)$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404272,7 +404872,9 @@ "id": "015652", "content": "求下列函数的值域: \\\\\n(1) $y=\\dfrac{x+1}{x+2}$;\\\\\n(2) $f(x)=\\begin{cases}x^2-x+1, & x<1, \\\\ \\dfrac{1}{x}, & x>1;\\end{cases}$\\\\\n(3) $y=x-2 \\sqrt{1-x}+2$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404292,7 +404894,9 @@ "id": "015653", "content": "求$y=\\dfrac{\\mathrm{e}^x+1}{\\mathrm{e}^x+2}$的值域.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404312,7 +404916,9 @@ "id": "015654", "content": "求$y=\\dfrac{x+1}{x^2+2}$的值域.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404332,7 +404938,9 @@ "id": "015655", "content": "设函数$f(x)=\\begin{cases}x^2+x,& x<0,\\\\ -x^2,& x \\geq 0.\\end{cases}$\\\\\n(1) 求函数$y=f(x)$的零点;\\\\\n(2) 若函数$f(f(a)) \\leq 2$, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404352,7 +404960,9 @@ "id": "015656", "content": "若函数$f(x)=2^x-\\dfrac{1}{20} x^2$($x<0$)的零点为$x_0$, 且$x_0 \\in(a,a+1)$, $a \\in \\mathbf{Z}$, 求$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404372,7 +404982,9 @@ "id": "015657", "content": "将函数$y=\\sqrt{4+6 x-x^2}-2$($x \\in[0,6]$)的图像绕坐标原点逆时针方向旋转角$\\theta$, $0 \\leq \\theta \\leq \\alpha$, 得到曲线$C$. 若对于每一个旋转角$\\theta$, 曲线$C$都是一个函数的图像, 则$\\alpha$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404392,7 +405004,9 @@ "id": "015658", "content": "设函数$f(x)=\\begin{cases}1-x^2,& x \\leq 2,\\\\ \\dfrac{1}{2} f(x-2),& x>2,\\end{cases}$则$f(f(6))=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404412,7 +405026,9 @@ "id": "015659", "content": "设$f(x)=\\begin{cases}x,& x \\in(-\\infty,a),\\\\ x^2,& x \\in[a,+\\infty).\\end{cases}$若$f(2)=4$, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404432,7 +405048,9 @@ "id": "015660", "content": "函数$f(x)=\\dfrac{\\sqrt{4-x^2}}{\\lg (x+1)}$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404452,7 +405070,9 @@ "id": "015661", "content": "已知函数$f(x)$的定义域为$(-1,0)$, 则函数$f(2 x-1)$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404472,7 +405092,9 @@ "id": "015662", "content": "函数$f(x)=\\begin{cases}\\log _{\\frac{1}{2}} x,& x \\geq 1,\\\\ 2^x,& x<1\\end{cases}$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404492,7 +405114,9 @@ "id": "015663", "content": "函数$f(x)=2^x|\\log _{0.5} x|-1$的零点个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404512,7 +405136,9 @@ "id": "015664", "content": "设函数$y=\\sqrt{4-x^2}$的定义域为$A$, 函数$y=\\ln (1-x)$的定义域为$B$, 则$A \\cap B$为\\bracket{20}.\n\\fourch{$(1,2)$}{$(1,2]$}{$(-2,1)$}{$[-2,1)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404532,7 +405158,9 @@ "id": "015665", "content": "函数$f(x)=\\log _3 x+x-3$的零点所在的区间是\\bracket{20}.\n\\fourch{$(0,1)$}{$(1,2)$}{$(2,3)$}{$(3,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404552,7 +405180,9 @@ "id": "015666", "content": "在如图所示的锐角三角形空地中, 欲建一个面积不小于$300 \\mathrm{m}^2$的内接矩形花园(阴影部分), 则其边长$x$(单位: $\\text{m}$) 的取值范围是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) coordinate (A) -- (3,0) coordinate (B)-- (1.7,3) coordinate (C) -- cycle;\n\\draw (0,-0.1) -- (0,-0.7) (3,-0.1) -- (3,-0.7);\n\\draw [<->] (0,-0.4)--(3,-0.4) node [midway, fill=white] {$40\\text{m}$};\n\\draw (3.1,0) -- (3.7,0) (1.8,3)--(3.7,3);\n\\draw [<->] (3.4,0) -- (3.4,3) node [midway, fill = white] {$40\\text{m}$};\n\\filldraw [pattern = north east lines] ($(A)!0.3!(C)$) -- ($(B)!0.3!(C)$) node [midway, above] {$x$} -- ($(A)!($(B)!0.3!(C)$)!(B)$) -- ($(A)!($(A)!0.3!(C)$)!(B)$)-- cycle;\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$[15,20]$}{$[12,25]$}{$[10,30]$}{$[20,30]$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404572,7 +405202,9 @@ "id": "015667", "content": "求函数$f(x)=\\dfrac{\\sqrt{x-2}}{x-3}+\\lg \\sqrt{4-x}$的定义域.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404592,7 +405224,9 @@ "id": "015668", "content": "设函数$f(x)=\\begin{cases}2^{-x},& x \\in(-\\infty,1],\\\\ \\log _{81} x,& x \\in(1,+\\infty),\\end{cases}$求满足$f(x)=\\dfrac{1}{4}$的$x$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404612,7 +405246,9 @@ "id": "015669", "content": "求下列函数的值域:\\\\\n(1) $y=\\sqrt{-x^2+x+1}$;\\\\\n(2) $y=2 x-3+\\sqrt{13-4 x}$;\\\\\n(3) $y=\\dfrac{2^x}{2^x+2}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404632,7 +405268,9 @@ "id": "015670", "content": "已知函数$f(x)=\\begin{cases}x^2-x+3,& x \\leq 1,\\\\ x+\\dfrac{2}{x},& x>1.\\end{cases}$设$a \\in \\mathbf{R}$, 若关于$x$的不等式$f(x) \\geq |\\dfrac{x}{2}+a|$在$\\mathbf{R}$上恒成立, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404652,7 +405290,9 @@ "id": "015671", "content": "已知函数$f(x)$为奇函数, 且当$x>0$时, $f(x)=x^2+\\dfrac{1}{x}$, 则$f(-1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404672,7 +405312,9 @@ "id": "015672", "content": "已知奇函数$f(x)$的定义域为$\\mathbf{R}$, 且满足$f(x+3)=f(x)$, $f(-1)=2$, 则$f(2023)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404692,7 +405334,9 @@ "id": "015673", "content": "下列函数中是增函数的为\\bracket{20}.\n\\fourch{$f(x)=-x$}{$f(x)=(\\dfrac{2}{3})^x$}{$f(x)=x^2$}{$f(x)=\\sqrt[3]{x}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404712,7 +405356,9 @@ "id": "015674", "content": "定义域为$\\mathbf{R}$的四个函数$y=x^3$, $y=2^x$, $y=x^2+1$, $y=2 \\sin x$中, 奇函数的个数是\\bracket{20}.\n\\fourch{$4$}{$3$}{$2$}{$1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404732,7 +405378,9 @@ "id": "015675", "content": "已知$f(x)$是定义在$[0,1]$上的函数, 那么``函数$f(x)$在$[0,1]$上严格递增''是``函数$f(x)$在$[0,1]$上的最大值为$f(1)$\" 的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404752,7 +405400,9 @@ "id": "015676", "content": "判断下列函数的奇偶性:\\\\\n(1) $f(x)=|2 x-3|+|2 x+3|$;\\\\\n(2) $f(x)=x^2+\\dfrac{1}{x}$;\\\\\n(3) $f(x)=\\begin{cases}x^2+x,& x<0,\\\\ -x^2+x,& x>0;\\end{cases}$\\\\\n(4) $f(x)=\\lg (x+\\sqrt{x^2+1})$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404772,7 +405422,9 @@ "id": "015677", "content": "已知函数$f(x)=x^3(a \\cdot 2^x-2^{-x})$是偶函数, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404792,7 +405444,9 @@ "id": "015678", "content": "设函数$f(x)=\\dfrac{1-x}{1+x}$, 则下列函数中为奇函数的是\\bracket{20}.\n\\fourch{$f(x-1)-1$}{$f(x-1)+1$}{$f(x+1)-1$}{$f(x+1)+1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404812,7 +405466,9 @@ "id": "015679", "content": "设函数$f(x)$是定义在$\\mathbf{R}$上的奇函数, 且当$x \\in(0,+\\infty)$时, $f(x)=1+\\lg x$.\\\\\n(1) 求函数$f(x)$的表达式;\\\\\n(2) 求$f(x)>0$的解集(用区间表示).", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404832,7 +405488,9 @@ "id": "015680", "content": "设常数$a \\geq 0$, 函数$f(x)=\\dfrac{2^x+a}{2^x-a}$, 根据$a$的不同取值, 讨论函数$y=f(x)$的奇偶性, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404852,7 +405510,9 @@ "id": "015681", "content": "设函数$f(x)$的定义域为$\\mathbf{R}$, $f(x+1)$为奇函数, $f(x+2)$为偶函数, 当$x \\in[1,2]$时, $f(x)=a x^2+b$. 若$f(0)+f(3)=6$, 则$f(\\dfrac{9}{2})=$\\bracket{20}.\n\\fourch{$-\\dfrac{9}{4}$}{$-\\dfrac{3}{2}$}{$\\dfrac{7}{4}$}{$\\dfrac{5}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404872,7 +405532,9 @@ "id": "015682", "content": "已知函数$f(x)$的定义域为$\\mathbf{R}$, $f(x+2)$为偶函数, $f(2 x+1)$为奇函数, 则\\bracket{20}.\n\\fourch{$f(-\\dfrac{1}{2})=0$}{$f(-1)=0$}{$f(2)=0$}{$f(4)=0$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -404892,7 +405554,9 @@ "id": "015683", "content": "设$f(x)$是定义在$\\mathbf{R}$上的偶函数, 其图像关于$x=1$对称, 对于任意的$x_1,x_2 \\in[0,\\dfrac{1}{2}]$, 都有$f(x_1+x_2)=f(x_1) \\cdot f(x_2)$.\\\\\n(1) 设$f(1)=2$, 求$f(\\dfrac{1}{2})$, $f(\\dfrac{1}{4})$;\\\\\n(2) 证明: $f(x)$是周期函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -404912,7 +405576,9 @@ "id": "015684", "content": "已知函数$f(x)=m x^2+2(m-1) x+1$为偶函数, 则$f(1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -404932,7 +405598,9 @@ "id": "015685", "content": "已知函数$f(x)$的周期为$1$, 且当$00,\\end{cases}$则$f(2022)$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405032,7 +405708,9 @@ "id": "015690", "content": "下列函数中, 既是奇函数又是减函数的是\\bracket{20}.\n\\fourch{$y=-3 x$}{$y=x^3$}{$y=\\log _3 x$}{$y=3^x$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -405052,7 +405730,9 @@ "id": "015691", "content": "函数$y=\\dfrac{10\\ln |x|}{x^2+2}$的图像大致为\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.5:4] plot (\\x,{-10*ln(abs(\\x))/(pow(\\x,2)+2)}); \n\\draw [domain = -4:-0.5] plot (\\x,{10*ln(abs(\\x))/(pow(\\x,2)+2)}); \n\\end{tikzpicture}}{\n\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.5:4] plot (\\x,{10*ln(abs(\\x))/(pow(\\x,2)+2)}); \n\\draw [domain = -4:-0.5] plot (\\x,{10*ln(abs(\\x))/(pow(\\x,2)+2)}); \n\\end{tikzpicture}}{\n\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.5:4] plot (\\x,{10*ln(abs(\\x))/(pow(\\x,2)+2)}); \n\\draw [domain = -4:-0.5] plot (\\x,{-10*ln(abs(\\x))/(pow(\\x,2)+2)}); \n\\end{tikzpicture}}{\n\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.5:4] plot (\\x,{-10*ln(abs(\\x))/(pow(\\x,2)+2)}); \n\\draw [domain = -4:-0.5] plot (\\x,{-10*ln(abs(\\x))/(pow(\\x,2)+2)}); \n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -405072,7 +405752,9 @@ "id": "015692", "content": "中国清朝数学家李善兰在$1859$年翻译《代数学》中首次将``function''译做``函数'', 沿用至今. 为什么这么翻译, 书中解释说: ``凡此变数中函彼变数者, 则此为彼之函数. ''这个解释说明了函数的内涵: 只要有一个法则, 使得取值范围中的每一个值$x$, 有一个确定的$y$和它对应就行了, 不管这个对应的法则是公式、图像、表格还是其他形式. 已知函数$f(x)$由下表给出, 则$f(f(-2)+1)$的值为\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline $x$ & $x \\leq 0$ & $02 x-1$;\\\\\n(2) 讨论函数$f(x)$的奇偶性, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405112,7 +405796,9 @@ "id": "015694", "content": "已知函数$f(x)$是定义在$\\mathbf{R}$上的奇函数, 且满足$f(x+1)=f(x-1)$, 当$0f(x_2)$的是\\bracket{20}.\n\\fourch{$f(x)=\\dfrac{1}{x}$}{$f(x)=(x-1)^2$}{$f(x)=\\mathrm{e}^x$}{$y=|x-1|+|2-x|$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -405232,7 +405928,9 @@ "id": "015700", "content": "已知奇函数$f(x)$是$(-\\infty,+\\infty)$上的严格减函数, 若$f(1)=-1$, 则满足$-1 \\leq f(x-2) \\leq 1$的$x$的取值范围是\\bracket{20}.\n\\fourch{$[-2,2]$}{$[-1,1]$}{$[0,4]$}{$[1,3]$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -405252,7 +405950,9 @@ "id": "015701", "content": "判断函数$f(x)=x+\\dfrac{4}{x}$, $x \\in[2,+\\infty)$的单调性.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405272,7 +405972,9 @@ "id": "015702", "content": "讨论函数$f(x)=x+\\dfrac{4}{x}$, $x \\in(0,+\\infty)$的单调性.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405292,7 +405994,9 @@ "id": "015703", "content": "如果$f(x)=x+\\dfrac{a}{x}$是$[2,+\\infty)$上的严格增函数, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405312,7 +406016,9 @@ "id": "015704", "content": "讨论函数$f(x)=x+\\dfrac{a}{x}$($a \\neq 0$)上的单调性.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405332,7 +406038,9 @@ "id": "015705", "content": "已知函数$f(x)=a x^3+\\dfrac{1}{x}$, 其中$a$为实数.\\\\\n(1) 根据$a$的不同取值, 判断函数$f(x)$的奇偶性, 并说明理由;\\\\\n(2) 若$a \\in(1,3)$, 判断函数$f(x)$在$[1,2]$上的单调性, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405352,7 +406060,9 @@ "id": "015706", "content": "已知函数$f(x)=|x-a|$, $g(x)=a x$($a \\in \\mathbf{R}$).\\\\\n(1) 讨论函数$f(x)$的奇偶性;\\\\\n(2) 若$a>0$, 记$F(x)=g(x)-f(x)$, 且$F(x)$在$(0,+\\infty)$上有最大值, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405372,7 +406082,9 @@ "id": "015707", "content": "设$f(x)$是$\\mathbf{R}$上的奇函数, 且对任意的实数$a$、$b$, 当$a+b \\neq 0$时, 都有$\\dfrac{f(a)+f(b)}{a+b}>0$.\\\\\n(1) 若$a>b$, 试比较$f(a),f(b)$的大小;\\\\\n(2) 若存在实数$x \\in[\\dfrac{1}{2},\\dfrac{3}{2}]$使得不等式$f(x-c)+f(x-c^2)>0$成立, 试求实数$c$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405392,7 +406104,9 @@ "id": "015708", "content": "若函数$f(x)=x^2+a x+\\dfrac{1}{x}$是区间$(\\dfrac{1}{2},+\\infty)$上的严格增函数, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405412,7 +406126,9 @@ "id": "015709", "content": "函数$y=-\\dfrac{1}{x+1}$的单调区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405432,7 +406148,9 @@ "id": "015710", "content": "函数$y=\\log _2(2 x^2-x)$的严格递减区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405452,7 +406170,9 @@ "id": "015711", "content": "若$f(x)=(m-1) x^2+m x+3$($x \\in \\mathbf{R})$是偶函数, 则$f(x)$的严格递增区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405472,7 +406192,9 @@ "id": "015712", "content": "已知函数$f(x)=\\begin{cases}2^x,& x \\geq 0,\\\\ 1,& x<0,\\end{cases}$若$f(1-a^2)>f(2 a)$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405492,7 +406214,9 @@ "id": "015713", "content": "已知函数$y=k x^2-4 x-8$是区间$[4,16]$上的严格减函数, 则实数$k$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405512,7 +406236,9 @@ "id": "015714", "content": "设$f(x)=\\dfrac{1}{x}-\\lg x$, 则不等式$f(\\dfrac{1}{x}-1)<1$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405532,7 +406258,9 @@ "id": "015715", "content": "下列函数中, 既是偶函数又在区间$(0,+\\infty)$上严格递增的函数为 \\bracket{20}.\n\\fourch{$y=x^3$}{$y=x^2+\\dfrac{1}{x^2}$}{$y=|x|$}{$y=-x^2$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -405552,7 +406280,9 @@ "id": "015716", "content": "下列函数中, 在区间$(-\\infty,0]$上为严格增函数的是\\bracket{20}.\n\\fourch{$y=x^2-2$}{$y=\\dfrac{3}{x}$}{$y=1-\\sqrt{2-x}$}{$y=-(x+2)^2$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -405572,7 +406302,9 @@ "id": "015717", "content": "已知$f(x)$是定义在$\\mathbf{R}$上的奇函数, 对任意两个不相等的正数$x_1$、$x_2$, 都有$\\dfrac{x_2 f(x_1)-x_1 f(x_2)}{x_1-x_2}<0$,\n则函数$g(x)=\\begin{cases}\\dfrac{f(x)}{x},& x \\neq 0,\\\\ 0,& x=0,\\end{cases}$\\bracket{20}.\n\\twoch{是偶函数, 且在$(0,+\\infty)$上严格递减}{是偶函数, 且在$(0,+\\infty)$上严格递增}{是奇函数, 且严格递减}{是奇函数, 且严格递增}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -405592,7 +406324,9 @@ "id": "015718", "content": "证明函数$f(x)=\\dfrac{a x}{x^2-1}(a>0)$是区间$(-1,1)$上的严格递减函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405612,7 +406346,9 @@ "id": "015719", "content": "已知函数$f(x)=(a+1) x^2+(a-1) x+(a^2-1)$, 其中$a \\in \\mathbf{R}$.\\\\\n(1) 当$f(x)$是奇函数时, 求实数$a$的值;\\\\\n(2) 当函数$f(x)$是区间$[2,+\\infty)$上严格增函数时, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405632,7 +406368,9 @@ "id": "015720", "content": "设函数$f(x)=x^2+|x-a|$, $a$为常数.\\\\\n(1) 若$f(x)$为偶函数, 求$a$的值;\\\\\n(2) 设$a>0$, $g(x)=\\dfrac{f(x)}{x}$是区间$(0,a]$的严格减函数, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405652,7 +406390,9 @@ "id": "015721", "content": "函数$y=\\begin{cases}x^2+2 x,& x \\geq 0,\\\\ -1,& x<0\\end{cases}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405672,7 +406412,9 @@ "id": "015722", "content": "函数$y=\\dfrac{2}{x-1}$的定义域为$[2,5)$, 则其值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405692,7 +406434,9 @@ "id": "015723", "content": "函数$y=|x|^3$, $x \\in[-2,-1]$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405712,7 +406456,9 @@ "id": "015724", "content": "函数$y=x-\\sqrt{1-2 x}$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405732,7 +406478,9 @@ "id": "015725", "content": "设函数$f(x)$的定义域为$\\mathbf{R}$, 有下列三个命题:\\\\\n\\textcircled{1} 若存在常数$M$, 使得对任意$x \\in \\mathbf{R}$, 有$f(x) \\leq M$, 则$M$是函数$f(x)$的最大值;\\\\\n\\textcircled{2} 若存在$x_0 \\in \\mathbf{R}$, 使得对任意$x \\in \\mathbf{R}$, 且$x \\neq x_0$, 有$f(x)20\\end{cases}$(万元).\\\\\n(1) 投入$A$、$B$两个项目的资金相同且$B$项目比$A$项目创造的利润高, 求投入$A$项目的资金$x$(万元)的取值范围;\\\\\n(2) 若该公司共有资金$30$万元, 全部用于投资$A$、$B$两个项目, 则该公司一年分别投入$A$、$B$两个项目多少万元, 创造的利润最大.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405852,7 +406610,9 @@ "id": "015731", "content": "已知函数$f(x)=x^2-2 a x+5$($a>1$).\\\\\n(1) 若$f(x)$的定义域和值域均是$[1,a]$, 求实数$a$的值;\\\\\n(2) 若$f(x)$在区间$(-\\infty,2]$上是减函数, 且对任意的$x_1,x_2 \\in[1,a+1]$, 总有$|f(x_1)-f(x_2)| \\leq 4$, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405872,7 +406632,9 @@ "id": "015732", "content": "已知$g(x)=-x^2-3$, $f(x)$是二次函数, 当$x \\in[-1,2]$时, $f(x)$的最小值为$1$, 且$f(x)+g(x)$为奇函数, 求$f(x)$的解析式.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -405892,7 +406654,9 @@ "id": "015733", "content": "函数$y=x+\\dfrac{4}{x}(x \\geq 4)$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405912,7 +406676,9 @@ "id": "015734", "content": "已知实数$x$、$y$满足$x^2+y^2=4$, 则$x^2 y^2$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405932,7 +406698,9 @@ "id": "015735", "content": "函数$f(x)=\\begin{cases}1-x,& x \\leq 0,\\\\ x^2+2 x,& x>0\\end{cases}$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405952,7 +406720,9 @@ "id": "015736", "content": "在一块半径为$R$半圆形铁皮上截出一矩形铁皮, 矩形的一边在半圆的直径上, 则这个矩形的最大面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -405972,7 +406742,9 @@ "id": "015737", "content": "若函数$f(x)=\\log _a x$($0g(x)$时, 求函数$\\dfrac{g(x)+1}{f(x)}$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406092,7 +406874,9 @@ "id": "015743", "content": "已知$\\dfrac{1}{3} \\leq a \\leq 1$, $f(x)=a x^2-2 x+1$在区间$[1,3]$上的最大值为$M(a)$, 最小值为$N(a)$, 设$g(a)=M(a)-N(a)$.\\\\\n(1) 求$g(a)$的解析式;\\\\\n(2) 判断$g(a)$的单调性, 并求出$g(a)$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406112,7 +406896,9 @@ "id": "015744", "content": "某地政府决定向当地纳税额在$4$万元至$8$万元(包括$4$万元和$8$万元)的小微企业发放补助款, 发放方案规定: 补助款随企业纳税额的增加而增加, 且补助款不低于纳税额的$50 \\%$. 设企业纳税额为$x$(单位: 万元), 补助款为$f(x)=\\dfrac{1}{4} x^2-b x+b+\\dfrac{1}{2}$(单位: 万元), 其中$b$为常数.\\\\\n(1) 分别判断$b=0$, $b=1$时, $f(x)$是否符合发放方案规定, 并说明理由;\\\\\n(2) 若函数$f(x)$符合发放方案规定, 求$b$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406132,7 +406918,9 @@ "id": "015745", "content": "幂函数$f(x)$的图像经过点$(3,\\dfrac{\\sqrt{3}}{3})$, 则$f(8)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406152,7 +406940,9 @@ "id": "015746", "content": "函数$y=2 x+\\dfrac{1}{x}$是区间的严格增区间为\\blank{50}, 是严格减区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406172,7 +406962,9 @@ "id": "015747", "content": "若$f(x)$是幂函数, 且满足$\\dfrac{f(4)}{f(2)}=8$, 则$f(\\dfrac{1}{2})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406192,7 +406984,9 @@ "id": "015748", "content": "下列幂函数中, 定义域为$\\mathbf{R}$的是\\bracket{20}.\n\\fourch{$y=x^{-1}$}{$y=x^{-\\frac{1}{2}}$}{$y=x^{\\frac{1}{3}}$}{$y=x^{\\frac{1}{2}}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -406212,7 +407006,9 @@ "id": "015749", "content": "函数$y=x+\\dfrac{9}{x}$, $x \\in[1,3]$, 则\\bracket{20}.\n\\twoch{有最大值$10$, 无最小值}{有最小值$6$, 无最大值}{有最大值$10$, 最小值$6$}{无最大最小值}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -406232,7 +407028,9 @@ "id": "015750", "content": "已知四个函数: \\textcircled{1} $y=-x$; \\textcircled{2} $y=-\\dfrac{1}{x}$; \\textcircled{3} $y=x^3$; \\textcircled{4} $y=x^{\\frac{1}{2}}$. 从中任选$2$个, 则事件``所选$2$个函数的图像有且仅有一个公共点''的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406252,7 +407050,9 @@ "id": "015751", "content": "下列函数中, 为偶函数的是\\bracket{20}.\n\\fourch{$y=x^{-2}$}{$y=x^{\\frac{1}{3}}$}{$y=x^{-\\frac{1}{2}}$}{$y=x^3$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -406272,7 +407072,9 @@ "id": "015752", "content": "若$f(x)=x^{\\frac{2}{3}}-x^{-\\frac{1}{2}}$, 则满足$f(x)<0$的$x$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406292,7 +407094,9 @@ "id": "015753", "content": "设$f(x)$是定义在$\\mathbf{R}$上以$2$为最小正周期的周期函数, 当$-1 \\leq x<1$的, $y=f(x)$的表达式是幂函数, 且经过点$(\\dfrac{1}{2},\\dfrac{1}{8})$, 求函数在$[2 k-1,2 k+1)$($k \\in \\mathbf{Z}$)上的表达式.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406312,7 +407116,9 @@ "id": "015754", "content": "已知$a \\in \\mathbf{R}$, 函数$f(x)=x+\\dfrac{a}{x+1}$, $x \\in[0,+\\infty)$, 求函数$f(x)$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406332,7 +407138,9 @@ "id": "015755", "content": "设$a$为实常数, $y=f(x)$是定义在$\\mathbf{R}$上的奇函数, 当$x<0$时, $f(x)=9 x+\\dfrac{a^2}{x}+7$, 若$f(x) \\geq a+1$对一切$x \\geq 0$成立, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406352,7 +407160,9 @@ "id": "015756", "content": "已知函数$f(x)=a x^2+\\dfrac{1}{x}$, 其中$a$为常数.\\\\\n(1) 根据$a$的不同取值, 判断函数$f(x)$的奇偶性, 并说明理由;\\\\\n(2) 若$a \\in(1,3)$, 判断函数$f(x)$在$[1,2]$上的单调性, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406372,7 +407182,9 @@ "id": "015757", "content": "已知函数$f(x)=x+\\dfrac{m}{x}+2$($m$为实常数, 且$m \\neq 0$).\\\\\n(1) 若函数$y=f(x)$图像上动点$P$到定点$Q(0,2)$的距离的最小值为$\\sqrt{2}$, 求实数$m$的值;\\\\\n(2) 若函数$y=f(x)$是区间$[2,+\\infty)$上的严格增函数, 试用函数单调性的定义求实数$m$的取值范围;\\\\\n(3) 设$m<0$, 若不等式$f(x) \\leq k x$在$x \\in[\\dfrac{1}{2},1]$有解, 求$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406392,7 +407204,9 @@ "id": "015758", "content": "函数$y=x^{-\\frac{2}{5}}$的严格递减区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406412,7 +407226,9 @@ "id": "015759", "content": "已知函数$f(x)=x^{\\frac{1}{3}}$, $x \\in(1,27)$的值域为$A$, 集合$B=\\{x | x^2-2 x<0,\\ x \\in \\mathbf{R}\\}$, 则$A \\cap B$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406432,7 +407248,9 @@ "id": "015760", "content": "已知$\\alpha \\in\\{-2,-1,-\\dfrac{1}{2},\\dfrac{1}{2},1,2,3\\}$, 若幂函数$f(x)=x^{\\alpha}$为奇函数, 且是区间$(0,+\\infty)$上的严格减函数, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406452,7 +407270,9 @@ "id": "015761", "content": "已知函数$f(x)=x+\\dfrac{a}{x}$($a \\in \\mathbf{R})$是区间$[1,+\\infty)$上的严格增函数, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406472,7 +407292,9 @@ "id": "015762", "content": "已知$f(x)=\\begin{cases}x^3,& x \\leq a,\\\\ x^2,& x>a\\end{cases}$($a \\in \\mathbf{R}$), 若存在实数$b$, 使函数$g(x)=f(x)-b$有两个零点, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406492,7 +407314,9 @@ "id": "015763", "content": "已知函数$f(x)=\\begin{cases}\\dfrac{2}{x},& x \\geq 2,\\\\ (x-1)^3,& x<2,\\end{cases}$若关于$x$的方程$f(x)=k$有两个不同的实根, 则实数$k$\n的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406512,7 +407336,9 @@ "id": "015764", "content": "函数$y=x^{\\frac{3}{5}}$, $x \\in[-1,1]$\\bracket{20}.\n\\twoch{是奇函数且是定义域上的严格增函数}{是偶函数且是定义域上的严格增函数}{是奇函数且是定义域上的严格减函数}{是偶函数且是定义域上的严格减函数}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -406532,7 +407358,9 @@ "id": "015765", "content": "设函数$f(x)=2 x+\\dfrac{1}{x}-1$($x<0$), 则$f(x)$\\bracket{20}.\n\\twoch{有最大值}{有最小值}{是$(-\\infty,0)$上的严格增函数}{是$(-\\infty,0)$上的严格减函数}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -406552,7 +407380,9 @@ "id": "015766", "content": "设$f(x)$是定义在$\\mathbf{R}$上的函数, 若存在两个不等实数$x_1,x_2 \\in \\mathbf{R}$, 使得$f(\\dfrac{x_1+x_2}{2})=\\dfrac{f(x_1)+f(x_2)}{2}$, 则称函数$f(x)$具有性质$P$, 那么以下函数: \\textcircled{1} $f(x)=\\begin{cases}\\dfrac{1}{x},& x \\neq 0,\\\\ 0,& x=0; \\end{cases}$\n\\textcircled{2} $f(x)=x^3$; \\textcircled{3} $f(x)=|x^2-1|$; \\textcircled{4} $f(x)=x^2$中, 不具有性质$P$的函数为\\bracket{20}.\n\\fourch{\\textcircled{1}}{\\textcircled{2}}{\\textcircled{3}}{\\textcircled{4}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -406572,7 +407402,9 @@ "id": "015767", "content": "设幂函数$f(x)=(a-1) x^k$($a \\in \\mathbf{R}$, $k \\in \\mathbf{Q}$)的图像经过点$(\\sqrt{2},2)$.\\\\\n(1) 求$a, k$的值;\\\\\n(2) 求函数$y=f(x)+\\dfrac{1}{f(x)}$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406592,7 +407424,9 @@ "id": "015768", "content": "考虑到高速公路行车安全需要, 一般要求高速公路的车速$v$(公里/小时) 控制在$[60,120]$范围内. 已知汽车以$v$公里/小时的速度在高速公路上均速行驶时, 每小时的油耗 (所需要的汽油量) 为$\\dfrac{1}{5}(v-k+\\dfrac{4500}{v})$升, 其中$k$为常数, 不同型号汽车$k$值不同, 且满足$60 \\leq k \\leq 120$.\\\\\n(1) 若某型号汽车以$120$公里/小时的速度行驶时, 每小时的油耗为$11.5$升, 欲使这种型号的汽车每小时的油耗不超过$9$升, 求车速$v$的取值范围;\\\\\n(2) 求不同型号汽车行驶$100$千米的油耗的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406612,7 +407446,9 @@ "id": "015769", "content": "已知函数$y=x+\\dfrac{a}{x}$有如下性质: 如果常数$a>0$, 那么该函数在$(0,\\sqrt{a}]$上是减函数, 在$[\\sqrt{a},+\\infty)$上是增函数.\\\\\n(1) 如果函数$y=x+\\dfrac{2^b}{x}(x>0)$在$(0,4]$上是减函数, 在$[4,+\\infty)$上是增函数, 求$b$的值;\\\\\n(2) 设常数$c \\in[1,4]$, 求函数$f(x)=x+\\dfrac{c}{x}$($1 \\leq x \\leq 2$)的最大值和最小值;\\\\\n(3) 当$n$是正整数时, 研究函数$g(x)=x^n+\\dfrac{c}{x^n}$($c>0$)的单调性, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406632,7 +407468,9 @@ "id": "015770", "content": "函数$y=(\\dfrac{1}{2})$的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406652,7 +407490,9 @@ "id": "015771", "content": "方程$|a^x-1|=2 a$($a>0$, $a \\neq 1$)有且仅有两解, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406672,7 +407512,9 @@ "id": "015772", "content": "已知函数$f(x)=3^x-(\\dfrac{1}{3})^x$, 则$f(x)$\\bracket{20}.\n\\twoch{是奇函数, 且在$\\mathbf{R}$上是严格增函数}{是偶函数, 且在$\\mathbf{R}$上是严格增函数}{是奇函数, 且在$\\mathbf{R}$上是严格减函数}{是偶函数, 且在$\\mathbf{R}$上是严格减函数}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -406692,7 +407534,9 @@ "id": "015773", "content": "下列函数中, 在区间$(-1,1)$上为减函数的是\\bracket{20}.\n\\fourch{$y=\\dfrac{1}{1-x}$}{$y=\\cos x$}{$y=\\ln (x+1)$}{$y=2^{-x}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -406712,7 +407556,9 @@ "id": "015774", "content": "已知函数$f(x)$满足: $f(x) \\geq |x|$且$f(x) \\geq 2^x$\\bracket{20}.\n\\twoch{若$f(a) \\leq |b|$, 则$a \\leq b$}{若$f(a) \\leq 2^b$, 则$a \\leq b$}{若$f(a) \\geq|b|$, 则$a \\geq b$}{若$f(a) \\geq 2^b$, 则$a \\geq b$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -406732,7 +407578,9 @@ "id": "015775", "content": "已知函数$f(x)=x^3(a \\cdot 2^x-2^{-x})$是偶函数, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406752,7 +407600,9 @@ "id": "015776", "content": "已知函数$f(x)$是定义在$\\mathbf{R}$上的周期为$2$的奇函数, 当$00$的解集为 \\bracket{20}.\n\\fourch{$(-\\infty,1)$}{$(1,+\\infty)$}{$(-\\infty,\\dfrac{1}{3})$}{$(\\dfrac{1}{3},+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -406792,7 +407644,9 @@ "id": "015778", "content": "定义域为$\\mathbf{R}$的函数$f(x)=2^x-2^{-x}$, $g(x)=2^x+2^{-x}$.\\\\\n(1) 请分别指出函数$y=f(x)$与函数$y=g(x)$的奇偶性、单调性、值域和零点(不必证明);\n(2) 设$h(x)=\\dfrac{f(x)}{g(x)}$, 请判断函数$y=h(x)$的奇偶性和单调性, 并证明你的结论. (必要时, 可以用(1)中的结论作为推理与证明的依据)", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406812,7 +407666,9 @@ "id": "015779", "content": "已知函数$f(x)=a^x+b^x$($a>0$, $b>0$, $a \\neq 1$, $b \\neq 1$).\\\\\n(1) 若$a=2$, $b=\\dfrac{1}{2}$, 求方程$f(x)=2$的根;\\\\\n(2) 若$a=2$, $b=\\dfrac{1}{2}$, 且对任意$x \\in \\mathbf{R}$, 不等式$f(2 x) \\geq m f(x)-6$恒成立, 求实数$m$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406832,7 +407688,9 @@ "id": "015780", "content": "设$a>0$, 函数$f(x)=\\dfrac{1}{1+a \\cdot 2^x}$\\\\\n(1) 若$a=1$, 求方程$f(x)=\\dfrac{1}{4}$的解;\\\\\n(2) 求函数$y=f(x) f(-x)$的最大值 (用$a$表示);\\\\\n(3) 设$g(x)=f(x)-f(x-1)$, 若对任意$x \\in(-\\infty,0]$, $g(x) \\geq g(0)$恒成立, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -406852,7 +407710,9 @@ "id": "015781", "content": "函数$y=\\dfrac{2^x}{2^x+1}$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406872,7 +407732,9 @@ "id": "015782", "content": "函数$y=\\sqrt{(\\dfrac{1}{2})^x-(\\dfrac{1}{3})^x}$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406892,7 +407754,9 @@ "id": "015783", "content": "不论$a$为何值时, 函数$y=(a-1) \\cdot 2^x-a$的图像过一定点, 则这个定点的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406912,7 +407776,9 @@ "id": "015784", "content": "函数$f(x)=a^x$($a>0$, $a \\neq 1$)在区间$[1,2]$上的最大值比最小值大$\\dfrac{a}{2}$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406932,7 +407798,9 @@ "id": "015785", "content": "已知$f(x)$是定义在$\\mathbf{R}$上的偶函数, 且是区间$(-\\infty,0)$上的严格增函数. 若实数$a$满足$f(2^{|a-1|})>f(-\\sqrt{2})$, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406952,7 +407820,9 @@ "id": "015786", "content": "已知函数$f(x)=\\begin{cases}a \\cdot 2^x,& x \\geq 0,\\\\ 2^{-x},& x<0,\\end{cases} a \\in \\mathbf{R}$, 若$f[f(-1)]=1$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -406972,7 +407842,9 @@ "id": "015787", "content": "指数函数$f(x)=a^x$($a>0$, 且$a \\neq 1$)是$\\mathbf{R}$上的严格减函数, 则函数$g(x)=(a-2) x^2$的单调性为\\bracket{20}.\n\\twoch{严格递增}{严格递减}{在$(-\\infty,0)$上递减, 在$(0,+\\infty)$上递增}{在$(-\\infty,0)$上递增, 在$(0,+\\infty)$上递减}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -406992,7 +407864,9 @@ "id": "015788", "content": "函数$f(x)=2^x$, $x_1,x_2 \\in \\mathbf{R}$且$x_1 \\neq x_2$, 则\\bracket{20}.\n\\twoch{$\\dfrac{1}{2}[f(x_1)+f(x_2)]=f(\\dfrac{x_1+x_2}{2})$}{$\\dfrac{1}{2}[f(x_1)+f(x_2)]>f(\\dfrac{x_1+x_2}{2})$}{$\\dfrac{1}{2}[f(x_1)+f(x_2)]0$, $x_2+x_3>0$, $x_3+x_1>0$则$f(x_1)+f(x_2)+f(x_3)$的值\\bracket{20}.\n\\fourch{一定等于零}{一定大于零}{一定小于零}{正负都有可能}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -407032,7 +407908,9 @@ "id": "015790", "content": "已知$f(x)=a-\\dfrac{1}{2^x-1}$是定义在$(-\\infty,-1] \\cup[1,+\\infty)$上的奇函数, 求函数$f(x)$的值域.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407052,7 +407930,9 @@ "id": "015791", "content": "已知函数$f(x)=\\mathrm{e}^x+\\mathrm{e}^{-x}$, 其中$e$是自然对数的底数.\\\\\n(1) 证明: $f(x)$是$\\mathbf{R}$上的偶函数;\\\\\n(2) 若关于$x$的不等式$m f(x) \\leq \\mathrm{e}^{-x}+m-1$在$(0,+\\infty)$上恒成立, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407072,7 +407952,9 @@ "id": "015792", "content": "已知$k \\in \\mathbf{R}$, $a>0$且$a \\neq 1$, $b>0$且$b \\neq 1$, 函数$f(x)=a^x+k \\cdot b^x$.\\\\\n(1) 如果实数$a$、$b$满足$a>1$, $a b=1$, 试判断函数$f(x)$的奇偶性, 并说明理由;\\\\\n(2) 若$a=2$, $b=\\dfrac{1}{2}$, 且$k>0$, 函数$f(x)$的图像是否存在一条垂直于$x$轴的对称轴, 若存在, 求出该对称轴的方程; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407092,7 +407974,9 @@ "id": "015793", "content": "若函数$f(x)=a^x$($a>0$, $a \\neq 1$)的图像过点$(-1,2)$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407112,7 +407996,9 @@ "id": "015794", "content": "若函数$f(x)=\\sqrt{\\log _{\\frac{1}{2}} x-1}$, 则函数$f(x)$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407132,7 +408018,9 @@ "id": "015795", "content": "函数$f(x)=\\lg x^2$的严格递减区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407152,7 +408040,9 @@ "id": "015796", "content": "下列函数中, 是定义域$\\mathbf{R}$上严格增函数的是\\bracket{20}.\n\\fourch{$y=\\mathrm{e}^{-x}$}{$y=x^3$}{$y=\\ln x$}{$y=|x|$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -407172,7 +408062,9 @@ "id": "015797", "content": "下列函数中, 既是$(0,+\\infty)$上的严格增函数, 又是偶函数的是\\bracket{20}.\n\\fourch{$y=\\dfrac{1}{x}$}{$y=2^x$}{$y=1-|x|$}{$y=\\lg |x|$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -407192,7 +408084,9 @@ "id": "015798", "content": "已知函数$f(x)=\\log _a x$($a>0$且$a \\neq 1$).\\\\\n(1) 若函数$f(x)$在$[2,3]$上的最大值与最小值的和为$2$, 求$a$的值;\\\\\n(2) 将函数$f(x)$图像上所有的点向左平移$2$个单位长度, 再向下平移$1$个单位长度, 所得函数图像不经过第二象限, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407212,7 +408106,9 @@ "id": "015799", "content": "已知函数$f(x)=\\log _a \\dfrac{1-m x}{x-1}$是奇函数 (其中$a>1$).\\\\\n(1) 求实数$m$的值;\\\\\n(2) 讨论函数$f(x)$的单调性.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407232,7 +408128,9 @@ "id": "015800", "content": "设函数$y=f(x)$定义在区间$(a,b)$上, 若对任意的$x_1$、$x_2$、$x_1'$、$x_2' \\in(a,b)$, 当$x_1+x_2=x_1'+x_2'$, 且$|x_1'-x_2'|<|x_1-x_2|$时, 不等式$f(x_1)+f(x_2)0$;\\\\\n(2) 若关于$x$的方程$f(x)-\\log _2[(a-4) x+2 a-5]=0$的解集中恰好有一个元素, 求$a$的取值范围;\\\\\n(3) 设$a>0$, 若对任意$t \\in[\\dfrac{1}{2},1]$, 函数$f(x)$在区间$[t,t+1]$上的最大值与最小值的差不超过$1$, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407272,7 +408172,9 @@ "id": "015802", "content": "若函数$f(x)=\\log _a[1-(2 a-1) x]$在区间$[2,4]$上为增函数, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407292,7 +408194,9 @@ "id": "015803", "content": "函数$f(x)=\\sqrt{\\log _{\\frac{1}{2}}(x-1)}$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407312,7 +408216,9 @@ "id": "015804", "content": "函数$f(x)=\\begin{cases}\\log _{\\frac{1}{2}} x,& x \\geq 1,\\\\ 2^x,& x<1\\end{cases}$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407332,7 +408238,9 @@ "id": "015805", "content": "已知函数$f(x)$的图像与$g(x)=(\\dfrac{1}{4})^x$的图像关于直线$y=x$对称, 则$f(2 x-x^2)$的严格递减区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407352,7 +408260,9 @@ "id": "015806", "content": "函数$f(x)=|\\log _3 x|$在区间$[a,b]$上的值域为$[0,1]$, 则$b-a$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407372,7 +408282,9 @@ "id": "015807", "content": "函数$f(x)=|\\log _{\\frac{1}{2}} x|$的严格递增区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407392,7 +408304,9 @@ "id": "015808", "content": "若函数$f(x)=\\begin{cases}x^2-2,& x \\leq 1,\\\\ \\lg |x-m|,& x>1\\end{cases}$在区间$[0,+\\infty)$上严格递增, 则实数$m$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407412,7 +408326,9 @@ "id": "015809", "content": "下列函数中, 其定义域和值域分别与函数$y=10^{\\lg x}$的定义域和值域相同的是\\bracket{20}.\n\\fourch{$y=x$}{$y=\\lg x$}{$y=2^x$}{$y=\\dfrac{1}{\\sqrt{x}}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -407432,7 +408348,9 @@ "id": "015810", "content": "若点$(a,b)$($a \\neq 1$)在$y=\\lg x$图像上, 则下列各点中也在此图像上的是\\bracket{20}.\n\\fourch{$(\\dfrac{1}{a},b)$}{$(10 a,1-b)$}{$(\\dfrac{10}{a},b+1)$}{$(a^2,2 b)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -407452,7 +408370,9 @@ "id": "015811", "content": "若函数$f(x)=(k-1) a^x-a^{-x}$($a>0$, $a \\neq 1$)在$\\mathbf{R}$上既是奇函数, 又是减函数, 则$g(x)=\\log _a(x+k)$的图像是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2.5) -- (-2,2.5);\n\\draw (-2,0) node [fill = white, below] {$-2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.8,\\x)-2},-\\x);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (-2,-2.5) -- (-2,2.5);\n\\draw (-2,0) node [fill = white, below] {$-2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.8,\\x)-2},\\x);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-0.5,0) -- (4.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (2,-2.5) -- (2,2.5);\n\\draw (2,0) node [fill = white, below] {$2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.4,\\x)+2},-\\x);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-0.5,0) -- (4.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (2,-2.5) -- (2,2.5);\n\\draw (2,0) node [fill = white, below] {$2$};\n\\draw [domain = -2.5:2.5, samples = 100] plot ({pow(1.4,\\x)+2},\\x);\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -407472,7 +408392,9 @@ "id": "015812", "content": "已知函数$f(x)=\\dfrac{1}{1-x}+\\lg \\dfrac{1+x}{1-x}$.\\\\\n(1) 求函数$f(x)$的定义域, 并判断它的单调性;\\\\\n(2) 解关于$x$的不等式$f[x(x+1)]>1$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407492,7 +408414,9 @@ "id": "015813", "content": "已知$a>0$且$a \\neq 1$, 求使方程$\\log _a(x-a k)=\\log _{a^2}(x^2-a^2)$有解的$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407512,7 +408436,9 @@ "id": "015814", "content": "已知$f(x)=\\log _2 \\dfrac{1+x}{1-x}$.\\\\\n(1) 解方程$f(x)=1$;\\\\\n(2) 设$x \\in(-1,1)$, $a>1$, 证明: $\\dfrac{a x-1}{a-x} \\in(-1,1)$且$f(\\dfrac{a x-1}{a-x})-f(x)=-f(\\dfrac{1}{a})$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407532,7 +408458,9 @@ "id": "015815", "content": "如果质点$A$按照规律$s=3 t^2$运动, 则在$t=3$时的瞬时速度为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407552,7 +408480,9 @@ "id": "015816", "content": "设函数$f(x)=a x+3$, 若$f'(1)=3$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407572,7 +408502,9 @@ "id": "015817", "content": "曲线$y=x^3+x+1$在点$(1,3)$处的切线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407592,7 +408524,9 @@ "id": "015818", "content": "已知函数$f(x)=m x^{m-n}$的导数为$f'(x)=8 x^3$, 则$m^n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407612,7 +408546,9 @@ "id": "015819", "content": "下列求导运算中正确的是\\bracket{20}.\n\\twoch{$(x+\\dfrac{1}{x})'=1+\\dfrac{1}{x^2}$}{$(\\lg x)'=\\dfrac{1}{x \\ln 10}$}{$(\\ln x)'=x$}{$(x^2 \\cos x)'=-2 x \\sin x$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -407632,7 +408568,9 @@ "id": "015820", "content": "设$f(x)=x^2$.\\\\\n(1) 函数$y=f(x)$从$x=1$到$x=1+\\Delta x$的平均变化率, 其中$\\Delta x$的值为: \\textcircled{1} $2$; \\textcircled{2} $1$; \\textcircled{3} $0.1$; \\textcircled{4} $0.01$;\\\\\n(2) 当$|\\Delta x|$越来越小时, 函数$f(x)$在区间$[1,1+\\Delta x]$上的平均变化率有怎样的变化趋势?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407652,7 +408590,9 @@ "id": "015821", "content": "求下列函数的导数:\\\\\n(1) $y=\\dfrac{1}{x}+\\dfrac{2}{x^2}+\\dfrac{3}{x^3}$;\\\\\n(2) $y=3^x-\\lg x$;\\\\\n(3) $y=x^2 \\sin x$;\\\\\n(4) $y=\\dfrac{\\sin x}{x}$;\\\\\n(5) $y=\\dfrac{x-1}{x+1}$;\\\\\n(6) $y=\\mathrm{e}^{-0.05 x+1}$;\\\\\n(7) $y=\\sin (\\pi x+\\varphi)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407672,7 +408612,9 @@ "id": "015822", "content": "已知直线$x-2 y-4=0$与抛物线$y^2=x$相交于$A$、$B$两点, $O$是坐标原点, 试在抛物线的$AB$弧上求一点$P$, 使$\\triangle ABP$的面积最大.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407692,7 +408634,9 @@ "id": "015823", "content": "已知曲线$C: y=3 x^4-2 x^3-9 x^2+4$.\\\\\n(1) 求曲线$C$在点$(1,-4)$的切线方程;\\\\\n(2) 对于(1)中的切线与曲线$C$是否还有其他公共点? 若有, 求出公共点; 若没有, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407712,7 +408656,9 @@ "id": "015824", "content": "已知函数$f(x)=2 x^3-\\dfrac{1}{2} x^2+m$($m$为常数) 图像上$A$处的切线与$x-y+3=0$的夹角为$\\dfrac{\\pi}{4}$, 则$A$点的横坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407732,7 +408678,9 @@ "id": "015825", "content": "已知函数$f(x)=-\\dfrac{2}{3} x^3+a x^2+4 x$, 又$y=f'(x)$的图像与$x$轴交于$(-k,0)$, $(2 k,0)$, $k>0$.\\\\\n(1) 求$a$的值;\\\\\n(2) 求过点$(0,0)$的曲线$y=f(x)$的切线方程.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407752,7 +408700,9 @@ "id": "015826", "content": "甲乙两厂经过治理, 污水的排放量$W$与时间$t$的关系如图所示, 试指出哪一个厂治污效果较好?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (3,0) node [below] {$t$};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$W$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,1.5) .. controls ++(1,0) and ++(-1,0) ..(2.5,1);\n\\path [name path = W1, draw] (0,2) .. controls ++(1,0) and ++(-1,0) ..(2.5,1);\n\\draw (0,2) node [left] {甲厂: $W_1(t)$};\n\\draw (0,1.5) node [left] {乙厂: $W_2(t)$};\n\\path [name path = l1] (1.5,0) -- (1.5,2);\n\\path [name intersections = {of = l1 and W1, by = T}];\n\\draw [dashed] (T)--(1.5,0) node [below] {$t_0-\\Delta t$} (2.5,1) -- (2.5,0) node [below] {$t_0$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407772,7 +408722,9 @@ "id": "015827", "content": "函数$y=(x-1)(x+1)^2$的导数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407792,7 +408744,9 @@ "id": "015828", "content": "若对任意$x \\in \\mathbf{R}$, $f'(x)=4 x^3$, $f(1)=-1$, 则$f(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407812,7 +408766,9 @@ "id": "015829", "content": "直线$y=3 x+1$是曲线$y=x^3-a$的一条切线, 则实数$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407832,7 +408788,9 @@ "id": "015830", "content": "已知$f(x)=\\dfrac{1}{3} x^3+3 x f'(0)$, 则$f'(1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407852,7 +408810,9 @@ "id": "015831", "content": "若曲线$f(x)=\\sqrt{x}$在点$(a,\\sqrt{a})$处的切线与两个坐标轴围成的三角形的面积为$2$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407872,7 +408832,9 @@ "id": "015832", "content": "$P$在曲线$y=x^3-x+\\dfrac{2}{3}$上移动, 在点$P$处的切线的倾斜角为$\\alpha$, 则$\\alpha$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -407892,7 +408854,9 @@ "id": "015833", "content": "曲线$y=x \\mathrm{e}^{x-1}$在点$(1,1)$处切线的斜率等于\\bracket{20}.\n\\fourch{$2 e$}{$e$}{$2$}{$1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -407912,7 +408876,9 @@ "id": "015834", "content": "曲线$y=x^3-x+3$在点$P$处的切线平行于直线$y=2 x-1$, 则$P$点的坐标为\\bracket{20}.\n\\fourch{$(1,3)$}{$(-1,3)$}{$(1,3)$和$(-1,3)$}{$(1,-3)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -407932,7 +408898,9 @@ "id": "015835", "content": "过点$(1,-1)$且与曲线$y=x^3-2 x$相切的切线方程为\\bracket{20}.\n\\fourch{$x-y-2=0$或$5 x+4 y-1=0$}{$x-y-2=0$}{$x-y+2=0$}{$x-y-2=0$或$4 x+5 y+1=0$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -407952,7 +408920,9 @@ "id": "015836", "content": "求下列函数的导数:\\\\\n(1) $y=\\mathrm{e}^x \\ln x$;\\\\\n(2) $y=\\dfrac{1+\\sin x}{1-\\cos x}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407972,7 +408942,9 @@ "id": "015837", "content": "已知曲线$y=5 \\sqrt{x}$, 求:\\\\\n(1) 曲线上与直线$y=2 x-4$平行的切线方程;\\\\\n(2) 求过点$P(0,5)$, 且与曲线相切的切线方程.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -407992,7 +408964,9 @@ "id": "015838", "content": "在曲线$y=x^2$上过哪一点的切线;\\\\\n(1) 平行于直线$y=4 x-5$;\\\\\n(2) 垂直于直线$2 x-6 y+5=0$;\\\\\n(3) 与$x$轴成$135{^ \\circ}$的倾斜角.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408012,7 +408986,9 @@ "id": "015839", "content": "函数$f(x)=a x^3+x$在$(-\\infty,+\\infty)$内是增函数, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408032,7 +409008,9 @@ "id": "015840", "content": "函数$y=x^3-3 x$在$[-\\dfrac{3}{2}, 0]$的最大值是\\blank{50}, 最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408052,7 +409030,9 @@ "id": "015841", "content": "函数$y=x-2 \\sin x$在$(0, \\pi)$上的严格递增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408072,7 +409052,9 @@ "id": "015842", "content": "$f(x)=x \\ln x$在$(0,5)$上是\\bracket{20}.\n\\onech{严格递增函数}{严格递减函数}{在$(0, \\dfrac{1}{\\mathrm{e}})$上严格递减, 在$(\\dfrac{1}{\\mathrm{e}}, 5)$上是严格递增}{在$(0, \\dfrac{1}{\\mathrm{e}})$上是递增函数, 在$(\\dfrac{1}{\\mathrm{e}}, 5)$上是递减函数}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -408092,7 +409074,9 @@ "id": "015843", "content": "$f'(x)$是函数$y=f(x)$的导数, 若$y=f'(x)$的图像如图所示, 则函数$y=f(x)$的图像可能是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:2] plot (\\x,{1.5*\\x*(\\x-1)});\n\\draw (1,0) node [below right] {$x_1$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.35:2] plot (\\x,{\\x*(\\x-1)*(\\x-1.5)});\n\\draw (1,0) node [below] {$x_1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -0.35:2] plot (\\x,{-\\x*(\\x-1)*(\\x-1.5)});\n\\draw (1,0) node [below] {$x_1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:2] plot (\\x,{pow(\\x,2)*(2*\\x-3)/4+0.5});\n\\draw [dashed] (1,0.25) -- (1,0) node [below] {$x_1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:2] plot (\\x,{-pow(\\x,2)*(2*\\x-3)/4+0.5});\n\\draw [dashed] (1,0.75) -- (1,0) node [below] {$x_1$};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -408112,7 +409096,9 @@ "id": "015844", "content": "求下列函数的单调区间.\\\\\n(1) $f(x)=x^2-\\ln x$;\\\\\n(2) $f(x)=\\dfrac{\\mathrm{e}^x}{x-2}$;\\\\\n(3) $f(x)=-x^3+3 x^2$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408132,7 +409118,9 @@ "id": "015845", "content": "已知函数$f(x)=a x^3+x^2$($a \\in \\mathbf{R}$)在$x=-\\dfrac{4}{3}$处取得极值.\\\\\n(1) 确定$a$的值;\\\\\n(2) 若$g(x)=f(x) \\mathrm{e}^x$, 讨论函数的单调性.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408152,7 +409140,9 @@ "id": "015846", "content": "已知函数$f(x)=-\\dfrac{1}{3} x^3+x^2+3 x+a$.\\\\\n(1) 求$f(x)$的严格递减区间;\\\\\n(2) 若$f(x)$在区间$[-3,4]$上的最小值为$\\dfrac{7}{3}$, 求$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408172,7 +409162,9 @@ "id": "015847", "content": "设函数$f(x)=-x^3-2 m x^2-m^2 x+1-m$(其中$m>-2$) 的图像在$x=2$处的切线与直线$f(x)=-5 x+12$平行.\\\\\n(1) 求$m$的值;\\\\\n(2) 求函数$f(x)$在区间$[0,1]$的最小值;\\\\\n(3) 若$a \\geq 0$, $b \\geq 0$, $c \\geq 0$, 且$a+b+c=1$, 试根据上述(1)、 (2)的结论证明: $\\dfrac{a}{1+a^2}+$$\\dfrac{b}{1+b^2}+\\dfrac{c}{1+c^2} \\leq \\dfrac{9}{10}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408192,7 +409184,9 @@ "id": "015848", "content": "已知函数$f(x)=(a x^2+x+2) \\mathrm{e}^x$($a>0$); 其中$\\mathrm{e}$是自然对数的底数.\n(1) 当$a=2$时, 求$f(x)$的极值;\\\\\n(2) 若$f(x)$在$[-2,2]$上是严格递增函数, 求$a$的取值范围;\\\\\n(3) 当$a=1$时, 求整数$t$的所有值, 使方程$f(x)=x+4$在$[t, t+1]$上有解.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408212,7 +409206,9 @@ "id": "015849", "content": "若函数$f(x)=a(x^3-x)$的递减区间为$(-\\dfrac{\\sqrt{3}}{3}, \\dfrac{\\sqrt{3}}{3})$, 则$a$的范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408232,7 +409228,9 @@ "id": "015850", "content": "函数$f(x)=a x^3+x$在$(-\\infty,+\\infty)$内是增函数, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408252,7 +409250,9 @@ "id": "015851", "content": "函数$y=2 x^3-3 x^2-12 x+5$在$[0,3]$上的最大值、最小值分别是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408272,7 +409272,9 @@ "id": "015852", "content": "已知函数$f(x)=x^3+a x^2+(a+6) x+1$有极大值和极小值, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408292,7 +409294,9 @@ "id": "015853", "content": "函数$y=x \\mathrm{e}^x$在其极值点处的切线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408312,7 +409316,9 @@ "id": "015854", "content": "用总长$14.8$米的钢条制作一个长方体容器的框架, 如果所制作容器的底面的一边比另一边长$0.5$米, 那么高为\\blank{50}米时, 容器的容积最大? 其最大容积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408332,7 +409338,9 @@ "id": "015855", "content": "函数$f(x)=\\ln x-a x$($a>0$)的严格递增区间为\\bracket{20}.\n\\fourch{$(0, \\dfrac{1}{a})$}{$(\\dfrac{1}{a},+\\infty)$}{$(0,+\\infty)$}{$(0, a)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -408352,7 +409360,9 @@ "id": "015856", "content": "若函数$f(x)=x+\\dfrac{4 m}{x}-m \\ln x$在$[1,2]$上为减函数, 则$m$的最小值为\\bracket{20}.\n\\fourch{$\\dfrac{3}{2}$}{$\\dfrac{3}{4}$}{$\\dfrac{2}{3}$}{$\\dfrac{4}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -408372,7 +409382,9 @@ "id": "015857", "content": "设函数$f(x)$在定义域内可导, $y=f(x)$的图像如图所示, 则导数$y=f'(x)$的图像可能为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:-0.2] plot (\\x,{0.4*\\x-0.3/\\x});\n\\draw [domain = 0.3:1.6] plot (\\x,{8*(\\x-0.5)*(\\x-1)*(\\x-1.3)});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:-0.4] plot (\\x,{-0.4+0.3/pow(\\x,2)});\n\\draw [domain = 0.3:1.6] plot (\\x,{8*(\\x-0.5)*(\\x-1)*(\\x-1.3)+0.1185});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:-0.5] plot (\\x,{0.4+0.3/pow(\\x,2)});\n\\draw [domain = 0.3:1.6] plot (\\x,{8*(\\x-0.5)*(\\x-1)*(\\x-1.3)+0.1185});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:-0.4] plot (\\x,{-0.4+0.3/pow(\\x,2)});\n\\draw [domain = 0.57:1.3] plot (\\x,{24*pow(\\x,2)-44.8*\\x+19.6});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:-0.5] plot (\\x,{0.4+0.3/pow(\\x,2)});\n\\draw [domain = 0.57:1.3] plot (\\x,{24*pow(\\x,2)-44.8*\\x+19.6});\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -408392,7 +409404,9 @@ "id": "015858", "content": "求下列函数的最值.\\\\\n(1) $f(x)=3 x-x^3$, $x \\in [-\\sqrt{3},3]$;\\\\\n(2) $f(x)=\\ln (1+x)-\\dfrac{1}{4} x^2$, $x \\in[0,2]$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408412,7 +409426,9 @@ "id": "015859", "content": "已知$a$为实数, $f(x)=(x^2-4) \\cdot(x-a)$.\\\\\n(1) 求导数$f'(x)$;\\\\\n(2) 若$f'(-1)=0$, 求$f(x)$在$[-2,2]$上的最大值和最小值;\\\\\n(3) 若$f(x)$在$(-\\infty,-2]$和$[2,+\\infty)$上都是递增的, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408432,7 +409448,9 @@ "id": "015860", "content": "已知函数$f(x)=\\dfrac{a}{x}-\\ln x$.\\\\\n(1) 若$f(x)$在$x=1$处取得极值, 求实数$a$的值;\\\\\n(2) 若$f(x) \\geq 5-3 x$. 恒成立, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408452,7 +409470,9 @@ "id": "015861", "content": "$625$的四次方根为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408472,7 +409492,9 @@ "id": "015862", "content": "函数$y=2-2^x$的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408492,7 +409514,9 @@ "id": "015863", "content": "函数$y=\\lg (16-2^{-x})$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408512,7 +409536,9 @@ "id": "015864", "content": "函数$y=a^x$在$[0,1]$上的最大值为$3$, 则$a$为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408532,7 +409558,9 @@ "id": "015865", "content": "已知$a \\in\\{-\\dfrac{2}{3},-\\dfrac{1}{3}-\\dfrac{1}{2}, \\dfrac{1}{2}, \\dfrac{1}{3}, \\dfrac{2}{3}\\}$, 若幂函数$f(x)=x^a$在区间$(0,+\\infty)$上是严格增函数, 且图像关于原点成中心对称, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408552,7 +409580,9 @@ "id": "015866", "content": "函数$f(x)=\\log _a x$($a>0$, $a \\neq 1$), 若$f(x_1)-f(x_2)=2$, 则$f(x_1^3)=f(x_2^3)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408572,7 +409602,9 @@ "id": "015867", "content": "已知函数$f(x)=|x+a|-|x-a|$($a \\neq 0$), 则$f(x)$的奇偶性为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408592,7 +409624,9 @@ "id": "015868", "content": "已知函数$f(x)=\\lg x$, $x_1, x_2 \\in(0,+\\infty)$($x_1 \\neq x_2$), 有如下结论:\n\\textcircled{1} $f(x_1+x_2)=f(x_1) f(x_2)$; \\textcircled{2} $f(x_1 x_2)=f(x_1)+f(x_2)$; \\textcircled{3} $\\dfrac{f(x_1)-f(x_2)}{x_1-x_2}>0$;\n\\textcircled{4} $f(\\dfrac{x_1+x_2}{2})<\\dfrac{f(x_1)+f(x_2)}{2}$. 其中正确结论的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408612,7 +409646,9 @@ "id": "015869", "content": "已知$f(x)$是奇函数, $g(x)$是偶函数, 且$f(-1)+g(1)=2$, $f(1)+g(-1)=4$, 则$g(1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408632,7 +409668,9 @@ "id": "015870", "content": "已知函数$f(x)$是$(-\\infty,+\\infty)$上的偶函数, 若对于$x \\geq 0$, 都有$f(x+2)=f(x)$, 且当$x \\in$$[0,2)$时, $f(x)=\\log _2(x+1)$, 则$f(2021)+f(-2022)$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408652,7 +409690,9 @@ "id": "015871", "content": "已知$a \\in \\mathbf{R}$, 函数$f(x)=\\begin{cases}x^2-4, & x>2, \\\\ |x-3|+a, & x \\leq 2 .\\end{cases}$若$f[f(\\sqrt{6})]=3$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408672,7 +409712,9 @@ "id": "015872", "content": "已知函数$f(x)=|\\lg x|-k x-2$, 给出下列四个结论: \\textcircled{1} 若$k=0$, 则$f(x)$有两个零点; \\textcircled{2} 存在$k<0$, 使得$f(x)$有一个零点; \\textcircled{3} 存在$k<0$, 使得$f(x)$有三个零点; \\textcircled{4} 存在$k>0$, 使得$f(x)$有三个零点. 以上结论中正确的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408692,7 +409734,9 @@ "id": "015873", "content": "设$y=f(x)$和$y=g(x)$是两个不同的幂函数, 集合$M=\\{(x, y) | f(x)=g(x)\\}$, 则集合$M$中元素的个数为\\bracket{20}.\n\\fourch{$1$或$2$或$0$}{$1$或$2$或$3$}{$1$或$2$或$3$或$4$}{$0$或$1$或$2$或$3$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -408712,7 +409756,9 @@ "id": "015874", "content": "若$f(x)=\\begin{cases}x^2, & x \\geq 0, \\\\ x, & x<0,\\end{cases} g(x)=\\begin{cases}x, & x \\geq 0, \\\\ -x^2, & x<0,\\end{cases}$则当$x<0$时, $f[g(x)]=$\\bracket{20}.\n\\fourch{$-x$}{$-x^2$}{$x$}{$x^2$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -408732,7 +409778,9 @@ "id": "015875", "content": "函数$y=|2^x-2|$\\bracket{20}.\n\\onech{在$(-\\infty,+\\infty)$上严格递增}{在$(-\\infty, 1]$上是减函数, 在$[1,+\\infty)$上是增函数}{在$(-\\infty, 1]$上是增函数, 在$[1,+\\infty)$上是减函数}{在$(-\\infty, 0]$上是减函数, 在$[0,+\\infty)$上是增函数}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -408752,7 +409800,9 @@ "id": "015876", "content": "已知函数$f(x)=x^2+\\dfrac{1}{4}$, $g(x)=\\sin x$, 则图像为下图的函数可能是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.2) -- (0,1.2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:2, samples = 100] plot (\\x,{sin(\\x/pi*180)/(pow(\\x,2)+1/4)});\n\\draw ({-pi/4},0.1) -- ({-pi/4},0) node [below] {$-\\dfrac{\\pi}{4}$};\n\\draw ({pi/4},0.1) -- ({pi/4},0) node [below] {$\\dfrac{\\pi}{4}$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$y=f(x)+g(x)-\\dfrac{1}{4}$}{$y=f(x)-g(x)-\\dfrac{1}{4}$}{$y=f(x) g(x)$}{$y=\\dfrac{g(x)}{f(x)}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -408772,7 +409822,9 @@ "id": "015877", "content": "已知函数$y=3^x-\\dfrac{9}{3^{|x|}}$.\\\\\n(1) 若$y>8$, 求$x$的取值范围;\\\\\n(2) 若$3^x+m y \\geq 0$对于$x \\in[2,3]$恒成立, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408792,7 +409844,9 @@ "id": "015878", "content": "上海某化学试剂厂以$x$千克/小时的速度生产某种产品(生产条件要求$1 \\leq x \\leq 10)$, 为了保证产品的质量, 需要一边生产一边运输, 这样按照目前的市场价格, 每小时可获得利润是$100(5 x+1-\\dfrac{3}{x})$元.\\\\\n(1) 要使生产运输该产品$2$小时获得的利润不低于$3000$元, 求$x$的取值范围;\\\\\n(2) 要使生产运输$900$千克该产品获得的利润最大, 问: 该工厂应该选取何种生产速度? 并求最大利润.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408812,7 +409866,9 @@ "id": "015879", "content": "设函数$y=\\dfrac{a x+1}{x-2}$.\\\\\n(1) 当$a=1$时, 在区间$[-2,2) \\cup(2,6]$上画出这个函数的图像;\\\\\n(2) 是否存在整数$a$, 使该函数在$[4,+\\infty)$上是严格减函数, 且当$x \\geq 4$时, $y \\leq 4$, 如果存在, 求出所有符合条件的$a$; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408832,7 +409888,9 @@ "id": "015880", "content": "已知$a>0$且$a \\neq 1$, 函数$f(x)=\\log _a(x+1)$, $g(x)=\\log _a \\dfrac{1}{1-x}$, 记$F(x)=2 f(x)+g(x)$.\\\\\n(1) 求函数$F(x)$的定义域$D$及其零点;\\\\\n(2) 若关于$x$的方程$F(x)-m=0$在区间$[0,1)$内仅有一解, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408852,7 +409910,9 @@ "id": "015881", "content": "已知函数$f(x)=x+\\dfrac{a}{x}+b$($x \\neq 0$), 其中$a$、$b$为实常数.\\\\\n(1) 若方程$f(x)=3 x+1$有且仅有一个实数解$x=2$, 求$a$、$b$的值;\\\\\n(2) 设$a>0, x \\in(0,+\\infty)$, 写出$f(x)$的单调区间, 并对严格递增区间用函数单调性定义进行证明;\\\\\n(3) 若对任意的$a \\in[\\dfrac{1}{2}, 2]$, 不等式$f(x) \\leq 10$在$x \\in[\\dfrac{1}{4}, 1]$上恒成立, 求实数$b$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408872,7 +409932,9 @@ "id": "015882", "content": "若$\\alpha=\\dfrac{62 \\pi}{3}$, 则$\\alpha$所在的象限是\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -408892,7 +409954,9 @@ "id": "015883", "content": "已知$\\alpha$为第三象限角, 则$\\dfrac{\\alpha}{2}$所在的象限是\\bracket{20}.\n\\fourch{第一或第二象限}{第二或第三象限}{第一或第三象限}{第二或第四象限}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -408912,7 +409976,9 @@ "id": "015884", "content": "若$\\alpha$、$\\beta$的终边互为反向延长线, 则\\bracket{20}.\n\\twoch{$\\alpha+\\beta=180{^ \\circ}$}{$\\alpha=\\beta+k \\cdot 360{^ \\circ}, k \\in \\mathbf{Z}$}{$\\alpha=\\beta+k \\cdot 180{^ \\circ}, k \\in \\mathbf{Z}$}{$\\alpha=\\beta+(2 k+1) \\cdot 180{^ \\circ}, k \\in \\mathbf{Z}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -408932,7 +409998,9 @@ "id": "015885", "content": "已知扇形的圆心角$\\alpha=120{^\\circ}$, 半径为$4$, 则该扇形的圆心角所对的弧长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408952,7 +410020,9 @@ "id": "015886", "content": "已知角$\\alpha$的终边经过点$(-4,3)$, 则$\\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -408972,7 +410042,9 @@ "id": "015887", "content": "已知一个扇形$OAB$的面积是$4 \\text{cm}^2$, 它的周长是$10 \\text{cm}$, 求它的中心角和弦$AB$的长.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -408992,7 +410064,9 @@ "id": "015888", "content": "已知一个扇形的周长为$2 a$($a>0$), 问这个扇形半径为何值时, 才能使这个扇形面积最大? 最大面积为多少? 并求此时扇形的中心角.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409012,7 +410086,9 @@ "id": "015889", "content": "已知$y=\\dfrac{\\sin \\alpha}{|\\sin \\alpha|}+\\dfrac{\\cos \\alpha}{|\\cos \\alpha|}+\\dfrac{\\tan \\alpha}{|\\tan \\alpha|}+\\dfrac{\\cot \\alpha}{|\\cot \\alpha|}$, 其中$\\alpha \\neq \\dfrac{k \\pi}{2}$($k \\in \\mathbf{Z}$), 请写出$y$所有可能的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409032,7 +410108,9 @@ "id": "015890", "content": "已知角$\\alpha$上一点$P(-\\sqrt{3}, y)$且$\\sin \\alpha=\\dfrac{\\sqrt{2}}{4} y$, 求$\\cos \\alpha$、$\\tan \\alpha$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409052,7 +410130,9 @@ "id": "015891", "content": "已知角$\\alpha$的终边经过点$P(3 m,-4 m)$($m<0$), 求角$\\alpha$的正弦、正切值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409072,7 +410152,9 @@ "id": "015892", "content": "已知角$\\alpha$的终边在直线$y=-3 x$上, 求$\\sin \\alpha$、$\\cos \\alpha$、$\\tan \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409092,7 +410174,9 @@ "id": "015893", "content": "如图所示, $A$、$B$是单位圆$O$上的点, 且$B$在第二象限, $C$是圆与$x$轴正半轴的交点, $A$点的坐标为$(\\dfrac{3}{5}, \\dfrac{4}{5})$, $\\triangle AOB$为正三角形.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0.6,0.8) node [right] {$A(\\dfrac 35,\\dfrac 45)$} coordinate (A);\n\\draw ($(O)!1!60:(A)$) node [above left] {$B$} coordinate (B);\n\\draw (O)--(A)--(B)--cycle;\n\\draw (0,0) circle (1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\sin \\angle COA$;\\\\\n(2) 求$\\cos \\angle COB$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409112,7 +410196,9 @@ "id": "015894", "content": "已知相互啮合的两个齿轮, 大轮有$48$齿, 小轮有$20$齿.\\\\\n(1) 当大轮转动一周时, 求小轮转动的角的大小;\\\\\n(2) 如果大轮的转速为$180$转/分, 小轮的半径为$10.5$厘米, 那么小轮的圆周上一点每$1$秒转过的弧长是多少?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409132,7 +410218,9 @@ "id": "015895", "content": "若一扇形的圆心角为$72{^ \\circ}$, 半径为$20 \\text{cm}$, 则扇形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409152,7 +410240,9 @@ "id": "015896", "content": "已知角$\\alpha$的终边经过点$P(3 m,-4 m)$, 则$\\tan \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409172,7 +410262,9 @@ "id": "015897", "content": "设$\\alpha$是三角形的一个内角, 在$\\sin \\alpha, \\cos \\alpha, \\tan \\alpha, \\cot \\dfrac{\\alpha}{2}$中, 有可能取负值的是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409192,7 +410284,9 @@ "id": "015898", "content": "已知角$\\theta$的顶点为坐标原点, 始边为$x$轴的正半轴. 若$P(4, y)$是角$\\theta$终边上一点, 且$\\sin \\theta=$$-\\dfrac{2 \\sqrt{5}}{5}$, 则$y=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409212,7 +410306,9 @@ "id": "015899", "content": "点$P$从$(1,0)$出发, 沿单位圆逆时针方向运动$\\dfrac{2 \\pi}{3}$弧长到达$Q$点, 则$Q$点的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409232,7 +410328,9 @@ "id": "015900", "content": "攒尖是古代中国建筑中屋顶的一种结构形式. 宋代称为撮尖, 清代称攒尖, 依其平面有圆形攒尖、三角攒尖、四角攒尖、六角攒尖等, 也有单檐和重檐之分, 多见于亭阁式建筑. 某园林建筑为六角攒尖, 它的主要部分的轮廓可近似看作一个正六棱锥, 若此正六棱锥的侧面等腰三角形的底角为$\\alpha$, 则侧棱与底面外接圆半径的比为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409252,7 +410350,9 @@ "id": "015901", "content": "给出下列命题: \\textcircled{1} 第二象限角大于第一象限角; \\textcircled{2} 三角形的内角是第一象限角或第二象限角; \\textcircled{3}不论用角度制还是用弧度制度量一个角, 它们与扇形所对半径的大小无关; \\textcircled{4}若$\\sin \\alpha=$$\\sin \\beta$, 则$\\alpha$、$\\beta$的终边相同; \\textcircled{5} 若$\\cos \\theta<0$, 则$\\theta$是第二或第三象限的角. 其中正确命题的个数是\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -409272,7 +410372,9 @@ "id": "015902", "content": "已知$\\cos \\theta=\\dfrac{3}{5}$, 且角$\\theta$在第一象限, 那么$2 \\theta$是\\bracket{20}.\n\\fourch{第一象限角}{第二象限角}{第三象限角}{第四象限角}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -409292,7 +410394,9 @@ "id": "015903", "content": "在直角$\\triangle POB$中(如图所示), $\\angle PBO=90{^ \\circ}$, 以$O$为圆心, $OB$为半径作圆弧交$OP$于点$A$. 若弧$AB$等分$\\triangle POB$的面积, 且$\\angle AOB=\\alpha$弧度, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [right] {$O$} coordinate (O);\n\\draw (-1,0) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (0,{tan(1.17/pi*180)}) node [above] {$P$} coordinate (P);\n\\draw (O)--(B)--(P)--cycle;\n\\draw ({180-1.17/pi*180}:1) node [right] {$A$} coordinate (A);\n\\draw (A) arc ({180-1.17/pi*180}:180:1);\n\\draw pic [draw,\"$\\alpha$\",scale = 0.5,angle eccentricity = 1.5] {angle = A--O--B};\n\\draw pic [draw, scale = 0.5] {right angle = O--B--P};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\tan \\alpha=\\alpha$}{$\\tan \\alpha=2 \\alpha$}{$\\sin \\alpha=2 \\cos \\alpha$}{$2 \\sin \\alpha=\\cos \\alpha$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -409312,7 +410416,9 @@ "id": "015904", "content": "已知角$\\alpha$终边上有一点$P$到原点的距离为$\\sqrt{10}$, 且$\\tan \\alpha=-\\dfrac{1}{3}$, 求点$P$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409332,7 +410438,9 @@ "id": "015905", "content": "一个扇形$OAB$的面积是$1 \\text{cm}^2$, 它的周长是$4 \\text{cm}$, 求圆心角的弧度数和弦长$AB$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409352,7 +410460,9 @@ "id": "015906", "content": "已知一个扇形的周长为$20 \\text{cm}$, 问这个扇形半径为何值时, 才能使这个扇形面积最大? 最大面积为多少? 并求此时扇形的中心角.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409372,7 +410482,9 @@ "id": "015907", "content": "已知一扇形的中心角是$\\alpha$, 所在圆的半径是$R$.\\\\\n(1) 若$\\alpha=60{^\\circ}, R=10 \\text{cm}$求扇形的弧长及该弧所在的弓形面积;\\\\\n(2) 若扇形的周长是一定值$C$($C>0$), 当$\\alpha$为多少弧度时, 该扇形有最大面积?", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409392,7 +410504,9 @@ "id": "015908", "content": "$(\\tan x+\\cot x) \\cos ^2 x=$\\bracket{20}.\n\\fourch{$\\tan x$}{$\\sin x$}{$\\cos x$}{$\\cot x$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -409412,7 +410526,9 @@ "id": "015909", "content": "若$\\sin \\alpha=-\\dfrac{5}{13}$, 且$\\alpha$为第四象限角, 则$\\tan \\alpha$的值等于\\bracket{20}.\n\\fourch{$\\dfrac{12}{5}$}{$-\\dfrac{12}{5}$}{$\\dfrac{5}{12}$}{$-\\dfrac{5}{12}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -409432,7 +410548,9 @@ "id": "015910", "content": "下列等式中恒成立的是\\bracket{20}.\n\\twoch{$\\sin (\\dfrac{\\pi}{2}+\\alpha)=-\\cos \\alpha$}{$\\cos (\\dfrac{\\pi}{2}+\\alpha)=\\sin \\alpha$}{$\\cot (\\dfrac{3 \\pi}{2}-\\alpha)=-\\tan \\alpha$}{$\\tan (3 \\pi-\\alpha)=-\\tan \\alpha$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -409452,7 +410570,9 @@ "id": "015911", "content": "若$\\cos \\alpha=\\dfrac{1}{3}$, 则$\\sin (\\alpha-\\dfrac{\\pi}{2})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409472,7 +410592,9 @@ "id": "015912", "content": "化简$\\cot \\alpha(\\sin \\alpha-\\cos \\alpha)+\\dfrac{\\cos \\alpha+\\cot \\alpha}{\\sec \\alpha+\\tan \\alpha}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409492,7 +410614,9 @@ "id": "015913", "content": "已知$\\cos \\alpha=\\dfrac{4}{5}$, 且$\\alpha$是第四象限角, 求$\\alpha$的正弦、正切的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409512,7 +410636,9 @@ "id": "015914", "content": "已知$\\cos \\alpha=\\dfrac{4}{5}$, 求$\\alpha$的正弦、正切的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409532,7 +410658,9 @@ "id": "015915", "content": "已知$\\cos \\alpha=a$($a<0$), 求$\\alpha$的正弦、正切的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409552,7 +410680,9 @@ "id": "015916", "content": "已知$\\sin \\alpha+2 \\cos \\alpha=0$, 求$\\tan (\\alpha+\\pi)+\\dfrac{\\sin (\\dfrac{5 \\pi}{2}+\\alpha)}{\\cos (\\dfrac{5 \\pi}{2}-\\alpha)}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409572,7 +410702,9 @@ "id": "015917", "content": "已知$\\sin \\alpha=\\dfrac{1}{2}+\\cos \\alpha$, 且$\\alpha \\in(0, \\dfrac{\\pi}{2})$, 求$\\dfrac{1-2 \\sin ^2 \\alpha}{\\sin \\alpha-\\cos \\alpha}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409592,7 +410724,9 @@ "id": "015918", "content": "已知$\\sin x+\\cos x=\\dfrac{1}{5}$, $0 \\leq x \\leq\\pi$, 求$\\tan x$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409612,7 +410746,9 @@ "id": "015919", "content": "在三角形$ABC$中,\\\\\n(1) 求证: $\\cos ^2 \\dfrac{A+B}{2}+\\cos ^2 \\dfrac{C}{2}=1$;\\\\\n(2) 若$\\cos (\\dfrac{\\pi}{2}+A) \\sin (\\dfrac{3 \\pi}{2}+B) \\tan (C-\\pi)<0$, 求证: 三角形$ABC$为钝角三角形.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409632,7 +410768,9 @@ "id": "015920", "content": "证明三角恒等式$\\dfrac{\\tan x}{1+\\tan x}+\\dfrac{\\cot x}{1-\\cot x}=\\dfrac{\\tan x+\\cot x}{\\tan x-\\cot x}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409652,7 +410790,9 @@ "id": "015921", "content": "求$\\cos 1{^ \\circ}+\\cos 2{^ \\circ}+\\cdots+\\cos 179{^ \\circ}$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409672,7 +410812,9 @@ "id": "015922", "content": "已知$\\alpha \\neq \\dfrac{n \\pi}{2}$, $n \\in \\mathbf{Z}$, 化简: $\\dfrac{\\sin (k \\pi-\\alpha)}{\\sin (k \\pi+\\alpha)}+\\dfrac{\\cos (k \\pi-\\alpha)}{\\cos (k \\pi+\\alpha)}+\\dfrac{\\tan (k \\pi-\\alpha)}{\\tan (k \\pi+\\alpha)}$, 其中$k \\in \\mathbf{Z}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -409692,7 +410834,9 @@ "id": "015923", "content": "$\\sin ^4 \\alpha+\\sin ^2 \\alpha \\cos ^2 \\alpha+\\cos ^2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409712,7 +410856,9 @@ "id": "015924", "content": "已知$\\alpha$为第二象限角, 且$\\sin \\alpha=\\dfrac{1}{3}$, 则$\\sin (\\dfrac{3 \\pi}{2}-\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409732,7 +410878,9 @@ "id": "015925", "content": "在$\\triangle ABC$中, 已知$\\cos A=\\dfrac{3}{5}$, 则$\\cos (B+C)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409752,7 +410900,9 @@ "id": "015926", "content": "已知$\\sin \\beta+\\cos \\beta=\\dfrac{3}{5}$, 则$\\sin \\beta \\cos \\beta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409772,7 +410922,9 @@ "id": "015927", "content": "已知$f(x)=a \\sin (\\pi x+\\alpha)+b \\cos (\\pi x+\\beta)$, 其中$a$、$b$、$\\alpha$、$\\beta$都是非零实数, 若$f(2021)=-1$, 则$f(2022)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409792,7 +410944,9 @@ "id": "015928", "content": "已知$\\alpha$是第三象限角, 化简$\\sqrt{\\dfrac{1+\\sin \\alpha}{1-\\sin \\alpha}}-\\sqrt{\\dfrac{1-\\sin \\alpha}{1+\\sin \\alpha}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -409812,7 +410966,9 @@ "id": "015929", "content": "已知角$\\alpha$的终边经过点$P(3,4)$, 将角$\\alpha$的终边绕原点$O$逆时针旋转$\\dfrac{\\pi}{2}$得到角$\\beta$的终边, 则$\\tan \\beta$等于\\bracket{20}.\n\\fourch{$-\\dfrac{4}{3}$}{$-\\dfrac{3}{4}$}{$\\dfrac{4}{5}$}{$-\\dfrac{5}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -409832,7 +410988,9 @@ "id": "015930", "content": "若$\\dfrac{\\pi}{2}=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) circle (1);\n\\draw (1,0) node [below right] {$C$} coordinate (C);\n\\draw (0.6,0.8) node [above right] {$A$} coordinate (A);\n\\draw ({60+atan(4/3)}:1) node [above left] {$B$} coordinate (B);\n\\draw (0,0) -- (A)--(B)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\dfrac{1+\\sin 2 \\alpha}{1+\\cos 2 \\alpha}$的值;\\\\\n(2) 求$|BC|^2$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -410872,7 +412132,9 @@ "id": "015982", "content": "已知$\\tan \\dfrac{\\alpha}{2}=2$, 则$\\tan (\\alpha+\\dfrac{\\pi}{4})$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -410892,7 +412154,9 @@ "id": "015983", "content": "已知$\\sin (x+y)=\\dfrac{1}{2}$, $\\sin (x-y)=\\dfrac{1}{3}$, 则$\\dfrac{\\tan x}{\\tan y}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -410912,7 +412176,9 @@ "id": "015984", "content": "化简$\\dfrac{1+\\cos 2 \\alpha}{\\cot \\dfrac{\\alpha}{2}-\\tan \\dfrac{\\alpha}{2}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -410932,7 +412198,9 @@ "id": "015985", "content": "若$\\cos x \\cos y+\\sin x \\sin y=\\dfrac{1}{2}$, $\\sin 2 x+\\sin 2 y=\\dfrac{2}{3}$, 则$\\sin (x+y)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -410952,7 +412220,9 @@ "id": "015986", "content": "在$\\triangle ABC$中, $\\angle C=60{^ \\circ}$, 则$\\cos A \\cos B$的取值范围为\\bracket{20}.\n\\fourch{$(-\\dfrac{1}{2}, \\dfrac{1}{4}]$}{$[-\\dfrac{3}{4}, \\dfrac{1}{4}]$}{$[0, \\dfrac{1}{4}]$}{以上都不对}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -410972,7 +412242,9 @@ "id": "015987", "content": "已知$3 \\sin \\beta=\\sin (2 \\alpha+\\beta)$, 求证: $\\tan (\\alpha+\\beta)=2 \\tan \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -410992,7 +412264,9 @@ "id": "015988", "content": "已知$\\sin \\beta=m \\sin (2 \\alpha+\\beta), m \\neq \\pm 1$, 求证: $\\tan (\\alpha+\\beta)=\\dfrac{1+m}{1-m} \\tan \\alpha$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411012,7 +412286,9 @@ "id": "015989", "content": "设$\\alpha$、$\\beta$是锐角且$3 \\sin ^2 \\alpha+2 \\sin ^2 \\beta=1,3 \\sin 2 \\alpha-2 \\sin 2 \\beta=0$, 求证: $\\alpha+2 \\beta=\\dfrac{\\pi}{2}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411032,7 +412308,9 @@ "id": "015990", "content": "在$\\triangle ABC$中, 已知三边$a$、$b$、$c$成等差数列, 求证: $\\tan \\dfrac{A}{2} \\tan \\dfrac{C}{2}=\\dfrac{1}{3}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411052,7 +412330,9 @@ "id": "015991", "content": "如图, 在矩形$ABCD$区域内, $D$处有一棵古树, 为保护古树, 以$D$为圆心, $DA$为半径划定圆$D$作为保护区域, 已知$AB=30 \\text{m}, AD=15 \\text{m}$, 点$E$为$AB$上的动点, 点$F$为$CD$上的动点, 满足$EF$与圆$D$相切.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (4,0) node [below] {$B$} coordinate (B);\n\\draw (4,2) node [above] {$C$} coordinate (C);\n\\draw (0,2) node [above] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\draw (A) arc (-90:0:2);\n\\draw (A) ++ ({2*tan(20)},0) node [below] {$E$} coordinate (E) --++ ({2/tan(40)},2) node [above] {$F$} coordinate (F);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\angle ADE=20{^ \\circ}$, 求$EF$的长;\\\\\n(2) 当$AE$多长时, 梯形$FEBC$的面积有最大值, 最大面积为多少? (长度精确到$0.1 \\text{m}$, 面积精确到$0.01 \\text{m}^2$)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411072,7 +412352,9 @@ "id": "015992", "content": "已知$\\sin \\alpha+\\sin \\beta=\\dfrac{1}{2}$, $\\cos \\alpha+\\cos \\beta=\\dfrac{1}{3}$, 求$\\cos (\\alpha-\\beta), \\tan \\dfrac{\\alpha+\\beta}{2}, \\sin (\\alpha+\\beta)$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411092,7 +412374,9 @@ "id": "015993", "content": "已知$\\tan \\dfrac{\\alpha}{2}=3$, 则$\\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411112,7 +412396,9 @@ "id": "015994", "content": "已知$\\dfrac{1+\\cos \\alpha}{\\sin \\alpha}=2$, 则$\\cos \\alpha-\\sin \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411132,7 +412418,9 @@ "id": "015995", "content": "化简$\\dfrac{\\sin (\\theta-\\alpha)+\\sin \\theta+\\sin (\\theta+\\alpha)}{\\cos (\\theta-\\alpha)+\\cos \\theta+\\cos (\\theta+\\alpha)}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411152,7 +412440,9 @@ "id": "015996", "content": "已知$\\tan \\dfrac{x}{2}=t$, 试用$t$表示$\\dfrac{\\cos 2 x}{1+\\sin 2 x}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411172,7 +412462,9 @@ "id": "015997", "content": "已知$\\cos ^2 \\alpha-\\cos ^2 \\beta=m$, 则$\\sin (\\alpha+\\beta) \\sin (\\alpha-\\beta)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411192,7 +412484,9 @@ "id": "015998", "content": "化简$2 \\sin (\\dfrac{\\pi}{4}+\\alpha) \\cdot \\sin (\\dfrac{\\pi}{4}-\\alpha)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411212,7 +412506,9 @@ "id": "015999", "content": "设$\\sin (x+y) \\sin (x-y)=m$, 则$\\cos ^2 x-\\cos ^2 y$的值为\\bracket{20}.\n\\fourch{$m$}{$m^2-1$}{$1-m^2$}{$-m$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -411232,7 +412528,9 @@ "id": "016000", "content": "已知$\\cos (x-\\dfrac{\\pi}{6})=m$, 则$\\cos x+\\cos (x-\\dfrac{\\pi}{3})=$\\bracket{20}.\n\\fourch{$2 m$}{$\\pm 2 m$}{$\\sqrt{3} m$}{$\\pm \\sqrt{3} m$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -411252,7 +412550,9 @@ "id": "016001", "content": "设$A$、$B$、$C \\in(0, \\dfrac{\\pi}{2})$, 且$\\sin A-\\sin C=\\sin B, \\cos A+\\cos C=\\cos B$, 则$B-A$等于\\bracket{20}.\n\\fourch{$-\\dfrac{\\pi}{3}$}{$\\dfrac{\\pi}{3}$}{$\\dfrac{\\pi}{6}$}{$\\dfrac{\\pi}{3}$或$-\\dfrac{\\pi}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -411272,7 +412572,9 @@ "id": "016002", "content": "求证: $\\dfrac{\\cos 2 x+\\cos 2 y}{1+\\cos 2(x+y)}=\\dfrac{\\cos (x-y)}{\\cos (x+y)}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411292,7 +412594,9 @@ "id": "016003", "content": "化简下列各式:\\\\\n(1) $\\sqrt{\\dfrac{1}{2}-\\dfrac{1}{2} \\sqrt{\\dfrac{1}{2}+\\dfrac{1}{2} \\cos 2 \\alpha}}$($\\alpha \\in(\\dfrac{3 \\pi}{2}, 2 \\pi)$);\\\\\n(2) $\\dfrac{\\cos ^2 \\alpha-\\sin ^2 \\alpha}{2 \\cot (\\dfrac{\\pi}{4}+\\alpha) \\cos ^2(\\dfrac{\\pi}{4}-\\alpha)}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411312,7 +412616,9 @@ "id": "016004", "content": "已知$0<\\alpha<\\pi$,\\\\\n(1) 求证: $2 \\sin 2 \\alpha \\leq \\cot \\dfrac{\\alpha}{2}$;\\\\\n(2) 当$\\alpha$为何值时, 等号成立.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411332,7 +412638,9 @@ "id": "016005", "content": "在$\\triangle ABC$中, $\\angle A=60{^ \\circ}$, $AB=5$且$S_{\\triangle}=5 \\sqrt{3}$, 则$BC$的长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411352,7 +412660,9 @@ "id": "016006", "content": "在$\\triangle ABC$中, $\\sin A: \\sin B: \\sin C=2: 3: 4$, 则$\\angle ACB=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411372,7 +412682,9 @@ "id": "016007", "content": "记$\\triangle ABC$的内角$A$、$B$、$C$的对边分别为$a$、$b$、$c$, 面积为$\\sqrt{3}$, $B=60{^ \\circ}$, $a^2+c^2=3 a c$, 则$\\triangle ABC$的外接圆的半径为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411392,7 +412704,9 @@ "id": "016008", "content": "在$\\triangle ABC$中, $\\angle A=\\dfrac{\\pi}{3}$, $AB=2$, $AC=3$, 则$\\triangle ABC$的外接圆的半径为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411412,7 +412726,9 @@ "id": "016009", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$的对边分别为$a$、$b$、$c$. 若$\\triangle ABC$为锐角三角形, 且满足$\\sin B(1+2 \\cos C)=2 \\sin A \\cos C+\\cos A \\sin C$, 则下列等式中成立的是\\bracket{20}.\n\\fourch{$a=2 b$}{$b=2 a$}{$A=2B$}{$B=2A$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -411432,7 +412748,9 @@ "id": "016010", "content": "在$\\triangle ABC$中, 已知$a=2 \\sqrt{3}$, $c=2$, $C=30{^ \\circ}$, 求$b, S_{\\triangle}$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411452,7 +412770,9 @@ "id": "016011", "content": "按下列条件解三角形时, 其中有唯一解的是\\bracket{20}.\n\\twoch{$a=20, b=28, A=40{^ \\circ}$}{$a=18, b=20, A=150{^ \\circ}$}{$b=20, c=34, B=70{^ \\circ}$}{$b=60, c=50, B=45{^ \\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -411472,7 +412792,9 @@ "id": "016012", "content": "已知在$\\triangle ABC$中, $c=2 b \\cos B$, $C=\\dfrac{2 \\pi}{3}$.\\\\\n(1) 求$B$的大小;\\\\\n(2) 在下列三个条件\\textcircled{1} $c=\\sqrt{2} b$; \\textcircled{2} 周长为$4+2 \\sqrt{3}$; \\textcircled{3} 面积为$S_{\\triangle ABC}=\\dfrac{3 \\sqrt{3}}{4}$中选择一个作为已知, 使$\\triangle ABC$存在且唯一确定, 并求出$BC$边上的中线的长度.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411492,7 +412814,9 @@ "id": "016013", "content": "设$\\triangle ABC$的三个内角$A$、$B$、$C$的对边分别为$a$、$b$、$c$, 满足$\\dfrac{a}{\\sqrt{3} \\cos A}=\\dfrac{b}{\\sin B}$.\\\\\n(1) 求角$A$的大小;\\\\\n(2) 若$2 \\sin ^2 \\dfrac{B}{2}+2 \\sin ^2 \\dfrac{C}{2}=1$, 试判断$\\triangle ABC$的形状, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411512,7 +412836,9 @@ "id": "016014", "content": "如图所示, 在一条海防警戒线上的点$A$、$B$、$C$处各有一个水声监测点, $B$、$C$两点到点$A$的距离分别为$20$千米和$50$千米, 某时刻, $B$收到发自静止目标$P$的一个声波信号, $8$秒后$A$、$C$同时接收到该声波信号, 已知声波在水中的传播速度是$1.5$千米/秒.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.1]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (-20,0) node [below] {$A$} coordinate (A);\n\\draw (30,0) node [below] {$C$} coordinate (C);\n\\draw (5,{4*sqrt(21)}) node [above] {$P$} coordinate (P);\n\\draw (A)--(P)--(C) (P)--(B) ($(A)!-0.1!(C)$)--($(C)!-0.1!(A)$);\n\\draw ($(A)!0.5!(B)$) node [below] {$20$};\n\\draw ($(B)!0.5!(C)$) node [below] {$30$};\n\\end{tikzpicture}\n\\end{center}\n(1) 设$A$到$P$的距离为$x$千米, 用$x$表示$B$、$C$到$P$的距离, 并求$x$的值;\\\\\n(2) 求$P$到海防警戒线$AC$的距离(结果精确到$0.01$千米).", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411532,7 +412858,9 @@ "id": "016015", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边分别为$a$、$b$、$c$, 且$\\cos A=\\dfrac{1}{3}$.\\\\\n(1) 求$\\sin ^2 \\dfrac{B+C}{2}+\\cos 2A$的值;\\\\\n(2) 若$a=\\sqrt{3}$, 求$bc$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411552,7 +412880,9 @@ "id": "016016", "content": "在锐角$\\triangle ABC$中, $AC=4$, $BC=3$, 三角形的面积等于$3 \\sqrt{3}$, 则$AB$的长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411572,7 +412902,9 @@ "id": "016017", "content": "在$\\triangle ABC$中, 已知$\\angle A=60{^ \\circ}$, $AB=5$, 面积为$10 \\sqrt{3}$, 则其周长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411592,7 +412924,9 @@ "id": "016018", "content": "$\\triangle ABC$的内角$A$、$B$、$C$的对边分别为$a$、$b$、$c$, 若$2 b \\cos B=a \\cos C+c \\cos A$, 则$B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411612,7 +412946,9 @@ "id": "016019", "content": "在$\\triangle ABC$中, 已知$a=3$, $c=3 \\sqrt{2}$, $\\angle A=30{^ \\circ}$, 则$\\angle C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411632,7 +412968,9 @@ "id": "016020", "content": "在$\\triangle ABC$中, $\\angle A=\\dfrac{\\pi}{3}$, $AB=2$, $AC=3$, 则$\\triangle ABC$的外接圆的半径为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411652,7 +412990,9 @@ "id": "016021", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对应的边分别为$a$、$b$、$c$, 已知$b \\cos C+c \\cos B=2 b$, 则$\\dfrac{a}{b}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411672,7 +413012,9 @@ "id": "016022", "content": "钝角三角形$ABC$的面积是$\\dfrac{1}{2}$, $AB=1$, $BC=\\sqrt{2}$, 则$AC$等于\\bracket{20}.\n\\fourch{$1$}{$2$}{$\\sqrt{5}$}{$5$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -411692,7 +413034,9 @@ "id": "016023", "content": "$\\triangle ABC$中, $a=8$, $B=60{^ \\circ}$, $C=75{^ \\circ}$, 则边$b$的长为\\bracket{20}.\n\\fourch{$4 \\sqrt{2}$}{$4 \\sqrt{6}$}{$4 \\sqrt{3}$}{$4 \\sqrt{7}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -411712,7 +413056,9 @@ "id": "016024", "content": "已知$D$、$C$、$B$三点在地面同一直线上, $DC=a$, $C$、$D$在点$B$的同侧且从$C$、$D$两点测得$A$的仰角分别为$\\alpha$、$\\beta$($\\alpha>\\beta$), 则$A$点离地面的高$AB$等于\\bracket{20}.\n\\fourch{$\\dfrac{a \\sin \\alpha \\sin \\beta}{\\sin (\\alpha-\\beta)}$}{$\\dfrac{a \\sin \\alpha \\sin \\beta}{\\cos (\\alpha-\\beta)}$}{$\\dfrac{a \\cos \\alpha \\cos \\beta}{\\sin (\\alpha-\\beta)}$}{$\\dfrac{a \\cos \\alpha \\cos \\beta}{\\cos (\\alpha-\\beta)}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -411732,7 +413078,9 @@ "id": "016025", "content": "记$\\triangle ABC$是内角$A$、$B$、$C$的对边分别为$a$、$b$、$c$. 已知$b^2=a c$, 点$D$在边$AC$上, $BD \\sin \\angle ABC=a \\sin C$.\\\\\n(1) 证明: $BD=b$;\\\\\n(2) 若$AD=2DC$, 求$\\cos \\angle ABC$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411752,7 +413100,9 @@ "id": "016026", "content": "已知$A$、$B$、$C$为$\\triangle ABC$的三个内角, $a$、$b$、$c$是其三条边, $a=2, \\cos C=-\\dfrac{1}{4}$.\\\\\n(1) 若$\\sin A=2 \\sin B$, 求$b$、$c$;\\\\\n(2) 若$\\cos (A-\\dfrac{\\pi}{4})=\\dfrac{4}{5}$, 求$c$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411772,7 +413122,9 @@ "id": "016027", "content": "某景区欲建造两条圆形观景步道$M_1$、$M_2$(宽度忽略不计), 如图所示, 已知$AB \\perp AC, AB=AC=AD=60$(单位: 米), 要求圆$M_1$与$AB$、$AD$. 分别相切于点$B$、$D$, 圆$M_2$与$AC$、$AD$分别相切于点$C$、$D$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (0,2) node [left] {$C$} coordinate (C);\n\\draw (60:2) node [below right] {$D$} coordinate (D);\n\\draw (C) -- (A) -- (B) (A) -- (D);\n\\filldraw (2,{2/sqrt(3)}) circle (0.02) coordinate (O1);\n\\draw (O1) circle ({2/sqrt(3)});\n\\draw (O1) ++ (30:{2/sqrt(3)}) node [above right] {$M_1$};\n\\filldraw (C) ++ ({2*tan(15)},0) circle (0.02) coordinate (O2);\n\\draw (O2) circle ({2*tan(15)});\n\\draw (O2) ++ (75:{2*tan(15)}) node [above right] {$M_2$};\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\angle BAD=60{^ \\circ}$, 求圆$M_1$、$M_2$的半径(结果精确到$0.1$米);\\\\\n(2) 若观景步道$M_1$与$M_2$的造价分别为每米$0.8$千元与每米$0.9$千元, 如何设计圆$M_1$、$M_2$的大小, 使总造价最低? 最低总造价是多少?(结果精确到$0.1$千元)", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -411792,7 +413144,9 @@ "id": "016028", "content": "$\\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, $\\sin \\alpha=\\dfrac{4}{5}$, 则$\\tan \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411812,7 +413166,9 @@ "id": "016029", "content": "函数$y=\\dfrac{\\sin x}{|\\sin x|}+\\dfrac{|\\cos x|}{\\cos x}+\\dfrac{\\tan x}{|\\tan x|}+\\dfrac{|\\cot x|}{\\cot x}$的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411832,7 +413188,9 @@ "id": "016030", "content": "若$\\cos \\alpha=\\dfrac{1}{3}$($0<\\alpha<\\pi$), 则$\\sin 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411852,7 +413210,9 @@ "id": "016031", "content": "在$\\triangle ABC$中, 已知$\\cos A=-\\dfrac{3}{5}$, 则$\\sin \\dfrac{A}{2}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411872,7 +413232,9 @@ "id": "016032", "content": "已知$\\tan (\\dfrac{\\pi}{4}+\\alpha)=3$, 则$\\sin 2 \\alpha-\\cos 2 \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411892,7 +413254,9 @@ "id": "016033", "content": "已知$\\sin \\alpha=\\dfrac{4}{5}$, $\\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, $\\tan (\\alpha-\\beta)=\\dfrac{1}{2}$, 则$\\tan (\\alpha-2 \\beta)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411912,7 +413276,9 @@ "id": "016034", "content": "在$\\triangle ABC$中, 若$a(2 \\cos ^2 \\dfrac{A}{2}-1)=b \\dfrac{1-\\tan ^2 \\dfrac{B}{2}}{1+\\tan ^2 \\dfrac{B}{2}}$, 则$\\triangle ABC$的形状是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411932,7 +413298,9 @@ "id": "016035", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边的长度分别为$a$、$b$、$c$, 且$a^2+b^2-c^2=\\sqrt{3} a b$, 则$\\angle C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411952,7 +413320,9 @@ "id": "016036", "content": "已知$\\triangle ABC$中, $\\cot A=-\\dfrac{12}{5}$, 则$\\cos A=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411972,7 +413342,9 @@ "id": "016037", "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边为$a$、$b$、$c$, 若$A=\\dfrac{\\pi}{3}$, $b=2 c$, 则$C=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -411992,7 +413364,9 @@ "id": "016038", "content": "若$f(x+2)=\\begin{cases}\\tan x, & x \\geq 0, \\\\ \\log _2(-x), & x<0,\\end{cases}$则$f(\\dfrac{\\pi}{4}+2) \\cdot f(-2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412012,7 +413386,9 @@ "id": "016039", "content": "若$\\dfrac{\\pi}{4}=latex,scale = 1.5]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (0,0) circle (1);\n\\draw (1,0) node [above right] {$A$} coordinate (A);\n\\draw (-0.6,0.8) node [above left] {$B$} coordinate (B);\n\\draw ($(A)+(B)$) node [above right] {$C$} coordinate (C);\n\\draw [->] (O)--(A);\n\\draw [->] (O)--(B);\n\\draw [->] (O)--(C);\n\\draw [dashed] (A)--(C)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 若点$B(-\\dfrac{3}{5}, \\dfrac{4}{5})$, 求$\\tan (2 \\theta+\\dfrac{\\pi}{4})$的值;\\\\\n(2) 若$\\overrightarrow{OA}+\\overrightarrow{OB}=\\overrightarrow{OC}$, 四边形$OACB$的面积用$S_{\\theta}$表示, 求$S_{\\theta}+\\overrightarrow{OA} \\cdot \\overrightarrow{OC}$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412212,7 +413606,9 @@ "id": "016049", "content": "函数$f(x)=\\tan 2 x$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412232,7 +413628,9 @@ "id": "016050", "content": "函数$y=\\cos (\\dfrac{\\pi}{3}-2 x)$的严格递减区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412252,7 +413650,9 @@ "id": "016051", "content": "函数$y=\\sin (x+\\dfrac{\\pi}{2})$是\\bracket{20}.\n\\fourch{周期为$2 \\pi$的偶函数}{周期为$2 \\pi$的奇函数}{周期为$\\pi$的偶函数}{周期为$\\pi$的奇函数}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -412272,7 +413672,9 @@ "id": "016052", "content": "函数$f(x)=\\sin \\dfrac{x}{3}+\\cos \\dfrac{x}{3}$的最小正周期和最大值分别是\\bracket{20}.\n\\fourch{$3 \\pi$和$\\sqrt{2}$}{$3 \\pi$和$2$}{$6 \\pi$和$\\sqrt{2}$}{$6 \\pi$和$2$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -412292,7 +413694,9 @@ "id": "016053", "content": "设函数$f(x)=\\cos x+b \\sin x$($b$为常数), 则``$b=0$''是``$f(x)$为偶函数''的 \\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -412312,7 +413716,9 @@ "id": "016054", "content": "下列区间中, 函数$f(x)=7 \\sin (x-\\dfrac{\\pi}{6})$严格递增的区间是 \\bracket{20}.\n\\fourch{$(0, \\dfrac{\\pi}{2})$}{$(\\dfrac{\\pi}{2}, \\pi)$}{$(\\pi, \\dfrac{3 \\pi}{2})$}{$(\\dfrac{3 \\pi}{2}, 2 \\pi)$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -412332,7 +413738,9 @@ "id": "016055", "content": "函数$f(x)=\\cos x-\\cos 2 x$, 试判断函数的奇偶性及最大值: \\bracket{20}.\n\\fourch{奇函数, 最大值为$2$}{偶函数, 最大值为$2$}{奇函数, 最大值为$\\dfrac{9}{8}$}{偶函数, 最大值为$\\dfrac{9}{8}$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -412352,7 +413760,9 @@ "id": "016056", "content": "判断函数$y=3 \\sin (2 x+\\varphi)$的奇偶性.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412372,7 +413782,9 @@ "id": "016057", "content": "设函数$f(x)=\\sin x+\\cos x$($x \\in \\mathbf{R}$).\\\\\n(1) 求函数$y=[f(x+\\dfrac{\\pi}{2})]^2$的最小正周期;\\\\\n(2) 求函数$y=f(x) f(x-\\dfrac{\\pi}{4})$在$[0, \\dfrac{\\pi}{2}]$上的最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412392,7 +413804,9 @@ "id": "016058", "content": "求函数$y=\\dfrac{\\sqrt{3}}{2} \\sin 2 x+\\cos ^2 x$的最小正周期.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412412,7 +413826,9 @@ "id": "016059", "content": "设函数$f(x)=\\sin 3 x+|\\sin 3 x|$, 则$f(x)$为\\bracket{20}.\n\\twoch{周期函数, 最小正周期为$\\dfrac{2 \\pi}{3}$}{周期函数, 最小正周期为$\\dfrac{\\pi}{3}$}{周期函数, 最小正周期为$2 \\pi$}{非周期函数}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -412432,7 +413848,9 @@ "id": "016060", "content": "已知函数$f(x)=2 \\sin x \\cos x+2 \\sqrt{3} \\cos ^2 x-\\sqrt{3}$, $x \\in \\mathbf{R}$.\n(1) 求函数$f(x)$的最小正周期和严格递增区间;\\\\\n(2) 在锐角$\\triangle ABC$中, 若$f(A)=1$, $\\overrightarrow{AB} \\cdot \\overrightarrow{AC}=\\sqrt{2}$, 求$\\triangle ABC$的面积.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412452,7 +413870,9 @@ "id": "016061", "content": "已知角$\\alpha$的顶点在原点, 始边与$x$轴的正半轴重合, 终边经过点$P(-3, \\sqrt{3})$.\\\\\n(1) 求$\\sin \\alpha \\cos \\alpha-\\tan \\alpha$的值;\\\\\n(2) 若函数$f(x)=\\cos (x+\\alpha) \\cos \\alpha+\\sin (x+\\alpha) \\sin \\alpha$($x \\in \\mathbf{R}$), 求函数$y=\\sqrt{3} f(\\dfrac{\\pi}{2}-2 x)+2 f^2(x)$的最大值, 并指出取到最大值时$x$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412472,7 +413892,9 @@ "id": "016062", "content": "设$f(x)=4 \\cos (\\omega x-\\dfrac{\\pi}{6}) \\sin \\omega x-\\cos (2 \\omega x+\\pi)$, 其中$\\omega>0$.\\\\\n(1) 求函数$y=f(x)$的值域;\\\\\n(2) 若$y=f(x)$在区间$[-\\dfrac{3 \\pi}{2}, \\dfrac{\\pi}{2}]$上为增函数, 求$\\omega$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412492,7 +413914,9 @@ "id": "016063", "content": "函数$y=\\sin 2 x \\cos 2 x$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412512,7 +413936,9 @@ "id": "016064", "content": "在函数: \\textcircled{1} $y=\\cos |2 x|$; \\textcircled{2} $y=|\\cos x|$; \\textcircled{3} $y=\\cos (2 x+\\dfrac{\\pi}{6})$; \\textcircled{4} $y=\\tan (2 x-\\dfrac{\\pi}{4})$中, 最小正周期为$\\pi$的函数的序号为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412532,7 +413958,9 @@ "id": "016065", "content": "已知函数$f(x)=(\\sin \\omega x+\\cos \\omega x)^2-1$的最小正周期为$\\pi$, 则$\\omega=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412552,7 +413980,9 @@ "id": "016066", "content": "函数$y=2 \\sin (\\dfrac{\\pi}{6}-2 x)(x \\in[0, \\pi])$为增函数的区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412572,7 +414002,9 @@ "id": "016067", "content": "对于函数$f(x)=|2 \\sin ^2 x-1|$, 若存在常数$c$, 使得对任意的$x \\in \\mathbf{R}$, $f(x+c)=f(x)$恒成立, 则$c$的最小正值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412592,7 +414024,9 @@ "id": "016068", "content": "设$\\omega>0$, 若函数$f(x)=2 \\sin \\omega x$在$[-\\dfrac{\\pi}{3}, \\dfrac{\\pi}{4}]$上严格递增, 则$\\omega$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412612,7 +414046,9 @@ "id": "016069", "content": "已知函数$f(x)=\\sin (\\pi x-\\dfrac{\\pi}{2})-1$, 则下列命题中正确的是\\bracket{20}.\n\\twoch{$f(x)$是周期为$1$的奇函数}{$f(x)$是周期为$2$的偶函数}{$f(x)$是周期为$1$的非奇非偶函数}{$f(x)$是周期为$2$的非奇非偶函数}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -412632,7 +414068,9 @@ "id": "016070", "content": "设$\\varphi \\in \\mathbf{R}$, 则``$\\varphi=0$''是``$f(x)=\\cos (x+\\varphi)$($x \\in \\mathbf{R}$)为偶函数''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分与不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -412652,7 +414090,9 @@ "id": "016071", "content": "已知函数$f(x)=\\sin 2 x+2 \\cos ^2 x$, 则在\\textcircled{1} $f(x)$的最大值为$3$; \\textcircled{2} $f(x)$的图像关于直线$x=\\dfrac{\\pi}{8}$对称; \\textcircled{3} $f(x)$的图像关于点$(-\\dfrac{\\pi}{8}, 1)$对称; \\textcircled{4} $f(x)$在$[-\\dfrac{\\pi}{4}, \\dfrac{\\pi}{4}]$上严格递增中, 正确的有\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412672,7 +414112,9 @@ "id": "016072", "content": "已知$f(x)=\\sin \\omega x$, $\\omega>0$.\\\\\n(1) 若$f(x)$的最小正周期$T=4 \\pi$求$\\omega$及方程$f(x)=\\dfrac{1}{2}$的解集;\\\\\n(2) 若$\\omega=1$, $g(x)=[f(x)]^2+\\sqrt{3} f(-x) \\cdot f(\\dfrac{\\pi}{2}-x)$, 求$x \\in[0, \\dfrac{\\pi}{4}]$时, $g(x)$的值域.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412692,7 +414134,9 @@ "id": "016073", "content": "已知函数$f(x)=\\cos 2 x+2 \\sqrt{3} \\sin x \\cos x$.\\\\\n(1) 求函数$f(x)$的最大值, 并指出取到最大值时对应的$x$的值;\\\\\n(2) 若$0<\\theta<\\dfrac{\\pi}{6}$, 且$f(\\theta)=\\dfrac{4}{3}$, 计算$\\cos 2 \\theta$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412712,7 +414156,9 @@ "id": "016074", "content": "已知函数$f(x)=a(2 \\cos ^2 \\dfrac{x}{2}+\\sin x)+b$.\\\\\n(1) 当$a=1$时, 求$f(x)$的严格递增区间;\\\\\n(2) 当$a>0$, 且$x \\in[0, \\pi]$时, $f(x)$的值域是$[3,4]$, 求$a$、$b$的值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412732,7 +414178,9 @@ "id": "016075", "content": "函数$y=\\cos (2 x+\\dfrac{\\pi}{2})$的垂直于$x$轴的对称轴方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412752,7 +414200,9 @@ "id": "016076", "content": "函数$y=A \\sin (\\omega x+\\varphi)$($A>0$, $\\omega>0$)图像上一个最高点为$P(\\dfrac{1}{2}, 1)$, 相邻的一个最低点为$Q(\\dfrac{1}{4},-1)$, 则$\\omega=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412772,7 +414222,9 @@ "id": "016077", "content": "若直线$x=a$与函数$f(x)=\\sin x$和$g(x)=\\cos x$的图像分别交于$M$、$N$两点, 则$|MN|$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412792,7 +414244,9 @@ "id": "016078", "content": "将函数$y=f(x)$的图像向右平移$\\dfrac{\\pi}{4}$个单位, 再向上平移$1$个单位后得到的函数对应的表达式为$y=2 \\sin ^2 x$, 则函数$f(x)$的表达式可以是\\bracket{20}.\n\\fourch{$2 \\sin x$}{$2 \\cos x$}{$\\sin 2 x$}{$\\cos 2 x$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -412812,7 +414266,9 @@ "id": "016079", "content": "方程$\\log _5 x=|\\sin x|$的解的个数为\\bracket{20}.\n\\fourch{$1$}{$3$}{$4$}{$5$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -412832,7 +414288,9 @@ "id": "016080", "content": "用``五点法''作出函数$y=2 \\sin (2 x+\\dfrac{\\pi}{3})$的草图.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412852,7 +414310,9 @@ "id": "016081", "content": "已知函数$f(x)=\\sqrt{3} \\sin 2 x-2 \\cos ^2 x-1$, 则下列结论中正确的是\\bracket{20}.\n\\onech{$f(x)$图像的一条对称轴方程为$x=\\dfrac{2 \\pi}{3}$}{$f(x)$图像的一个对称中心为$(\\dfrac{\\pi}{12}, 0)$}{将曲线$y=2 \\sin (x-\\dfrac{\\pi}{6})$上各点的横坐标缩短到原来的$\\dfrac{1}{2}$(纵坐标不变), 再向下平移$2$个单位长度, 可得到$y=f(x)$的图像}{将$f(x)$的图像向右平移$\\dfrac{\\pi}{6}$个单位长度, 得到的曲线关于$y$轴对称}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -412872,7 +414332,9 @@ "id": "016082", "content": "把函数$y=f(x)$图像上所有点的横坐标缩短到原来的$\\dfrac{1}{2}$倍, 纵坐标不变, 再把所得曲线向右平移$\\dfrac{\\pi}{3}$个单位长度, 得到函数$y=\\sin (x-\\dfrac{\\pi}{4})$的图像, 则$f(x)=$\\bracket{20}.\n\\fourch{$\\sin (\\dfrac{x}{2}-\\dfrac{7 x}{12})$}{$\\sin (\\dfrac{x}{2}+\\dfrac{\\pi}{12})$}{$\\sin (2 x-\\dfrac{7 \\pi}{12})$}{$\\sin (2 x+\\dfrac{\\pi}{12})$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -412892,7 +414354,9 @@ "id": "016083", "content": "函数$f(x)=3 \\sin x+4 \\cos x$在区间$[0, \\pi]$上的对称轴为$x=\\varphi$, 则$\\cos \\varphi=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412912,7 +414376,9 @@ "id": "016084", "content": "已知函数$f(x)=2 \\cos (\\omega x+\\varphi)$的部分图像如图所示, 则$f(\\dfrac{\\pi}{2})=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-1,0) -- (4.5,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] (0,2) -- ({13*pi/12},2) -- ({13*pi/12},0);\n\\draw [domain = -0.5:4, samples = 100] plot (\\x,{2*sin(2*\\x/pi*180+60)});\n\\draw (0,2) node [left] {$2$};\n\\draw ({pi/3},0) node [below left] {$\\dfrac\\pi 3$};\n\\draw ({13*pi/12},0) node [below] {$\\dfrac{13\\pi}{12}$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -412932,7 +414398,9 @@ "id": "016085", "content": "若关于$x$的方程$\\sin x+\\cos x=k$在区间$[0, \\pi]$上有两个不同的实数解$x_1$、$x_2$, 求实数$k$的取值范围, 并求$x_1+x_2$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412952,7 +414420,9 @@ "id": "016086", "content": "已知函数$f(x)=\\sqrt{3} \\sin \\omega x+\\cos \\omega x+c$($\\omega>0$, $x \\in \\mathbf{R}$, $c$是实数常数)的图像上的一个最高点是$(\\dfrac{\\pi}{6}, 1)$, 与该最高点最近的一个最低点是$(\\dfrac{2 \\pi}{3},-3)$.\\\\\n(1) 求函数$f(x)$的解析式及其严格递增区间;\\\\\n(2) 在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边分别为$a$、$b$、$c$, 且$\\overrightarrow{AB} \\cdot \\overrightarrow{BC}=-\\dfrac{1}{2} a c$, 角$A$的取值范围是区间$M$, 当$x \\in M$时, 试求函数$f(x)$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412972,7 +414442,9 @@ "id": "016087", "content": "已知定义在区间$[-\\dfrac{\\pi}{2}, \\pi]$上的函数$y=f(x)$的图像关于直线$x=\\dfrac{\\pi}{4}$对称, 当$x \\geq \\dfrac{\\pi}{4}$时, 函数$f(x)=\\sin x$.\\\\\n(1) 求$f(-\\dfrac{\\pi}{2})$, $f(-\\dfrac{\\pi}{4})$的值;\\\\\n(2) 求$y=f(x)$的函数关系式;\\\\\n(3) 如果关于$x$的方程$f(x)=a$有解, 那么将方程在$a$取某一确定值时, 所求得的所有解的和记为$M_a$. 求$M_a$的所有可能值及相应的$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -412992,7 +414464,9 @@ "id": "016088", "content": "函数$f(x)=\\sin x+2|\\sin x|$, $x \\in[0,2 \\pi]$的图像与直线$y=k$有且仅有两个不同的交点, 则$k$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413012,7 +414486,9 @@ "id": "016089", "content": "若函数$y=\\cos 2 x+a \\sin 2 x$的图像关于直线$x=-\\dfrac{\\pi}{8}$对称, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413032,7 +414508,9 @@ "id": "016090", "content": "若将函数$f(x)=\\sin 2 x+\\cos 2 x$的图像向右平移$\\varphi$个单位, 所得图像关于$y$轴对称, 则$\\varphi$的最小正值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413052,7 +414530,9 @@ "id": "016091", "content": "已知函数$f(x)=\\sqrt{3} \\sin \\omega x+\\cos \\omega x$($\\omega>0$, $x \\in \\mathbf{R}$), 在曲线$y=f(x)$与直线$y=1$的交点中, 若相邻交点距离的最小值为$\\dfrac{\\pi}{3}$, 则$f(x)$的最小正周期为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413072,7 +414552,9 @@ "id": "016092", "content": "如图为函数$f(x)=2 \\sin (\\omega x+\\varphi)$($\\omega>0$, $0 \\leq \\varphi \\leq \\pi$)的部分图像, 其中$A$、$B$两点之间的距离为$5$, 那么$f(-1)=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw [->] (-2,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.5:2.5] plot (\\x,{2*sin(60*\\x+150)});\n\\draw [dashed] (-1,0) -- (-1,2) node [above] {$A$} -- (0,2) node [right] {$2$};\n\\draw (0,1) node [right] {$1$};\n\\draw [dashed] (0,-2) -- (2,-2) node [below] {$B$} -- (2,0);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413092,7 +414574,9 @@ "id": "016093", "content": "已知$f(x)=\\sin \\omega x-\\sqrt{3} \\cos \\omega x$($\\omega>0$)的图像与$x$轴的两个相邻交点的距离为$\\pi$, 把$f(x)$图像上每一点的横坐标缩小到原来的一半, 再沿$x$轴向左平移$\\dfrac{\\pi}{3}$个单位长度, 然后纵坐标扩大到原来的$2$倍, 得到$g(x)$的图像, 若$g(x)$在$[-a, a]$上严格递增, 则$a$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413112,7 +414596,9 @@ "id": "016094", "content": "函数$f(x)=\\lg \\dfrac{1+\\sin x}{\\cos x}$($x \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$)的图像大致是\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.55:1.55, samples = 100] plot (\\x,{log10((1+sin(\\x/pi*180))/(cos(\\x/pi*180)))});\n\\draw [dashed] ({-pi/2},-2)--++(0,4) ({pi/2},-2)--++(0,4);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.55:1.55, samples = 100] plot (\\x,{-log10((1+sin(\\x/pi*180))/(cos(\\x/pi*180)))});\n\\draw [dashed] ({-pi/2},-2)--++(0,4) ({pi/2},-2)--++(0,4);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.55:1.55, samples = 100] plot (\\x,{abs(log10((1+sin(\\x/pi*180))/(cos(\\x/pi*180))))});\n\\draw [dashed] ({-pi/2},-2)--++(0,4) ({pi/2},-2)--++(0,4);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.8]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.55:1.55, samples = 100] plot (\\x,{-abs(log10((1+sin(\\x/pi*180))/(cos(\\x/pi*180))))});\n\\draw [dashed] ({-pi/2},-2)--++(0,4) ({pi/2},-2)--++(0,4);\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -413132,7 +414618,9 @@ "id": "016095", "content": "将函数$y=\\sin x$的图像向左平移$\\dfrac{\\pi}{2}$个单位, 得到函数$y=f(x)$的图像, 则下列说法中正确的是\\bracket{20}.\n\\twoch{$y=f(x)$是奇函数}{$y=f(x)$的周期为$\\pi$}{$y=f(x)$的图像关于直线$x=\\dfrac{\\pi}{2}$对称}{$y=f(x)$的图像关于点$(-\\dfrac{\\pi}{2}, 0)$对称}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -413152,7 +414640,9 @@ "id": "016096", "content": "方程$2^x=\\cos x$的解的个数为\\bracket{20}.\n\\fourch{$0$个}{$1$个}{$2$个}{无穷多个}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -413172,7 +414662,9 @@ "id": "016097", "content": "已知函数$f(x)=2 \\sin (\\omega x-\\dfrac{\\pi}{6})$的图像的一条对称轴为$x=\\pi$, 其中$\\omega$为常数, 且$\\omega \\in(0,1)$, 以下结论中正确的有\\blank{50}.\\\\\n\\textcircled{1} 函数$f(x)$的最小正周期为$3 \\pi$;\\\\\n\\textcircled{2} $f(\\dfrac{3}{4} \\pi)=\\sqrt{3}$;\\\\\n\\textcircled{3} 将函数$f(x)$的图像向左平移$\\dfrac{\\pi}{6}$所得图像关于原点对称;\\\\\n\\textcircled{4} 函数$f(x)$在区间$(0,100 \\pi)$上有$67$个零点.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413192,7 +414684,9 @@ "id": "016098", "content": "设函数$f(x)=\\sin (\\omega x-\\dfrac{\\pi}{6})+\\sin (\\omega x-\\dfrac{\\pi}{2})$, 其中$0<\\omega<3$. 已知$f(\\dfrac{\\pi}{6})=0$.\\\\\n(1) 求$\\omega$;\\\\\n(2) 将函数$y=f(x)$的图像上各点的横坐标伸长为原来的$2$倍(纵坐标不变), 再将得到的图像向左平移$\\dfrac{\\pi}{4}$个单位, 得到函数$y=g(x)$的图像, 求$g(x)$在$[-\\dfrac{\\pi}{4}, \\dfrac{3 \\pi}{4}]$上的最小值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -413212,7 +414706,9 @@ "id": "016099", "content": "设函数$f(x)=\\sin (2 x+\\varphi)$($-\\pi<\\varphi<0$), $y=f(x)$图像的一条对称轴是直线$x=\\dfrac{\\pi}{8}$.\\\\\n(1) 求$\\varphi$;\\\\\n(2) 求函数$y=f(x)$的严格递增区间;\\\\\n(3) 画出函数$y=f(x)$在区间$[0, \\pi]$上的图像.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -413232,7 +414728,9 @@ "id": "016100", "content": "设单位圆上的点$P(x, y)$, 求过点$P$斜率为$-1$的直线在$y$轴上截距的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413252,7 +414750,9 @@ "id": "016101", "content": "函数$y=-\\sin x-\\cos x$, $x \\in[\\pi, \\dfrac{3 \\pi}{2}]$, 则此函数的最大值为最小值为\\blank{50} , 最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413272,7 +414772,9 @@ "id": "016102", "content": "函数$y=\\sin \\dfrac{x}{2}+\\cos \\dfrac{x}{2}$, $x \\in(-2 \\pi, 2 \\pi)$的严格递增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413292,7 +414794,9 @@ "id": "016103", "content": "函数$f(x)=\\sin ^2 2x$的最小正周期是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413312,7 +414816,9 @@ "id": "016104", "content": "已知$\\sin ^2(\\dfrac{\\pi}{4}+\\alpha)=\\dfrac{2}{3}$, 则$\\sin 2 \\alpha$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413332,7 +414838,9 @@ "id": "016105", "content": "若函数$f(x)=\\sin (x+\\varphi)+\\cos x$的最大值为$2$, 则常数$\\varphi$的一个取值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413352,7 +414860,9 @@ "id": "016106", "content": "在平面直角坐标系中, 函数$y=\\cos x$和函数$y=\\tan x$的定义域都是$(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, 它们的交点为$P$, 则点$P$的纵坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413372,7 +414882,9 @@ "id": "016107", "content": "若函数具有以下性质:\\textcircled {1} 关于$y$轴对称;\\textcircled {2} 对于任意$x \\in \\mathbf{R}$, 都有$f(4+x)=f(4-x)$, 则$f(x)$的解析式可以为 \\blank{50}(只须写出满足条件的一个解析式即可).", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413392,7 +414904,9 @@ "id": "016108", "content": "函数$y=\\sin (\\omega x+\\varphi)$($\\omega>0$, $0<\\varphi<\\pi$)的最小正周期为$\\pi$, 且函数图像关于点$(-\\dfrac{3 \\pi}{8}, 0)$对称, 则函数的解析式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413412,7 +414926,9 @@ "id": "016109", "content": "已知$2 \\cos (x-\\dfrac{\\pi}{4})=\\sqrt{2}$且$x \\in(0, \\pi)$, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413432,7 +414948,9 @@ "id": "016110", "content": "已知$2 \\tan \\theta-\\tan (\\theta+\\dfrac{\\pi}{4})=7$, 则$\\tan \\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413452,7 +414970,9 @@ "id": "016111", "content": "关于函数$f(x)=\\sin x+\\dfrac{1}{\\sin x}$有如下四个命题:\n\\textcircled{1} $f(x)$的图像关于$y$轴对称;\\textcircled {2} $f(x)$的图像关于原点对称;\\textcircled {3} $f(x)$的图像关于直线$x=\\dfrac{\\pi}{2}$对称;\\textcircled {4} $f(x)$的最小值为$2$. 其中所有真命题的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413472,7 +414992,9 @@ "id": "016112", "content": "若$\\alpha$为第四象限角, 则\\bracket{20}.\n\\fourch{$\\cos 2 \\alpha>0$}{$\\cos 2 \\alpha<0$}{$\\sin 2 \\alpha>0$}{$\\sin 2 \\alpha<0$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -413492,7 +415014,9 @@ "id": "016113", "content": "已知函数$f(x)=A \\cos (\\omega x+\\varphi)$($A>0$, $\\omega>0$, $\\varphi \\in \\mathbf{R})$, 则``$f(x)$是奇函数''是``$\\varphi=\\dfrac{\\pi}{2}$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -413512,7 +415036,9 @@ "id": "016114", "content": "已知$\\sin \\theta=-\\dfrac{1}{3}$且$\\theta \\in(-\\pi,-\\dfrac{\\pi}{2})$, 则$\\theta$可表示成\\bracket{20}.\n\\fourch{$-\\arcsin (-\\dfrac{1}{3})$}{$-\\dfrac{\\pi}{2}+\\arcsin (-\\dfrac{1}{3})$}{$-\\pi+\\arcsin (-\\dfrac{1}{3})$}{$-\\pi-\\arcsin (-\\dfrac{1}{3})$}", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -413532,7 +415058,9 @@ "id": "016115", "content": "已知函数$f(x)=\\sin (\\omega x+\\varphi)$($\\omega>0$, $|\\varphi|<\\dfrac{\\pi}{2}$)的部分图像如图所示, 则下列说法中正确的有\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (3.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [dashed] ({pi/12},1) -- ({pi/12},0) node [below] {$\\dfrac{\\pi}{12}$};\n\\draw ({5*pi/6},0) node [below right] {$\\dfrac{5\\pi}{6}$};\n\\draw [domain = -1:3.5, samples = 100] plot (\\x,{sin(2*\\x/pi*180+60)});\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} $f(x)=f(\\pi+x)$;\\textcircled {2} $f(x)=-f(\\pi+x)$;\\textcircled {3} $f(x)=f(\\dfrac{2 \\pi}{3}-x)$;\\textcircled {4} $f(x)=-f(\\dfrac{2 \\pi}{3}-x)$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413552,7 +415080,9 @@ "id": "016116", "content": "在\\textcircled{1} $a c=\\sqrt{3}$;\\textcircled {2} $c \\sin A=3$;\\textcircled {3} $c=\\sqrt{3} b$这三个条件中任选一个, 补充在下面问题中, 若问题中的三角形存在, 求$c$的值; 若问题中的三角形不存在, 说明理由.\\\\\n问题: 是否存在$\\triangle ABC$, 它的内角$A$、$B$、$C$的对边分别为$a$、$b$、$c$, 且$\\sin A=\\sqrt{3} \\sin B$, $C=\\dfrac{\\pi}{6}$, \\underline{\\hbox to 20pt{}}.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -413572,7 +415102,9 @@ "id": "016117", "content": "已知函数$f(x)=4 \\cos \\omega x \\sin (\\omega x+\\dfrac{\\pi}{4})$($\\omega>0$)的最小正周期为$\\pi$.\\\\\n(1) 求$\\omega$的值;\\\\\n(2) 讨论$f(x)$在区间$[0, \\dfrac{\\pi}{2}]$上的单调性.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -413592,7 +415124,9 @@ "id": "016118", "content": "函数$f(x)=3 \\sin (2 x+\\dfrac{\\pi}{6})$的部分图像如图所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-2,0) -- (7,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [dashed] (0,3) node [left] {$y_0$} -- ({pi*7/6},3) --++ (0,-3) node [below] {$x_0$};\n\\draw [domain = -1:6, samples = 100] plot (\\x,{3*sin(2*\\x/pi*180+30)});\n\\end{tikzpicture}\n\\end{center}\n(1) 写出$f(x)$的最小正周期及图中$x_0$、$y_0$的值;\\\\\n(2) 求$f(x)$在区间$[-\\dfrac{\\pi}{2},-\\dfrac{\\pi}{12}]$上的最大值和最小值.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -413612,7 +415146,9 @@ "id": "016119", "content": "已知函数$f(x)=2 \\sin ^2(\\dfrac{\\pi}{4}+x)-2 \\sqrt{3} \\cos ^2 x+3 \\sqrt{3}$, $x \\in[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2}]$.\\\\\n(1) 求$f(x)$的最大值和最小值;\\\\\n(2) 若不等式$|f(x)-m|<2$在$x \\in[\\dfrac{\\pi}{4}, \\dfrac{\\pi}{2}]$上恒成立, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -413632,7 +415168,9 @@ "id": "016120", "content": "设函数$f(x)=2 \\sin x \\cos ^2 \\dfrac{\\varphi}{2}+\\cos x \\sin \\varphi-\\sin x$($0<\\varphi<\\pi$)在$x=\\pi$处取最小值.\\\\\n(1) 求$\\varphi$的值;\\\\\n(2) 在$\\triangle ABC$中, $a$、$b$、$c$分别是角$A$、$B$、$C$的对边, 已知$a=1$, $b=\\sqrt{2}$, $f(A)=\\dfrac{\\sqrt{3}}{2}$, 求角$C$.", "objs": [], - "tags": [], + "tags": [ + "第三单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -413652,7 +415190,9 @@ "id": "016121", "content": "下列命题中真命题是\\bracket{20}.\n\\onech{任何两个非零向量的单位向量都是相等的向量}{任何两个非零向量的单位向量是相等的向量或互为负向量}{一个非零向量的单位向量有两个, 它们互为负向量}{任何非零向量的单位向量的模长相等}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -413672,7 +415212,9 @@ "id": "016122", "content": "下列命题中正确的是\\bracket{20}.\n\\onech{$\\overrightarrow {a}$与$\\overrightarrow {b}$共线, $\\overrightarrow {b}$与$\\overrightarrow {c}$共线, 则$\\overrightarrow {a}$与$\\overrightarrow {c}$也共线}{任意两个相等的非零向量的始点与终点是一平行四边形的四顶点}{向量$\\overrightarrow {a}$与$\\overrightarrow {b}$不共线, 则$\\overrightarrow {a}$与$\\overrightarrow {b}$都是非零向量}{有相同起点的两个非零向量不平行}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -413692,7 +415234,9 @@ "id": "016123", "content": "设$\\overrightarrow {a}, \\overrightarrow {b}$是两个不平行的向量, 若实数$\\lambda, \\mu$使得``$\\lambda \\overrightarrow {a}+\\mu \\overrightarrow {b}=\\overrightarrow{0}$'', 则$\\lambda=$\\blank{50}, $\\mu=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413712,7 +415256,9 @@ "id": "016124", "content": "已知$G$是$\\triangle ABC$的重心, $D$、$E$、$F$分别为$AB$、$AC$、$BC$中点, 则$\\overrightarrow{GD}+\\overrightarrow{GE}+\\overrightarrow{GF}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413732,7 +415278,9 @@ "id": "016125", "content": "$G$是$\\triangle ABC$的重心, 下列各向量中与向量$\\overrightarrow{AB}$平行的向量为\\bracket{20}.\n\\fourch{$\\overrightarrow{AB}+\\overrightarrow{BC}+\\overrightarrow{AC}$}{$\\overrightarrow{AG}+\\overrightarrow{GB}+\\overrightarrow{BC}$}{$\\overrightarrow{AG}+\\overrightarrow{BG}+\\overrightarrow{CG}$}{$3 \\overrightarrow{AG}+\\overrightarrow{AC}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -413752,7 +415300,9 @@ "id": "016126", "content": "已知非零向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$, 则``$\\overrightarrow {a} \\cdot \\overrightarrow {c}=\\overrightarrow {b} \\cdot \\overrightarrow {c}$''是``$\\overrightarrow {a}=\\overrightarrow {b}$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -413772,7 +415322,9 @@ "id": "016127", "content": "在$\\triangle ABC$中, $AD$为$BC$边上的中线, $E$为$AD$的中点, 则$\\overrightarrow{EB}=$\\bracket{20}.\n\\fourch{$\\dfrac{3}{4} \\overrightarrow{AB}-\\dfrac{1}{4} \\overrightarrow{AC}$}{$\\dfrac{1}{4} \\overrightarrow{AB}-\\dfrac{3}{4} \\overrightarrow{AC}$}{$\\dfrac{3}{4} \\overrightarrow{AB}+\\dfrac{1}{4} \\overrightarrow{AC}$}{$\\dfrac{1}{4} \\overrightarrow{AB}+\\dfrac{3}{4} \\overrightarrow{AC}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -413792,7 +415344,9 @@ "id": "016128", "content": "设两个非零向量$\\overrightarrow{e_1}$、$\\overrightarrow{e_2}$不共线, 如果$\\overrightarrow{AB}=2 \\overrightarrow{e_1}+3 \\overrightarrow{e_2}$, $\\overrightarrow{BC}=6 \\overrightarrow{e_1}+23 \\overrightarrow{e_2}$, $\\overrightarrow{CD}=4 \\overrightarrow{e_1}-8 \\overrightarrow{e_2}$, 求证: $A$、$B$、$D$三点共线.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -413812,7 +415366,9 @@ "id": "016129", "content": "设$\\overrightarrow{e_1}$、$\\overrightarrow{e_2}$是两个不共线的向量, 已知$\\overrightarrow{AB}=2 \\overrightarrow{e_1}+k \\overrightarrow{e_2}$, $\\overrightarrow{CB}=\\overrightarrow{e_1}+3 \\overrightarrow{e_2}$, $\\overrightarrow{CD}=2 \\overrightarrow{e_1}-\\overrightarrow{e_2}$, 若$A$、$B$、$D$三点共线, 求$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -413832,7 +415388,9 @@ "id": "016130", "content": "已知$O$为坐标原点, 对于函数$f(x)=a \\sin x+b \\cos x$, 称向量$\\overrightarrow{OM}=(a, b)$为函数$f(x)$的伴随向量, 同时称函数$f(x)$为向量$\\overrightarrow{OM}$的伴随函数.\\\\\n(1) 设函数$g(x)=\\sin (\\dfrac{\\pi}{2}+x)+2 \\cos (\\dfrac{\\pi}{2}-x)$, 试求$g(x)$的伴随向量$\\overrightarrow{OM}$的模;\\\\\n(2) 记$\\overrightarrow{ON}=(1, \\sqrt{3})$的伴随函数为$h(x)$, 求使得关于$x$的方程$h(x)-t=0$在$[0, \\dfrac{\\pi}{2}]$内恒有两个不相等实数解的实数$t$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -413852,7 +415410,9 @@ "id": "016131", "content": "非零向量$\\overrightarrow {a}, \\overrightarrow {b}$的方向相反, 且$\\overrightarrow {a}=k \\overrightarrow {b}$, 则实数$k$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413872,7 +415432,9 @@ "id": "016132", "content": "在平行四边形$ABCD$中, $\\overrightarrow{AB}=\\overrightarrow {a}$, $\\overrightarrow{AD}=\\overrightarrow {b}$, $\\overrightarrow{AN}=3 \\overrightarrow{NC}$, $M$为$BC$的中点, 则$\\overrightarrow{MN}=$\\blank{50}(用$\\overrightarrow {a}$、$\\overrightarrow {b}$表示).", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413892,7 +415454,9 @@ "id": "016133", "content": "如图, $E$、$F$是四边形$ABCD$对角线$AC$、$BD$的中点, $\\overrightarrow{AB}=\\overrightarrow {a}$, $\\overrightarrow{CD}=\\overrightarrow {c}$, 则$\\overrightarrow{EF}=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (3,0) node [below] {$B$} coordinate (B);\n\\draw (0.5,1.6) node [above] {$D$} coordinate (D);\n\\draw (2.1,1.4) node [above] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(C)$) node [left] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(D)$) node [right] {$F$} coordinate (F);\n\\draw (A)--(B)--(C)--(D)--cycle (A)--(C)(B)--(D)(E)--(F);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413912,7 +415476,9 @@ "id": "016134", "content": "在正六边形$ABCDEF$中, 对角线$BD, CF$相交于点$P$. 若$\\overrightarrow{AP}=x \\overrightarrow{AB}+y \\overrightarrow{AF}$, 则$x+y=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413932,7 +415498,9 @@ "id": "016135", "content": "在$\\triangle ABC$中, $P$为$BC$的中点, 内角$A$、$B$、$C$的对边分别为$a$、$b$、$c$, 若$c \\overrightarrow{AC}+a \\overrightarrow{PA}+b \\overrightarrow{PB}=\\overrightarrow{0}$, 则$\\triangle ABC$的形状为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413952,7 +415520,9 @@ "id": "016136", "content": "如图, 在$\\triangle ABC$中, 点$D, E$是线段$BC$上两个动点, 且$\\overrightarrow{AD}+\\overrightarrow{AE}=x \\overrightarrow{AB}+y \\overrightarrow{AC}$, 则$\\dfrac{1}{x}+\\dfrac{4}{y}$的最小值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (3,0) node [below] {$C$} coordinate (C);\n\\draw (2,1.5) node [above] {$A$} coordinate (A);\n\\draw ($(B)!0.25!(C)$) node [below] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw [->] (A)--(B);\n\\draw [->] (A)--(C);\n\\draw [->] (A)--(D);\n\\draw [->] (A)--(E);\n\\draw (B)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -413972,7 +415542,9 @@ "id": "016137", "content": "两个非零向量的模相等是两个向量相等的\\bracket{20}条件.\n\\fourch{充分非必要}{必要非充分}{充要}{既不充分也不必要}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -413992,7 +415564,9 @@ "id": "016138", "content": "若$D$为$\\triangle ABC$的边$AB$的中点, 则$\\overrightarrow{CB}=$\\bracket{20}.\n\\fourch{$2 \\overrightarrow{CD}-\\overrightarrow{CA}$}{$2 \\overrightarrow{CA}-\\overrightarrow{CD}$}{$2 \\overrightarrow{CD}+\\overrightarrow{CA}$}{$2 \\overrightarrow{CA}+\\overrightarrow{CD}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -414012,7 +415586,9 @@ "id": "016139", "content": "设非零向量$\\overrightarrow {a}, \\overrightarrow {b}$满足$|\\overrightarrow {a}+\\overrightarrow {b}|=|\\overrightarrow {a}-\\overrightarrow {b}|$, 则\\bracket{20}.\n\\fourch{$\\overrightarrow {a} \\perp \\overrightarrow {b}$}{$|\\overrightarrow {a}|=|\\overrightarrow {b}|$}{$\\overrightarrow {a}\\parallel \\overrightarrow {b}$}{$|\\overrightarrow {a}|>|\\overrightarrow {b}|$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -414032,7 +415608,9 @@ "id": "016140", "content": "如图, 在平行四边形$ABCD$中, $E$、$F$分别是$BC$、$DC$的中点, $G$为交点, 若$\\overrightarrow{AB}=\\overrightarrow {a}, \\overrightarrow{AD}=\\overrightarrow {b}$, 试以$\\overrightarrow {a}$、$\\overrightarrow {b}$为基底表示$\\overrightarrow{DE}$、$\\overrightarrow{BF}$、$\\overrightarrow{CG}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (3,0) node [below] {$B$} coordinate (B);\n\\draw (1,2) node [above] {$D$} coordinate (D);\n\\draw (4,2) node [above] {$C$} coordinate (C);\n\\draw ($(D)!0.5!(C)$) node [above] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(C)$) node [right] {$E$} coordinate (E);\n\\draw ($(B)!{2/3}!(F)$) node [below left] {$G$} coordinate (G);\n\\draw [->] (B)--(F);\n\\draw [->] (D)--(E);\n\\draw [->] (C)--(G);\n\\draw [->] (A)--(B);\n\\draw [->] (A)--(D);\n\\draw (B)--(C)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414052,7 +415630,9 @@ "id": "016141", "content": "在$\\triangle ABC$中, $\\angle B=120{^ \\circ}$, $\\overrightarrow{AB}=\\overrightarrow {a}$, $\\overrightarrow{BC}=\\overrightarrow {b}$, 且$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=3$, 试用$\\overrightarrow {a}$、$\\overrightarrow {b}$表示与$\\overrightarrow{AC}$同方向的单位向量$\\overrightarrow{e_0}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414072,7 +415652,9 @@ "id": "016142", "content": "已知函数$y=f(x)=k x+b$的图像与$x$、$y$轴分别相交于点$A$、$B$, $\\overrightarrow{AB}=2 \\overrightarrow {i}+2 \\overrightarrow {j}$($\\overrightarrow {i}, \\overrightarrow {j}$分别是与$x$、$y$轴正半轴同方向的单位向量), 函数$g(x)=x^2-x-6$.\\\\\n(1) 求$k$、$b$的值;\\\\\n(2) 当$x$满足$f(x)>g(x)$时, 求函数$\\dfrac{g(x)+1}{f(x)}$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414092,7 +415674,9 @@ "id": "016143", "content": "向量$\\overrightarrow {a}$的模为$10$, 它与向量$\\overrightarrow {b}$的夹角为$150{^ \\circ}$, 则它在$\\overrightarrow {b}$方向上的数量投影为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414112,7 +415696,9 @@ "id": "016144", "content": "已知向量$\\overrightarrow{BA}=(\\dfrac{1}{2}, \\dfrac{\\sqrt{3}}{2})$, $\\overrightarrow{BC}=(\\dfrac{\\sqrt{3}}{2}, \\dfrac{1}{2})$, 则$\\angle ABC=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414132,7 +415718,9 @@ "id": "016145", "content": "与向量$\\overrightarrow {a}=(3,4)$垂直的单位向量为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414152,7 +415740,9 @@ "id": "016146", "content": "设向量$\\overrightarrow{e_1}, \\overrightarrow{e_2}$是两个互相垂直的单位向量, 且$\\overrightarrow {a}=2 \\overrightarrow{e_1}-\\overrightarrow{e_2}$, $\\overrightarrow {b}=\\overrightarrow{e_2}$, 则$|\\overrightarrow {a}+2 \\overrightarrow {b}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414172,7 +415762,9 @@ "id": "016147", "content": "已知向量$\\overrightarrow {a}, \\overrightarrow {b}$, $|\\overrightarrow {a}|=6$, $|\\overrightarrow {b}|=3$, 且$\\overrightarrow {a} \\cdot \\overrightarrow {b}=-12$, 则向量$\\overrightarrow {a}$在$\\overrightarrow {b}$方向上的投影为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414192,7 +415784,9 @@ "id": "016148", "content": "在矩形$ABCD$中, $AB=1$, $AD=2$, 动点$P$在以点$C$为圆心且与$BD$相切的圆上. 若$\\overrightarrow{AP}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AD}$, 则$\\lambda+\\mu$的最大值为\\bracket{20}.\n\\fourch{$3$}{$2 \\sqrt{2}$}{$\\sqrt{5}$}{$2$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -414212,7 +415806,9 @@ "id": "016149", "content": "已知非零向量$\\overrightarrow {a}, \\overrightarrow {b}$满足$|\\overrightarrow {a}|=2|\\overrightarrow {b}|$, 且$(\\overrightarrow {a}-\\overrightarrow {b}) \\perp \\overrightarrow {b}$, 则$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{6}$}{$\\dfrac{\\pi}{3}$}{$\\dfrac{2 \\pi}{3}$}{$\\dfrac{5 \\pi}{6}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -414232,7 +415828,9 @@ "id": "016150", "content": "已知$\\overrightarrow{e_1}, \\overrightarrow{e_2}$是互相垂直的单位向量. 若$\\sqrt{3} \\overrightarrow{e_1}-\\overrightarrow{e_2}$与$\\overrightarrow{e_1}+\\lambda \\overrightarrow{e_2}$的夹角为$60{^ \\circ}$, 则实数$\\lambda$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414252,7 +415850,9 @@ "id": "016151", "content": "已知$A$、$B$是半径为$\\sqrt{2}$的圆$O$上的两个点, $\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=1$, $\\odot O$所在平面上有一点$C$满足$|\\overrightarrow{OA}+\\overrightarrow{OB}-\\overrightarrow{OC}|=1$, 则向量$\\overrightarrow{OC}$的模的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414272,7 +415872,9 @@ "id": "016152", "content": "如图, 在同一个平面内, 向量$\\overrightarrow{OA}$、$\\overrightarrow{OB}$、$\\overrightarrow{OC}$的模分别为$1$、$1$、$\\sqrt{2}$, $\\overrightarrow{OA}$与$\\overrightarrow{OC}$的夹角为$\\alpha$, 且$\\tan \\alpha=7$, $\\overrightarrow{OB}$与$\\overrightarrow{OC}$的夹角为$45{^ \\circ}$. 若$\\overrightarrow{OC}=m \\overrightarrow{OA}+n \\overrightarrow{OB}$($m$、$n \\in \\mathbf{R}$), 则$m+n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414292,7 +415894,9 @@ "id": "016153", "content": "如图, 在平面斜坐标系$xOy$中, $\\angle x O y=60{^\\circ}$, 平面上任一点$P$关于斜坐标系的斜坐标是这样定义的: 若$\\overrightarrow{OP}=x \\cdot \\overrightarrow{e_1}+y \\cdot \\overrightarrow{e_2}$, 其中$\\overrightarrow{e_1}$、$\\overrightarrow{e_2}$分别为与$x$轴、$y$轴同方向的单位向量, 则$P$点的斜坐标为$(x, y)$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (2,0) node [below] {$x$};\n\\draw [->] (60:-1) -- (60:1.5) node [left] {$y$};\n\\draw (0,0) node [below] {$O$};\n\\filldraw (0,0) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n(1) 若点$P$的斜坐标为$(2,-2)$, 求$P$到$O$的距离$|PO|$;\\\\\n(2) 求以$O$为圆心. $1$为半径的圆在斜坐标系$x O y$中的方程.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414312,7 +415916,9 @@ "id": "016154", "content": "已知向量$\\overrightarrow {a}=(1,0,2)$, $\\overrightarrow {b}=(2,1,0)$, 则$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414332,7 +415938,9 @@ "id": "016155", "content": "已知向量$\\overrightarrow {a}=(1,2)$, $\\overrightarrow {b}=(2,-2)$, $\\overrightarrow {c}=(1, \\lambda)$. 若$\\overrightarrow {c}\\parallel(2 \\overrightarrow {a}+\\overrightarrow {b})$, 则$\\lambda=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414352,7 +415960,9 @@ "id": "016156", "content": "如图, 以长方体$ABCD-A_1B_1C_1D_1$的顶点$D$为坐标原点, 过$D$的三条棱所在的直线为坐标轴, 建立空间直角坐标系, 若$\\overrightarrow{DB_1}$的坐标为$(4,3,2)$, 则$\\overrightarrow{AC_1}$的坐标为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{3}\n\\def\\m{4}\n\\def\\n{2}\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [above right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [->] (D1) --++ (0,1,0) node [right] {$z$};\n\\draw [->] (C) --++ (1,0,0) node [below] {$y$};\n\\draw [->] (A) --++ (225:1) node [right] {$x$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414372,7 +415982,9 @@ "id": "016157", "content": "在平面直角坐标系中, 已知点$A(-1,0)$、$B(2,0)$, $E$、$F$是$y$轴上的两个动点, 且$|\\overrightarrow{EF}|=2$, 则$\\overrightarrow{AE} \\cdot \\overrightarrow{BF}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414392,7 +416004,9 @@ "id": "016158", "content": "过曲线$y^2=4 x$的焦点$F$并垂直于$x$轴的直线分别与曲线$y^2=4 x$交于$A$、$B$, $A$在$B$上方, $M$为抛物线上一点, $\\overrightarrow{OM}=\\lambda \\overrightarrow{OA}+(\\lambda-2) \\overrightarrow{OB}$, 则$\\lambda=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414412,7 +416026,9 @@ "id": "016159", "content": "在$\\triangle ABC$中, $\\angle C=\\dfrac{\\pi}{2}$, $AC=BC=2$, $M$为$AC$的中点, $P$在$AB$上, 则$\\overrightarrow{MP} \\cdot \\overrightarrow{CP}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414432,7 +416048,9 @@ "id": "016160", "content": "设向量$\\overrightarrow {a}=(1,2)$, $\\overrightarrow {b}=(m,-1)$, 且$(\\overrightarrow {a}+\\overrightarrow {b}) \\perp \\overrightarrow {a}$, 则实数$m=$\\bracket{20}.\n\\fourch{$-3$}{$\\dfrac{3}{2}$}{$-2$}{$-\\dfrac{3}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -414452,7 +416070,9 @@ "id": "016161", "content": "设$\\overrightarrow {m}$、$\\overrightarrow {n}$为非零向量, 则``存在负数$\\lambda$, 使得$\\overrightarrow {m}=\\lambda \\overrightarrow {n}$''是``$\\overrightarrow {m} \\cdot \\overrightarrow {n}<0$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -414472,7 +416092,9 @@ "id": "016162", "content": "如图所示, 正八边形$A_1A_2A_3A_4A_5A_6A_7A_8$的边长为$2$, 若$P$为该正八边形边上的动点, 则$\\overrightarrow{A_1A_3} \\cdot \\overrightarrow{A_1P}$的取值范围为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A_1$} coordinate (A_1) --++ (0:1) node [below] {$A_2$} coordinate (A_2) --++ (45:1) node [right] {$A_3$} coordinate (A_3) --++ (90:1) node [right] {$A_4$} coordinate (A_4) --++ (135:1) node [above] {$A_5$} coordinate (A_5) --++ (180:1) node [above] {$A_6$} coordinate (A_6) --++ (225:1) node [left] {$A_7$} coordinate (A_7) --++ (270:1) node [left] {$A_8$} coordinate (A_8) --cycle;\n\\draw ($(A_4)!0.4!(A_5)$) node [above right] {$P$} coordinate (P);\n\\draw [->] (A_1)--(A_3);\n\\draw [->] (A_1) -- (P);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$[0,8+6 \\sqrt{2}]$}{$[-2 \\sqrt{2}, 8+6 \\sqrt{2}]$}{$[-8-6 \\sqrt{2}, 2 \\sqrt{2}]$}{$[-8-6 \\sqrt{2}, 8+6 \\sqrt{2}]$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -414492,7 +416114,9 @@ "id": "016163", "content": "已知$\\overrightarrow {a}$、$\\overrightarrow {b}$是非零向量, 且满足$|\\overrightarrow {a}|=2$, $(\\overrightarrow {a}-\\overrightarrow {b}) \\cdot(\\overrightarrow {a}+\\overrightarrow {b})=1$.\\\\\n(1) 求$(\\overrightarrow {a}-\\overrightarrow {b})^2+(\\overrightarrow {a}+\\overrightarrow {b})^2$;\\\\\n(2) 若$\\overrightarrow {a} \\cdot \\overrightarrow {b}=-\\sqrt{3}$, 求$\\overrightarrow {a}$、$\\overrightarrow {b}$的夹角.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414512,7 +416136,9 @@ "id": "016164", "content": "已知$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=3$, $|3 \\overrightarrow {a}-\\overrightarrow {b}|=6$.\\\\\n(1) 求$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角;\\\\\n(2) 求$|\\overrightarrow {a}+3 \\overrightarrow {b}|$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414532,7 +416158,9 @@ "id": "016165", "content": "如图, 在等腰直角三角形$AOB$中, $OA=OB=1$, $\\overrightarrow{AB}=4 \\overrightarrow{AC}$, 求$\\overrightarrow{OC} \\cdot(\\overrightarrow{OB}-\\overrightarrow{OA})$\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (2,0) node [right] {$A$} coordinate (A);\n\\draw (0,2) node [left] {$B$} coordinate (B);\n\\draw [->] (O)--(A);\n\\draw [->] (O)--(B);\n\\draw [->] (O)--($(A)!0.25!(B)$) node [above right] {$C$} coordinate (C);\n\\draw (A)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414552,7 +416180,9 @@ "id": "016166", "content": "已知$O$为原点, 点$A$、$B$的坐标分别为$(a, 0)$、$(0, a)$, 其中$a$是正常数, 点$P$在线段$AB$上且$\\overrightarrow{AP}=t \\cdot \\overrightarrow{AB}$($0 \\leq t \\leq 1$), 求$\\overrightarrow{OA} \\cdot \\overrightarrow{OP}$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414572,7 +416202,9 @@ "id": "016167", "content": "已知平面上$A$、$B$两点的坐标分别是$(3,5)$、$(0,1)$, $P$为直线$AB$上一点, 且$\\overrightarrow{AP}=\\dfrac{1}{5} \\overrightarrow{PB}$, 则点$P$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414592,7 +416224,9 @@ "id": "016168", "content": "已知$\\triangle ABC$的三个顶点$A$、$B$、$C$的坐标分别是$(1,2)$、$(2,3)$ 、$(3,7)$, 则此三角形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414612,7 +416246,9 @@ "id": "016169", "content": "已知$P_1(-2,4), P_2(5,3)$, 点$P$在$\\overrightarrow{P_1P_2}$的延长线上, 且$|\\overrightarrow{P_1P}|=2|\\overrightarrow{P_2P}|$, 则$P$点的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414632,7 +416268,9 @@ "id": "016170", "content": "$\\triangle ABC$中, 若$\\overrightarrow{AD}=2 \\overrightarrow{DB}, \\overrightarrow{CD}=\\dfrac{1}{3} \\overrightarrow{CA}+\\lambda \\overrightarrow{CB}$, 则$\\lambda=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414652,7 +416290,9 @@ "id": "016171", "content": "平面向量$\\overrightarrow {a}$, $\\overrightarrow {b}$共线的充要条件是\\bracket{20}.\n\\twoch{$\\overrightarrow {a}, \\overrightarrow {b}$方向相同}{$\\overrightarrow {a}, \\overrightarrow {b}$两向量中至少有一个为零向量}{存在$\\lambda \\in \\mathbf{R}$, $\\overrightarrow {b}=\\lambda \\overrightarrow {a}$}{存在不全为零的实数$\\lambda_1, \\lambda_2$, $\\lambda_1 \\overrightarrow {a}+\\lambda_2 \\overrightarrow {b}=\\overrightarrow{0}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -414672,7 +416312,9 @@ "id": "016172", "content": "如图, 已知$DB$是平行四边形$ABCD$的对角线, $F$为边$DC$的中点, $AF$与$BD$相交于点$E$, 用向量法证明$E$是$BD$的三等分点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (3,0) node [below] {$B$} coordinate (B);\n\\draw (1,2) node [above] {$D$} coordinate (D);\n\\draw ($(D)+(B)-(A)$) node [above] {$C$} coordinate (C);\n\\draw ($(C)!0.5!(D)$) node [above] {$F$} coordinate (F);\n\\draw ($(A)!{2/3}!(F)$) node [right] {$E$} coordinate (E);\n\\draw (A)--(F) (B)--(D) (A)--(B)--(C)--(D)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414692,7 +416334,9 @@ "id": "016173", "content": "在边长为$1$的等边三角形$ABC$中, $D$为线段$BC$上的动点, $DE \\perp AB$且交$AB$于点$E$. $DF\\parallel AB$且交$AC$于点$F$, 则$|2 \\overrightarrow{BE}+\\overrightarrow{DF}|$的值为\\blank{50}; $(\\overrightarrow{DE}+\\overrightarrow{DF})\\cdot \\overrightarrow{DA}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414712,7 +416356,9 @@ "id": "016174", "content": "已知向量$\\overrightarrow {a}=(\\cos \\dfrac{3}{2} x, \\sin \\dfrac{3}{2} x)$, $\\overrightarrow {b}=(\\cos \\dfrac{x}{2},-\\sin \\dfrac{x}{2})$, 且$x \\in[0, \\dfrac{\\pi}{2}]$.\\\\\n(1) 求$\\overrightarrow {a} \\cdot \\overrightarrow {b}$及$|\\overrightarrow {a}+\\overrightarrow {b}|$;\\\\\n(2) 若$f(x)=\\overrightarrow {a} \\cdot \\overrightarrow {b}-2 \\lambda|\\overrightarrow {a}+\\overrightarrow {b}|$的最小值为$-\\dfrac{3}{2}$, 求实数$\\lambda$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414732,7 +416378,9 @@ "id": "016175", "content": "已知两个力 (单位: $\\text{N}$)$\\overrightarrow{f_1}$与$\\overrightarrow{f_2}$的夹角为$60{^ \\circ}$, 其中$\\overrightarrow{f_1}=(2,0)$. 某质点在这两个力的共同作用下, 由点$A(1,1)$移动至点$B(6,6)$(单位: $\\text{m}$) .\\\\\n(1) 求$\\overrightarrow{f_2}$;\\\\\n(2) 求$\\overrightarrow{f_1}$与$\\overrightarrow{f_2}$的合力对质点所做的功.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414752,7 +416400,9 @@ "id": "016176", "content": "已知$\\triangle ABC$的三个顶点$A$、$B$、$C$的坐标分别是$(1,1)$、$(4,2)$、$(-2,-6)$, 则此三角形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414772,7 +416422,9 @@ "id": "016177", "content": "若两个非零向量$\\overrightarrow {a}, \\overrightarrow {b}$满足$|\\overrightarrow {a}+\\overrightarrow {b}|=|\\overrightarrow {a}-\\overrightarrow {b}|=2|\\overrightarrow {a}|$, 则向量$\\overrightarrow {a}+\\overrightarrow {b}$与$\\overrightarrow {a}-\\overrightarrow {b}$的夹角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414792,7 +416444,9 @@ "id": "016178", "content": "如图所示, 三个边长为$2$的等边三角形有一条边在同一直线上, 边$B_3C_3$上有$10$个不同的点$P_1, P_2, \\cdots, P_{10}$, 记$M_i=\\overrightarrow{AB_2} \\cdot \\overrightarrow{AP_i}$($i=1,2, \\cdots, 10$), 则$M_1+M_2+\\cdots+M_{10}=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) coordinate (A) node [below] {$A$};\n\\draw (2,0) coordinate (C_1) node [below] {$C_1$};\n\\draw (4,0) coordinate (C_2) node [below] {$C_2$};\n\\draw (6,0) coordinate (C_3) node [below] {$C_3$};\n\\foreach \\i in {1,2,3}\n{\\draw (C_\\i) ++ (120:2) coordinate (B_\\i) node [above] {$B_\\i$};};\n\\draw (A) -- ($(B_3)!0.9!(C_3)$) node [right] {$P_1$};\n\\draw (A) -- ($(B_3)!0.7!(C_3)$) node [right] {$P_2$};\n\\draw (A) -- ($(B_3)!0.1!(C_3)$) node [right] {$P_{10}$};\n\\draw ($(B_3)!0.4!(C_3)$) node [right]{$\\cdots$};\n\\draw (A)--(C_3)--(B_3)--(C_2)--(B_2)--(C_1)--(B_1)--(A) (A)--(B_2);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414812,7 +416466,9 @@ "id": "016179", "content": "已知$P$为椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}{2}=1$上任意一点, $Q$与$P$关于$x$轴对称, $F_1$、$F_2$为椭圆的左右焦点, 若$\\overrightarrow{F_1P} \\cdot \\overrightarrow{F_2P} \\leq 1$, 则向量$\\overrightarrow{F_1P}$与$\\overrightarrow{F_2Q}$的夹角的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414832,7 +416488,9 @@ "id": "016180", "content": "已知$A_1$、$A_2$、$A_3$、$A_4$、$A_5$五个点, 满足$\\overrightarrow{A_n A_{n+1}} \\cdot \\overrightarrow{A_{n+1} A_{n+2}}=0$($n=1,2,3$), $|\\overrightarrow{A_n A_{n+1}}| \\cdot|\\overrightarrow{A_{n+1} A_{n+2}}|=n+1(n=1,2,3)$, 则$|\\overrightarrow{A_1A_5}|$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414852,7 +416510,9 @@ "id": "016181", "content": "已知$\\overrightarrow{a_1}$、$\\overrightarrow{a_2}$、$\\overrightarrow{b_1}$、$\\overrightarrow{b_2}$、$\\cdots$、$\\overrightarrow{b_k}$($k \\in \\mathbf{N}$, $k \\ge 1$)是平面内两两互不相等的向量, 满足$|\\overrightarrow{a_1}-\\overrightarrow{a_2}|=1$, 且$|\\overrightarrow{a_i}-\\overrightarrow{b_j}| \\in\\{1,2\\}$(其中$i=1,2$, $j=1,2, \\cdots, k$), 则$k$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -414872,7 +416532,9 @@ "id": "016182", "content": "已知$\\triangle ABC$的三个顶点分别是$A(1, \\dfrac{3}{2})$、$B(4,-2)$、$C(1, y)$, 重心$G(x,-1)$, 则$x$、$y$的值分别是\\bracket{20}.\n\\fourch{$x=2$, $y=5$}{$x=1$, $y=-\\dfrac{5}{2}$}{$x=1$, $y=-1$}{$x=2$, $y=-\\dfrac{5}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -414892,7 +416554,9 @@ "id": "016183", "content": "已知$A$、$B$为平面上的两个定点, 且$|\\overrightarrow{AB}|=2$. 该平面上的动线段$PQ$的端点$P$、$Q$, 满足$|\\overrightarrow{AP}| \\leq 5$, $\\overrightarrow{AP} \\cdot \\overrightarrow{AB}=6$, $\\overrightarrow{AQ}=-2 \\overrightarrow{AP}$, 则动线段$PQ$所形成图形的面积为\\bracket{20}.\n\\fourch{$36$}{$60$}{$81$}{$108$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -414912,7 +416576,9 @@ "id": "016184", "content": "在$\\triangle ABC$中, $D$为$BC$中点, $E$为$AD$中点, 则以下结论:\\textcircled {1} 存在$\\triangle ABC$, 使得$\\overrightarrow{AB} \\cdot \\overrightarrow{CE}=0$;\\textcircled {2} 存在$\\triangle ABC$, 使得$\\overrightarrow{CE}\\parallel(\\overrightarrow{CB}+\\overrightarrow{CA})$; 其中成立的是\\bracket{20}.\n\\fourch{\\textcircled{1}成立,\\textcircled {2} 成立}{\\textcircled{1}成立,\\textcircled {2} 不成立}{\\textcircled{1}不成立,\\textcircled {2} 成立}{\\textcircled{1}不成立,\\textcircled {2}不成立}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -414932,7 +416598,9 @@ "id": "016185", "content": "已知向量$\\overrightarrow {a}=(-\\dfrac{1}{2}, \\dfrac{\\sqrt{3}}{2})$, $\\overrightarrow {b}=(2 \\cos \\theta, 2 \\sin \\theta)$, $0<\\theta<\\pi$.\\\\\n(1) 若$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 求$\\cos \\theta$的值;\\\\\n(2) 若$|\\overrightarrow {a}+\\overrightarrow {b}|=|\\overrightarrow {b}|$, 求$\\sin (\\theta+\\dfrac{\\pi}{6})$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414952,7 +416620,9 @@ "id": "016186", "content": "已知平面向量$\\overrightarrow {a}=(\\sqrt{3},-1)$, $\\overrightarrow {b}=(\\dfrac{1}{2}, \\dfrac{\\sqrt{3}}{2})$.\\\\\n(1) 求$\\overrightarrow {a} \\cdot \\overrightarrow {b}$;\\\\\n(2) 设$\\overrightarrow {c}=\\overrightarrow {a}+(x-3) \\overrightarrow {b}$, $\\overrightarrow {d}=-y \\overrightarrow {a}+x \\overrightarrow {b}$(其中$x \\neq 0$), 著$\\overrightarrow {c} \\perp \\overrightarrow {d}$, 试求函数关系式$y=f(x)$并解不等式$f(x)>7$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414972,7 +416642,9 @@ "id": "016187", "content": "已知抛物线$C: y^2=4 x$, $O$为坐标原点, 直线$l: k x-y-1=0$与抛物线$C$交于不同两点$A$、$B$.\\\\\n(1) 当$k=2$时, 求$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}$的值;\\\\\n(2) 当$k$变化时, 求$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -414992,7 +416664,9 @@ "id": "016188", "content": "已知$\\overrightarrow {a}=(3,4)$、$\\overrightarrow {b}=(\\sin \\alpha, \\cos \\alpha)$, 且$\\overrightarrow {a}$与$\\overrightarrow {b}$平行, 则$\\tan \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415012,7 +416686,9 @@ "id": "016189", "content": "已知$\\overrightarrow {a}=(2,1)$、$\\overrightarrow {b}=(m, 3)$, 若$\\overrightarrow {a}$、$\\overrightarrow {b}$的夹角是锐角, 则$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415032,7 +416708,9 @@ "id": "016190", "content": "已知$A$、$B$、$C$三点共线, 且$A$、$B$、$C$三点的纵坐标分别为$2$、$5$、$10$, 则点$A$分$\\overrightarrow{BC}$所成的比是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415052,7 +416730,9 @@ "id": "016191", "content": "点$A$的坐标是$(3,-2)$, 点$B$在$y$轴上, 且$|\\overrightarrow{AB}|=3 \\sqrt{2}$, 则$\\overrightarrow{AB}$的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415072,7 +416752,9 @@ "id": "016192", "content": "直角坐标平面$xOy$中, 若定点$A(2,2)$与动点$P(x, y)$满足$(\\overrightarrow{OP}-\\overrightarrow{OA}) \\cdot \\overrightarrow{OP}=4$, 则点$P$的轨迹方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415092,7 +416774,9 @@ "id": "016193", "content": "已知的$\\overrightarrow {a}$与$\\overrightarrow {b}$夹角为$\\dfrac{2 \\pi}{3}$, $|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=3$, 若$(3 \\overrightarrow {a}-2 \\overrightarrow {b}) \\perp(2 \\overrightarrow {a}+k \\overrightarrow {b})$, 则实数$k$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415112,7 +416796,9 @@ "id": "016194", "content": "如图, 正六边形$ABCDEF$的边长为$1$, 则$\\overrightarrow{AC} \\cdot \\overrightarrow{DB}=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A) --++ (0:1) node [below] {$B$} coordinate (B) --++ (60:1) node [right] {$C$} coordinate (C) --++ (120:1) node [above] {$D$} coordinate (D) --++ (180:1) node [above] {$E$} coordinate (E) --++ (240:1) node [left] {$F$} coordinate (F) --cycle;\n\\draw [->] (A)--(C);\n\\draw [->] (D)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415132,7 +416818,9 @@ "id": "016195", "content": "$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$是平面上的非零向量, 且两两不平行, 下列结论:\\\\\n\\textcircled{1} $(\\overrightarrow {a} \\cdot \\overrightarrow {b}) \\overrightarrow {c}=\\overrightarrow {a}(\\overrightarrow {b} \\cdot \\overrightarrow {c})$;\\\\\n\\textcircled{2} 若$\\overrightarrow {a} \\cdot \\overrightarrow {b}=\\overrightarrow {c} \\cdot \\overrightarrow {b}$, 则$\\overrightarrow {a}=\\overrightarrow {c}$;\\\\\n\\textcircled{3} $|\\overrightarrow {a} \\cdot \\overrightarrow {b}| \\leq|\\overrightarrow {a}||\\overrightarrow {b}|$;\\\\\n\\textcircled{4} 若$\\overrightarrow {a} \\cdot \\overrightarrow {b}=0$, 则$\\overrightarrow {a}=\\overrightarrow{0}$或$\\overrightarrow {b}=\\overrightarrow{0}$;\\\\\n\\textcircled{5} 若$k \\overrightarrow {a}=\\overrightarrow{0}$, 则$k=0$;\\\\\n\\textcircled{6} 对任意实数$t$, 都有$|\\overrightarrow {a}-t \\overrightarrow {b}| \\geq|\\overrightarrow {a}-\\overrightarrow {b}|$成立, 则$(\\overrightarrow {a}-\\overrightarrow {b}) \\perp \\overrightarrow {b}$.\\\\\n其中所有正确结论的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415152,7 +416840,9 @@ "id": "016196", "content": "如图, 在$\\triangle ABC$中, 点$O$是$BC$的中点, 过点$O$的直线分别交直线$AB$、$AC$于不同的两点$M$、$N$, 若$\\overrightarrow{AB}=m \\overrightarrow{AM}$, $\\overrightarrow{AC}=n \\overrightarrow{AN}$, 则$m+n$的值为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0) node [right] {$C$} coordinate (C);\n\\draw (1.2,2) node [above] {$A$} coordinate (A);\n\\draw ($(B)!0.5!(C)$) node [below] {$O$} coordinate (O);\n\\draw (A)--(B)--(C)--cycle;\n\\draw ($(A)!1.5!(B)$) node [left] {$M$} coordinate (M);\n\\draw ($(C)!0.25!(A)$) node [right] {$N$} coordinate (N);\n\\draw (M)--(N)(B)--(M);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415172,7 +416862,9 @@ "id": "016197", "content": "点$O$是平面上一定点, $A$、$B$、$C$是此平面上不共线的三个点, 动点$P$满足$\\overrightarrow{OP}=\\overrightarrow{OA}+\\lambda(\\dfrac{\\overrightarrow{AB}}{|\\overrightarrow{AB}|}+\\dfrac{\\overrightarrow{AC}}{|\\overrightarrow{AC}|}), \\lambda \\in[0,+\\infty)$, 则点$P$的轨迹一定通过$\\triangle ABC$的\\blank{50}心.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415192,7 +416884,9 @@ "id": "016198", "content": "已知$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=1$且$\\overrightarrow {a} \\cdot \\overrightarrow {b}=0$, 若向量$\\overrightarrow {c}$满足$(\\overrightarrow {c}-\\overrightarrow {a}) \\cdot(\\overrightarrow {c}-\\overrightarrow {b})=0$ , 则$|\\overrightarrow {c}|$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415212,7 +416906,9 @@ "id": "016199", "content": "如图, $O$是线段$AB$外一点, $|OA|=3$, $|OB|=2$, $P$是线段$AB$的垂直平分线$l$上的动点. 求$\\overrightarrow{OP} \\cdot \\overrightarrow{AB}=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0) node [below] {$O$} coordinate (O);\n\\draw (-2,0) node [left] {$B$} coordinate (B);\n\\draw (50:3) node [above] {$A$} coordinate (A);\n\\path [name path = arc1] (B)++ (3,0) arc (0:-15:3);\n\\path [name path = arc2] (A)++ (-120:3) arc (-120:-105:3);\n\\path [name intersections = {of = arc1 and arc2, by = P}];\n\\draw (O)--(P) node [right] {$P$};\n\\draw ($(A)!0.5!(B)$) coordinate (Q);\n\\draw (P)--($(P)!1.5!(Q)$);\n\\draw (A)--(B)--(O)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415232,7 +416928,9 @@ "id": "016200", "content": "若$\\overrightarrow{AB} \\cdot \\overrightarrow{BC}+\\overrightarrow{AB}^2=0$, 则$\\triangle ABC$必定是\\bracket{20}.\n\\fourch{锐角三角形}{直角三角形}{钝角三角形}{等腰直角三角形}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -415252,7 +416950,9 @@ "id": "016201", "content": "设向量$\\overrightarrow {a}=(x-1,1)$, $\\overrightarrow {b}=(3, x+1)$, 则``$\\overrightarrow {a}\\parallel \\overrightarrow {b}$''是``$x=2$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充分必要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -415272,7 +416972,9 @@ "id": "016202", "content": "设$A_1$、$A_2$、$A_3$、$A_4$是平面上给定的$4$个不同的点, 则使$\\overrightarrow{MA_1}+\\overrightarrow{MA_2}+\\overrightarrow{MA_3}+\\overrightarrow{MA_4}=\\overrightarrow{0}$成立的点$M$的个数为\\bracket{20}.\n\\fourch{$0$}{$1$}{$2$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -415292,7 +416994,9 @@ "id": "016203", "content": "在$\\triangle ABC$中, 有命题: \\textcircled {1} $\\overrightarrow{AB}-\\overrightarrow{AC}=\\overrightarrow{BC}$; \\textcircled {2} $\\overrightarrow{AB}+\\overrightarrow{BC}+\\overrightarrow{CA}=\\overrightarrow{0}$; \\textcircled {3} 若$(\\overrightarrow{AB}+\\overrightarrow{AC}) \\cdot(\\overrightarrow{AB}-\\overrightarrow{AC})=0$, 则$\\triangle ABC$为等腰三角形; \\textcircled {4} 若$\\overrightarrow{AC} \\cdot \\overrightarrow{AB}>0$, 则$\\triangle ABC$为锐角三角形. 上述命题中正确的是\\bracket{20}.\n\\fourch{\\textcircled {2}\\textcircled {3}}{\\textcircled {1}\\textcircled {4}}{\\textcircled {1}\\textcircled {2}}{\\textcircled {2}\\textcircled {3}\\textcircled {4}}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -415312,7 +417016,9 @@ "id": "016204", "content": "在直角坐标系$xOy$中, 已知点$P(2 \\cos x, \\cos 2 x+1)$和点$Q(\\sin x,-1)$, 其中$x \\in[0, \\pi]$. 若向量$\\overrightarrow{OP}$与$\\overrightarrow{OQ}$垂直, 求$x$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415332,7 +417038,9 @@ "id": "016205", "content": "已知向量$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为$60{^ \\circ}$, 且$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=2$, 设$\\overrightarrow {m}=3 \\overrightarrow {a}-\\overrightarrow {b}$, $\\overrightarrow {n}=t \\overrightarrow {a}+2 \\overrightarrow {b}$.\\\\\n(1) 若$\\overrightarrow {m} \\perp \\overrightarrow {n}$, 求实数$t$的值;\\\\\n(2) 若$\\overrightarrow {m}\\parallel \\overrightarrow {n}$, 求实数$t$的值;\\\\\n(3) 当$t=2$时, 求$\\overrightarrow {m}, \\overrightarrow {n}$的夹角.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415352,7 +417060,9 @@ "id": "016206", "content": "已知$M(2 \\cos x, 1)$, $N(\\cos x, \\sqrt{3} \\sin 2 x+a)$, $a \\in \\mathbf{R}$, 且$f(x)=\\overrightarrow{OM} \\cdot \\overrightarrow{ON}$.\\\\\n(1) 若$x \\in \\mathbf{R}$, 求$f(x)$的严格递增区间;\\\\\n(2) 若$x \\in[0, \\dfrac{\\pi}{2}], f(x)$的最大值是$4$, 求$a$的值;\\\\\n(3) 在 (2) 的条件下, 求满足$f(x)=1$, $x \\in[-\\pi, \\pi]$的$x$的集合.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415372,7 +417082,9 @@ "id": "016207", "content": "若向量$\\overrightarrow {a}=(\\cos \\alpha, \\sin \\alpha)$, $\\overrightarrow {b}=(\\cos \\beta, \\sin \\beta)$, 且$|k \\overrightarrow {a}+\\overrightarrow {b}|=\\sqrt{3}|\\overrightarrow {a}-k \\overrightarrow {b}|$($k \\neq 0$).\\\\\n(1) 用$k$表示$\\overrightarrow {a} \\cdot \\overrightarrow {b}$;\\\\\n(2) 当$k>0$时, 求$\\overrightarrow {a} \\cdot \\overrightarrow {b}$的最小值, 并求此时$\\overrightarrow {a}$与$\\overrightarrow {b}$夹角大小;\\\\\n(3) 是否存在$k \\in \\mathbf{R}$, 使$\\overrightarrow {a}$与$\\overrightarrow {b}$的夹角为钝角? 若存在, 求$k$的取值范围; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415392,7 +417104,9 @@ "id": "016208", "content": "已知向量$\\overrightarrow {a}=(x^2+1, p+2)$, $\\overrightarrow {b}=(3, x)$, $f(x)=\\overrightarrow {a} \\cdot \\overrightarrow {b}$, $p$是实数.\\\\\n(1) 若存在唯一实数$x$, 使$\\overrightarrow {a}+\\overrightarrow {b}$与$\\overrightarrow {c}=(1,2)$平行, 试求$p$的值;\\\\\n(2) 若函数$y=f(x)$是偶函数, 试求函数$y=|f(x)-15|$在区间$[-1,3]$上的值域;\\\\\n(3) 若函数$f(x)$在区间${[-\\dfrac{1}{2},+\\infty)}$上是增函数, 试讨论方程$f(x)+\\sqrt{x}-p=0$解的个数, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415412,7 +417126,9 @@ "id": "016209", "content": "已知复数$z$满足$z+2 \\overline {z}=6+\\mathrm{i}$, 则$z$的实部为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415432,7 +417148,9 @@ "id": "016210", "content": "给出以下式子: \\textcircled{1} $5 \\mathrm{i}>\\mathrm{i}$; \\textcircled{2} $|2+3 \\mathrm{i}|>|-1+4 \\mathrm{i}|$; \\textcircled{3} $\\mathrm{i}^2>-\\mathrm{i}$; \\textcircled{4} $|2+\\mathrm{i}|>2 \\cdot \\mathrm{i}^8$, 其中正确的不等式的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415452,7 +417170,9 @@ "id": "016211", "content": "有下列命题: \\textcircled{1} 两个复数一定不能比较大小; \\textcircled{2} $a+b \\mathrm{i}=1+\\mathrm{i}$($a$、$b \\in \\mathbf{C}$)的充要条件是$a=b=1$; \\textcircled{3} 若实数$x$与复数$x \\mathrm{i}$对应, 则实数集与纯虚数集一一对应; \\textcircled{4} 若$a$、$b$是复数, 则$a^2+b^2=0$的充要条件是$a=b=0$. 其中错误命题的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415472,7 +417192,9 @@ "id": "016212", "content": "在复平面上, 平行于$y$轴的非零向量所对应的复数一定是\\bracket{20}.\n\\fourch{实数}{虚数}{纯虚数}{实数或纯虚数}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -415492,7 +417214,9 @@ "id": "016213", "content": "复数$\\dfrac{2-\\mathrm{i}}{1-3 \\mathrm{i}}$在复平面内对应的点所在的象限为\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -415512,7 +417236,9 @@ "id": "016214", "content": "当且仅当$a$取何实数值时, 复数$z=\\dfrac{a^2-a-6}{a+3}+(a^2-2 a-15) \\mathrm{i}$是:\\\\\n(1) 实数;\\\\\n(2) 虚数;\\\\\n(3) 纯虚数.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415532,7 +417258,9 @@ "id": "016215", "content": "已知$x$是实数, $y$是纯虚数, 且满足$(2 x-1)+(3-y) \\mathrm{i}=y-\\mathrm{i}$, 求$x$、$y$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415552,7 +417280,9 @@ "id": "016216", "content": "已知$x$、$y \\in \\mathbf{R}$, 且满足$(2 x-1)+(3-y) \\mathrm{i}=y-\\mathrm{i}$, 求$x$、$y$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415572,7 +417302,9 @@ "id": "016217", "content": "已知复数$z_1$满足$(1+\\mathrm{i}) z_1=-1+5 \\mathrm{i}$, $z_2=a-2-\\mathrm{i}$, 其中$\\mathrm{i}$为虚数单位, $a \\in \\mathbf{R}$, 若$|z_1-\\overline{z_2}|<|z_1|$, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415592,7 +417324,9 @@ "id": "016218", "content": "设虚数$z$满足$|2 z+5|=|\\overline {z}+10|$.\\\\\n(1) 求$|z|$的值;\\\\\n(2) 若$\\dfrac{z}{m}+\\dfrac{m}{z}$为实数, 求实数$m$的值;\\\\\n(3) 若$(1-2 \\mathrm{i}) z$在复平面上对应的点在第一、三象限的角平分线上, 求复数$z$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415612,7 +417346,9 @@ "id": "016219", "content": "设全集$U=\\mathbf{C}$, $A=\\{z|||z|-1|=1-| z |,\\ z \\in \\mathbf{C}\\}$, $B=\\{z|| z |<1,\\ z \\in \\mathbf{C}\\}$, 若$z \\in A \\cap \\overline {B}$, 求复数$z$在复平面内所对应的点的轨迹.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415632,7 +417368,9 @@ "id": "016220", "content": "已知$z=1-3 \\mathrm{i}$, 则$|\\overline {z}-\\mathrm{i}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415652,7 +417390,9 @@ "id": "016221", "content": "已知$a \\in \\mathbf{R}$, $(1+a \\mathrm{i}) \\mathrm{i}=3+\\mathrm{i}$, ($\\mathrm{i}$为虚数单位), 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415672,7 +417412,9 @@ "id": "016222", "content": "已知$|z_1|>|z_2|$, 当$z_1=a$, $z_2=-2+\\mathrm{i}$时, 实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415692,7 +417434,9 @@ "id": "016223", "content": "设$z=(1+\\mathrm{i}) \\sin \\alpha-(1+\\mathrm{i} \\cos \\alpha)$($\\alpha \\in \\mathbf{R}$), 当$\\alpha=$\\blank{50} 时, $z \\in \\mathbf{R}$; 当$\\alpha=$\\blank{50}时, $z$是纯虚数.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415712,7 +417456,9 @@ "id": "016224", "content": "已知$z \\in \\mathbf{C}$, 且$1<|z| \\leq 2$, 则$z$在复平面内对应的点所表示的图形为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415732,7 +417478,9 @@ "id": "016225", "content": "已知复数$z$满足$z+\\dfrac{3}{z}=0$, 则$|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415752,7 +417500,9 @@ "id": "016226", "content": "复数$z=(1-t+t^2)-\\sqrt{1+t^2} \\mathrm{i}$($t \\in \\mathbf{R}$)在复平面内所对应的点在\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -415772,7 +417522,9 @@ "id": "016227", "content": "下列四个命题: \\textcircled{1} 满足$z=\\dfrac{1}{z}$的复数只有$\\pm 1$, $\\pm \\mathrm{i}$; \\textcircled{2} 若$a$、$b \\in \\mathbf{R}$, 且$a=b$, 则$(a-b)+(a+b)\\mathrm{i}$是纯虚数; \\textcircled{3} 复数$z \\in \\mathbf{R}$的充要条件是$z=\\overline {z}$; \\textcircled{4} 若$z_1, z_2 \\in \\mathbf{C}$, 则$z_1^2+z_2^2=0 \\Leftrightarrow z_1=z_2=0$. 其中正确的有\\bracket{20}.\n\\fourch{1 个}{2 个}{3 个}{4 个}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -415792,7 +417544,9 @@ "id": "016228", "content": "在复平面上, 若正方形$OMNP$(按逆时针方向排列, $O$为原点) 中, 顶点$M$对应的复数为$1+2 \\mathrm{i}$, 则$\\overrightarrow{NP}$对应的复数为\\bracket{20}.\n\\fourch{$1-2 \\mathrm{i}$}{$-1+2 \\mathrm{i}$}{$-1-2 \\mathrm{i}$}{$-2-\\mathrm{i}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -415812,7 +417566,9 @@ "id": "016229", "content": "已知复数$z$满足$|z-\\mathrm{i}|=|z-1|$, 求复数$z$所对应的点$Z$的轨迹的直角坐标方程, 并指出点$Z$的轨迹.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415832,7 +417588,9 @@ "id": "016230", "content": "已知复数$z=(x-2)+y \\mathrm{i}$($x, y \\in \\mathbf{R}$)的模为$\\sqrt{3}$, 求$\\dfrac{y}{x}$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415852,7 +417610,9 @@ "id": "016231", "content": "设$z=\\log _2(1+m)+\\mathrm{i}\\log_{\\frac{1}{2}}(3-m)$($m \\in \\mathbf{R}$).\\\\\n(1) 若$z$是虚数, 求$m$的取值范围;\\\\\n(2) 若$z$在复平面内对应的点在第三象限, 求$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415872,7 +417632,9 @@ "id": "016232", "content": "在复平面内, 复数$z$满足$(1-\\mathrm{i}) z=2$, 则$z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415892,7 +417654,9 @@ "id": "016233", "content": "已知复数$z=\\dfrac{1+3 \\mathrm{i}}{1-\\mathrm{i}}$, 则$z$的实部为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415912,7 +417676,9 @@ "id": "016234", "content": "若复数$z$满足$(3-4 \\mathrm{i}) z=|4+3 \\mathrm{i}|$, 则$z$的虚部为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415932,7 +417698,9 @@ "id": "016235", "content": "设$2(z+\\overline {z})+3(z-\\overline {z})=4+6 \\mathrm{i}$, 则$z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415952,7 +417720,9 @@ "id": "016236", "content": "若$\\omega=-\\dfrac{1}{2}+\\dfrac{\\sqrt{3}}{2} \\mathrm{i}$, 则$(1+\\omega)(1+\\omega-\\omega^2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -415972,7 +417742,9 @@ "id": "016237", "content": "计算:\\\\\n(1) $\\dfrac{3-4 \\mathrm{i}}{1+2 \\mathrm{i}}+(2+\\mathrm{i}^{15})-(1-\\mathrm{i})^6$;\\\\\n(2) $\\dfrac{(-1+\\sqrt{3} \\mathrm{i})^3}{(1+\\mathrm{i})^6}-\\dfrac{-2+\\mathrm{i}}{1+2 \\mathrm{i}}$;\\\\\n(3) $|\\dfrac{-\\mathrm{i}(3-4 \\mathrm{i})^5}{(1-2 \\mathrm{i})^{10}}|$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -415992,7 +417764,9 @@ "id": "016238", "content": "已知$z=1-2 \\mathrm{i}$, 求:\\\\\n(1) $\\dfrac{2}{z+\\mathrm{i}}-(\\overline {z}+1)$的值;\\\\\n(2) 求适合不等式$\\dfrac{|a z-\\mathrm{i}|}{\\sqrt{a+1}} \\geq\\sqrt{\\dfrac{1}{2}}$的实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416012,7 +417786,9 @@ "id": "016239", "content": "已知$z=-\\dfrac{1+\\mathrm{i}}{\\sqrt{2}}$, 求$z^{100}+z^{50}+1$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416032,7 +417808,9 @@ "id": "016240", "content": "已知$z=-\\dfrac{2}{1+\\sqrt{3} \\mathrm{i}}$, 求$1+z+z^2+\\cdots+z^9$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416052,7 +417830,9 @@ "id": "016241", "content": "设$z$是虚数, $\\omega=z+\\dfrac{1}{z}$是实数, 且$-1<\\omega<2$.\\\\\n(1) 求$|z|$的值及$z$的实部的取值范围;\\\\\n(2) 设$\\mu=\\dfrac{1-z}{1+z}$, 求证$\\mu$为纯虚数;\\\\\n(3) 求$\\omega-\\mu^2$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416072,7 +417852,9 @@ "id": "016242", "content": "已知$z=2-\\mathrm{i}$, 则$z(\\overline {z}+\\mathrm{i})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416092,7 +417874,9 @@ "id": "016243", "content": "复数$(1-\\mathrm{i})^3$的虚部为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416112,7 +417896,9 @@ "id": "016244", "content": "设复数$z$满足$(1-\\mathrm{i}) z=2 \\mathrm{i}$, 则$z=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416132,7 +417918,9 @@ "id": "016245", "content": "已知复数$z=\\dfrac{5 \\mathrm{i}}{1+2 \\mathrm{i}}$($\\mathrm{i}$是虚数单位), 则$|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416152,7 +417940,9 @@ "id": "016246", "content": "设复数: $z_1=1+\\mathrm{i}$, $z_2=x+2 \\mathrm{i}$($x \\in \\mathbf{R}$), 若$z_1 z_2$为实数, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416172,7 +417962,9 @@ "id": "016247", "content": "已知$z \\cdot \\overline {z}=4$, 则$|1+\\sqrt{3} \\mathrm{i}+z|$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416192,7 +417984,9 @@ "id": "016248", "content": "已知$a$、$b \\in \\mathbf{R}$, $\\mathrm{i}$是虚数单位, 若$a+\\mathrm{i}=2-b \\mathrm{i}$, 则$(a+b \\mathrm{i})^2=$\\bracket{20}.\n\\fourch{$3-4 \\mathrm{i}$}{$3+4 \\mathrm{i}$}{$4-3 \\mathrm{i}$}{$4+3 i$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -416212,7 +418006,9 @@ "id": "016249", "content": "在复平面内, $O$为坐标原点, 已知向量$\\overrightarrow{OA}$对应的复数为$1-\\mathrm{i}$, 若将向量$\\overrightarrow{OA}$向右平移$1$个单位后得到$\\overline{O'A'}$, 则点$A'$所对应的复数为\\bracket{20}.\n\\fourch{$-\\mathrm{i}$}{$1$}{$1-2 \\mathrm{i}$}{$2-\\mathrm{i}$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -416232,7 +418028,9 @@ "id": "016250", "content": "欧拉公式$\\mathrm{e}^{\\mathrm{i} \\theta}=\\cos \\theta+\\mathrm{i} \\sin \\theta$($\\mathrm{e}$是自然对数的底数, $\\mathrm{i}$是虚数单位) 是由瑞士著名数学家欧拉发现的, 它将三角函数的定义域扩大到复数, 建立了三角函数和指数函数的关系, 它在复变函数论里占有非常重要的地位, 当$\\theta=\\pi$时, 就有$\\mathrm{e}^{\\mathrm{i} \\pi}+1=0$, 根据上述背景知识试判断$\\mathrm{e}^{-\\frac{\\pi}{3} \\mathrm{i}}$表示的复数在复平面对应的点位于\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -416252,7 +418050,9 @@ "id": "016251", "content": "设$\\omega=z+a \\mathrm{i}$, 其中$a$是实数, $\\mathrm{i}$是虚数单位, $z=\\dfrac{(1-4 \\mathrm{i})(1+\\mathrm{i})+2+4 \\mathrm{i}}{3+4 \\mathrm{i}}$, 且$|\\omega| \\leq \\sqrt{2}$, 求$a$的范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416272,7 +418072,9 @@ "id": "016252", "content": "设$z \\in \\mathbf{C}$, $z+|\\overline {z}|=2+\\mathrm{i}$, 求$z$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416292,7 +418094,9 @@ "id": "016253", "content": "设$z+\\dfrac{4}{z} \\in \\mathbf{R}$, $|z-2|=2$, 求$z$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416312,7 +418116,9 @@ "id": "016254", "content": "设$z=a+b \\mathrm{i}$($a$、$b \\in \\mathbf{R}$, $|a| \\neq 1$), $|z|=1$.\\\\\n(1) 求证: $u=\\dfrac{z+1}{z-1}$是纯虚数;\\\\\n(2) 求$|z+2 \\overline {z}+2|$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416332,7 +418138,9 @@ "id": "016255", "content": "在复数范围内, 方程$x^2+2=0$的解是\\blank{50}; 方程$x^2-4 x+5=0$的解为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416352,7 +418160,9 @@ "id": "016256", "content": "设$m \\in \\mathbf{R}$, 若$z$是关于$x$的方程$x^2+m x+m^2-1=0$的一个虚根, 则$|\\overline {z}|$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416372,7 +418182,9 @@ "id": "016257", "content": "实系数方程$x^2+a x+b=0$的一个根是$\\dfrac{2}{\\mathrm{i}}$, 则$a=$\\blank{50}, $b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416392,7 +418204,9 @@ "id": "016258", "content": "若关于$x$的实系数一元二次方程的两个根分别是$x_1=1+\\sqrt{3} \\mathrm{i}$, $x_2=1-\\sqrt{3} \\mathrm{i}$, 则这个方程可以是\\bracket{20}.\n\\fourch{$x^2-2 x+2=0$}{$x^2-2 x+4=0$}{$x^2+3 x+2=0$}{$x^2+2 x+4=0$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -416412,7 +418226,9 @@ "id": "016259", "content": "二次方程$x^2-3 \\mathrm{i} x-3=0$的根的情况是\\bracket{20}.\n\\fourch{有两个不相等的实根}{有两个虚根}{有一对共轭虚根}{有一个实根一个虚根}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -416432,7 +418248,9 @@ "id": "016260", "content": "已知方程$x^2+a x+b=0$($a$、$b \\in \\mathbf{R}$)的一个根是$1+\\sqrt{3} \\mathrm{i}$, 求$a$、$b$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416452,7 +418270,9 @@ "id": "016261", "content": "已知方程$x^2-4 x+k=0$有一个虚根为$1-2 \\mathrm{i}$, 求$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416472,7 +418292,9 @@ "id": "016262", "content": "已知方程$x^2-4 \\mathrm{i} x+k=0$有实根, 求实数$k$的值的集合.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416492,7 +418314,9 @@ "id": "016263", "content": "证明: 在复数范围内, 方程$|z|^2+(1-\\mathrm{i}) \\overline {z}-(1+\\mathrm{i}) z=\\dfrac{5-5 \\mathrm{i}}{2+\\mathrm{i}}(\\mathrm{i}$为虚数单位) 无解.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416512,7 +418336,9 @@ "id": "016264", "content": "方程$x^2-2 \\sqrt{2} x+m=0$的两个虚根为$\\alpha$、$\\beta$, 且$|\\alpha-\\beta|=3$, 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416532,7 +418358,9 @@ "id": "016265", "content": "方程$x^2-2 \\sqrt{2} x+m=0$的两个复数根为$\\alpha, \\beta$, 且$|\\alpha-\\beta|=3$, 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416552,7 +418380,9 @@ "id": "016266", "content": "方程$x^2-2 \\sqrt{2} x+m=0$的两个复数根为$\\alpha$、$\\beta$, 且$|\\alpha|+|\\beta|=3$, 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416572,7 +418402,9 @@ "id": "016267", "content": "已知方程$x^2-2 \\sqrt{2} x+m=0$至少有一个模为$\\sqrt{2}$的根, 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416592,7 +418424,9 @@ "id": "016268", "content": "若复数$z=1+\\mathrm{i}$($\\mathrm{i}$为虚数单位) 是方程$x^2+c x+d=0$($c$、$d$均为实数) 的一个根, 则$|c+d \\mathrm{i}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416612,7 +418446,9 @@ "id": "016269", "content": "在复数范围内分解因式: $2 x^3+2 x^2+5 x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416632,7 +418468,9 @@ "id": "016270", "content": "在复数集中, 方程$x(x-2)+4=0$的解是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416652,7 +418490,9 @@ "id": "016271", "content": "实系数一元二次方程$x^2+a x+b=0$的一根为$x_1=\\dfrac{3+\\mathrm{i}}{1+\\mathrm{i}}$(其中$\\mathrm{i}$为虚数单位), 则$a+b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416672,7 +418512,9 @@ "id": "016272", "content": "已知复数$z=a+b \\mathrm{i}$($a,b \\in (0,+\\infty)$)($\\mathrm{i}$是虚数单位) 是方程$x^2-4 x+5=0$的根. 复数$w=u+3 \\mathrm{i}$($u \\in \\mathbf{R}$)满足$|w-z|<2 \\sqrt{5}$, 则$u$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416692,7 +418534,9 @@ "id": "016273", "content": "已知实系数一元二次方程$x^2-2 x+m=0$的两根为$x_1, x_2$, 且$|x_1|+|x_2|=8$. 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416712,7 +418556,9 @@ "id": "016274", "content": "以下三个命题: \\textcircled{1} 在复数范围内, 方程$a x^2+b x+c=0$($a$、$b$、$c \\in \\mathbf{R}$, $a \\neq 0)$总有两个根; \\textcircled{2} 若$1+2 \\mathrm{i}$是方程$x^2+p x+q=0$的一个根, 则这个方程的另一根是$1-2 \\mathrm{i}$; \\textcircled{3} 实系数一元二次方程的两个根满足韦达定理. 其中真命题的个数是\\bracket{20}.\n\\fourch{0}{1}{2}{3}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -416732,7 +418578,9 @@ "id": "016275", "content": "给出下列命题, 其中正确的命题是\\bracket{20}.\n\\twoch{若$z \\in \\mathbf{C}$, 且$z^2<0$, 那么$z$一定是纯虚数}{若$z_1, z_2 \\in \\mathbf{C}$且$z_1-z_2>0$, 则$z_1>z_2$}{若$z \\in \\mathbf{R}$, 则$z \\cdot \\overline {z}=|z|^2$不成立}{若$x \\in \\mathbf{C}$, 则方程$x^3=2$只有一个根}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -416752,7 +418600,9 @@ "id": "016276", "content": "若$a \\in \\mathbf{R}$, 则``关于$x$的方程$x^2+a x+1=0$无实根''是``$z=(2 a-1)+(a-1) \\mathrm{i}$(其中$\\mathrm{i}$表示虚数单位) 在复平面上对应的点位于第四象限''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -416772,7 +418622,9 @@ "id": "016277", "content": "已知实数$p$满足不等式$\\dfrac{2 x+1}{x+2}<0$, 试判断方程$z^2-2 z+5-p^2=0$有无实根, 并证明.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416792,7 +418644,9 @@ "id": "016278", "content": "在复数范围内解方程$|z|^2+(z+\\overline {z}) \\mathrm{i}=\\dfrac{3-\\mathrm{i}}{2+\\mathrm{i}}$($\\mathrm{i}$为虚数单位).", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416812,7 +418666,9 @@ "id": "016279", "content": "已知复数$w$满足$w-4=(3-2 w) \\mathrm{i}$($\\mathrm{i}$为虚数单位), $z=\\dfrac{5}{w}+|w-2|$, 求一个以$z$为根的实系数一元二次方程.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416832,7 +418688,9 @@ "id": "016280", "content": "关于$z$的方程$z^2+(x+y \\mathrm{i}) z+1=0$($x, y \\in \\mathbf{R}$)在$z \\in[\\dfrac{1}{2}, 2]$内有解, 求$x+y$的范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -416852,7 +418710,9 @@ "id": "016281", "content": "计算$\\dfrac{\\mathrm{i}-2}{1+2 \\mathrm{i}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416872,7 +418732,9 @@ "id": "016282", "content": "$3-4 \\mathrm{i}$的平方根为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416892,7 +418754,9 @@ "id": "016283", "content": "若复数$z$满足$z=\\mathrm{i}(2-z)$($\\mathrm{i}$是虚数单位), 则$\\overline {z}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416912,7 +418776,9 @@ "id": "016284", "content": "设$x$、$y$为实数, 且$\\dfrac{x}{1-\\mathrm{i}}+\\dfrac{y}{1-2 \\mathrm{i}}=\\dfrac{5}{1-3 \\mathrm{i}}$, 则$x+y=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416932,7 +418798,9 @@ "id": "016285", "content": "若复数$z=\\dfrac{(\\sqrt{2}+\\sqrt{2} \\mathrm{i})^3(4+5 \\mathrm{i})}{5-4 \\mathrm{i}}$, 则$|z|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416952,7 +418820,9 @@ "id": "016286", "content": "关于未知数$x$的实系数一元二次方程$x^2-b x+c=0$的一个根是$1+3 \\mathrm{i}$(其中$\\mathrm{i}$为虚数单位), 这个一元二次方程可以为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416972,7 +418842,9 @@ "id": "016287", "content": "设$x=2+\\mathrm{i}$($\\mathrm{i}$为虚数单位), $M=1-\\mathrm{C}_{12}^1 x+\\mathrm{C}_{12}^2 x^2-\\mathrm{C}_{12}^3 x^3+\\cdots-\\mathrm{C}_{12}^{11} x^{11}+\\mathrm{C}_{12}^{12} x^{12}$, 则$M=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -416992,7 +418864,9 @@ "id": "016288", "content": "若复数$z$满足$z^2+9=0$, 则$z$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417012,7 +418886,9 @@ "id": "016289", "content": "设$z=1+\\mathrm{i}$($\\mathrm{i}$是虚数单位), 则$\\dfrac{2}{z}+z^2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417032,7 +418908,9 @@ "id": "016290", "content": "已知复数$z_1=3-\\mathrm{i}$, $z_2=2 \\mathrm{i}-1$, 则复数$\\dfrac{\\mathrm{i}}{z_1}-\\dfrac{\\overline{z_2}}{4}$的虚部等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417052,7 +418930,9 @@ "id": "016291", "content": "设复数$z=(a-2 \\sin ^2 \\theta)+(1+2 \\cos \\theta) \\mathrm{i}, a \\in \\mathbf{R}$, $\\theta \\in(0, \\pi)$. 已知$z$是方程$x^2-2 x+5=0$的一个根, 且$z$在复平面内对应的点位于第一象限, 则$\\theta=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417072,7 +418952,9 @@ "id": "016292", "content": "已知复数集合$A=\\{x+y \\mathrm{i}|| x|\\leq 1, \\ | y |\\leq 1,\\ x, y \\in \\mathbf{R}\\}$, $B=\\{z_2 | z_2=(\\dfrac{3}{4}+\\dfrac{3}{4} \\mathrm{i}) z_1,\\ z_1 \\in A\\}$, 其中$\\mathrm{i}$为虚数单位. 若复数$z \\in A \\cap B$, 则$z$对应的点$Z$在复平面内所形成图形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417092,7 +418974,9 @@ "id": "016293", "content": "设$a$、$b \\in \\mathbf{R}, \\mathrm{i}$是虚数单位, 则``$a b=0$''是``复数$a+\\dfrac{b}{\\mathrm{i}}$为纯虚数''的\\bracket{20}.\n\\twoch{充要条件}{充分不必要条件}{必要不充分条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -417112,7 +418996,9 @@ "id": "016294", "content": "已知$z_1$、$z_2$、$z_3 \\in \\mathbf{C}$, 下列结论中正确的是\\bracket{20}.\n\\twoch{若$z_1^2+z_2^2+z_3^2=0$, 则$z_1=z_2=z_3=0$}{若$z_1^2+z_2^2+z_3^2>0$, 则$z_1^2+z_2^2>-z_3^2$}{若$z_1^2+z_2^2>-z_3^2$, 则$z_1^2+z_2^2+z_3^2>0$}{若$z_1+z_1=0$, 则$z_1$是纯虚数}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -417132,7 +419018,9 @@ "id": "016295", "content": "在复数集内, 下列结论中正确的是\\bracket{20}.\n\\onech{若$|z| \\leq a$($a>0$), 则$-a \\leq z \\leq a$}{若$z^n=1$, 则$n=0$}{$z_1 \\overline{z_2}+\\overline{z_1} z_2$一定是实数}{若方程$a x^2+b x+c=0$有实数根, 则$\\Delta=b^2-4 a c \\geq 0$}", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -417152,7 +419040,9 @@ "id": "016296", "content": "已知复数$z=\\cos \\theta+\\mathrm{i} \\sin \\theta$($-\\dfrac{\\pi}{2}<\\theta<\\dfrac{\\pi}{2}$)(其中$\\mathrm{i}$为虚数单位)下列说法中正确的有\\blank{50}.\\\\\n\\textcircled{1} 复数$z$在复平面上对应的点可能落在第二象限; \\textcircled{2} $z$可能为实数; \\textcircled{3} $|z|=1$; \\textcircled{4} $\\dfrac{1}{z}$的虚部为$\\sin \\theta$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417172,7 +419062,9 @@ "id": "016297", "content": "已知$z \\in \\mathbf{C}$, 且$|z|-\\mathrm{i}=\\overline {z}+2+3 \\mathrm{i}(\\mathrm{i}$为虚数单位), 求$\\dfrac{z}{2+\\mathrm{i}}$.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417192,7 +419084,9 @@ "id": "016298", "content": "已知复数$z_0=\\sqrt{2 a+1}+a \\mathrm{i}$, $z=z_0-|z_0|+1-(1+\\sqrt{2}) \\mathrm{i}$, 试问: 是否存在实数$a$, 使得$z$为纯虚数? 若存在, 求出$a$的值; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417212,7 +419106,9 @@ "id": "016299", "content": "已知复数$z_1=\\sin x+\\lambda \\mathrm{i}, z_2=(\\sin x+\\sqrt{3} \\cos x)-\\mathrm{i}$($\\lambda, x \\in \\mathbf{R}$, $\\mathrm{i}$为虚数单位).\\\\\n(1) 若$2 z_1=z_2 \\mathrm{i}$, 且$x \\in(0, \\pi)$, 求$x$与$\\lambda$的值;\\\\\n(2) 设复数$z_1, z_2$在复平面上对应的向量分别为$\\overrightarrow{OZ}_1, \\overrightarrow{OZ}_2$, 若$\\overrightarrow{OZ_1} \\perp \\overrightarrow{OZ_2}$, 且$\\lambda=f(x)$, 求$f(x)$的最小正周期和严格递减区间.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417232,7 +419128,9 @@ "id": "016300", "content": "已知集合$A=\\{z_1|| z_1-2 | \\leq 2, \\ z_1 \\in \\mathbf{C}\\}$, $B=\\{z | z=\\dfrac{1}{2} z_1 \\mathrm{i}+b, \\ z_1 \\in A,\\ b \\in \\mathbf{R}\\}$.\\\\\n(1) 当$b=0$时, 写出集合$B$在复平面内所表示的区域;\\\\\n(2) 当$A \\cap B=\\varnothing$时, 求$b$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417252,7 +419150,9 @@ "id": "016301", "content": "对任意复数$z=x+y \\mathrm{i}$($x, y \\in \\mathbf{R}$), 定义$g(z)=3^x(\\cos y+\\mathrm{i}\\sin y)$.\\\\\n(1) 若$g(z)=3$, 求相应的复数$z$;\\\\\n(2) 计算$g(2+\\dfrac{\\pi}{4} \\mathrm{i})$, $g(-1+\\dfrac{\\pi}{4} \\mathrm{i})$, $g(1+\\dfrac{\\pi}{2} \\mathrm{i})$, 并构造它们之间的一个等式, 由此发现一个更一般的等式, 并加以证明.", "objs": [], - "tags": [], + "tags": [ + "第五单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417272,7 +419172,9 @@ "id": "016302", "content": "在等差数列$\\{a_n\\}$中, 已知$a_1=\\dfrac{1}{3}$, $a_2+a_5=4$, $a_n=33$, 则$n$是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417292,7 +419194,9 @@ "id": "016303", "content": "在数列$\\{a_n\\}$中, $a_1=3$, 且对任意大于$1$的正整数$n$, 点$(\\sqrt{a_n}, \\sqrt{a_{n-1}})$在直线$x-y-\\sqrt{3}=0$, 则$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417312,7 +419216,9 @@ "id": "016304", "content": "等差数列$\\{a_n\\}$中, 公差为$\\dfrac{1}{2}$, $a_1+a_3+a_5+\\cdots+a_{99}=60$, 则$a_2+a_4+a_6+\\cdots+a_{100}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417332,7 +419238,9 @@ "id": "016305", "content": "数列$\\{a_n\\}$对任意的正整数$p$、$q$, 满足$a_{p+q}=a_p+a_q$, 且$a_2=-6$, 那么$a_{10}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417352,7 +419260,9 @@ "id": "016306", "content": "已知方程$(x^2-2 x+m)(x^2-2 x+n)=0$的四个根组成一个首项为$\\dfrac{1}{4}$的等差数列, 则$|m-n|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417372,7 +419282,9 @@ "id": "016307", "content": "已知等差数列$\\{a_n\\}$中, $a_1=1$, $a_3=-3$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 若数列$\\{a_n\\}$的前$k$项和$S_k=-35$, 求$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417392,7 +419304,9 @@ "id": "016308", "content": "记$S_n$是公差不为$0$的等差数列$\\{a_n\\}$的前$n$项和, 若$a_3=S_5$, $a_2 a_4=S_4$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式$a_n$;\\\\\n(2) 求使$S_n>a_n$成立的$n$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417412,7 +419326,9 @@ "id": "016309", "content": "已知数列$\\{a_n\\}$的各项均为正数, 记$S_n$为$\\{a_n\\}$的前$n$项和, 从下面 \\textcircled{1} \\textcircled{2} \\textcircled{3} 中选取两个作为条件, 证明另外一个成立. \\textcircled{1} 数列$\\{a_n\\}$是等差数列; \\textcircled{2} 数列$\\{\\sqrt{S_n}\\}$是等差数列; \\textcircled{3} $a_2=3 a_1$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417432,7 +419348,9 @@ "id": "016310", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, $a_1=-\\dfrac{9}{4}$, 且$4S_{n+1}=3S_n-9$.\\\\\n(1) 求数列$\\{a_n\\}$的通项;\\\\\n(2) 设数列$\\{b_n\\}$满足, $3 b_n+(n-4) a_n=0$($n$为正整数) 记$\\{b_n\\}$的前$n$项和为$T_n$, 若$T_n \\leq \\lambda b_n$对任意正整数$n$恒成立, 求实数$\\lambda$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417452,7 +419370,9 @@ "id": "016311", "content": "设$\\{a_n\\}$和$\\{b_n\\}$是两个等差数列, 记$c_n=\\max \\{b_1-a_1 n, b_2-a_2 n, \\cdots, b_n-a_n n\\}$($n=1,2,3, \\cdots$), 其中$\\max \\{x_1, x_2, \\cdots, x_s\\}$表示$x_1, x_2, \\cdots, x_s$这$s$个数中最大的数. 若$a_n=n$, $b_n=2 n-1$, 求$c_1, c_2, c_3$的值, 并证明$\\{c_n\\}$是等差数列.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417472,7 +419392,9 @@ "id": "016312", "content": "在等差数列$\\{a_n\\}$中, 若$a_1+a_2+a_3+a_4=30$, 则$a_2+a_3=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417492,7 +419414,9 @@ "id": "016313", "content": "$\\{a_n\\}$和$\\{b_n\\}$是两个等差数列, 其中$\\dfrac{a_k}{b_k}$($1 \\leq k \\leq 5$)为常值, $a_1=288$, $a_5=96$, $b_1=192$, 则$b_3=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417512,7 +419436,9 @@ "id": "016314", "content": "在等差数列$\\{a_n\\}$中, $a_1=7$, 公差为$d$, 前$n$项和为$S_n$, 当且仅当$n=8$时$S_n$取得最大值, 则$d$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417532,7 +419458,9 @@ "id": "016315", "content": "等差数列$\\{a_n\\}$中, $a_{15}=10$, $a_{45}=90$, 则$a_{60}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417552,7 +419480,9 @@ "id": "016316", "content": "在等差数列$\\{a_n\\}$中, 已知$a_2=\\dfrac{1}{10}$, $a_{10}=\\dfrac{1}{2}$, 则$a_1+a_2+\\cdots+a_{39}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417572,7 +419502,9 @@ "id": "016317", "content": "等差数列$\\{a_n\\}$中, 已知$a_1=\\dfrac{1}{3}$, $a_2+a_5=4$, $a_n=33$, 则$n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417592,7 +419524,9 @@ "id": "016318", "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n, S_{m-1}=-2, S_m=0, S_{m+1}=3$, 则$m=$\\bracket{20}.\n\\fourch{$3$}{$4$}{$5$}{$6$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -417612,7 +419546,9 @@ "id": "016319", "content": "数列$\\{a_n\\}$是严格递增的整数数列, 且$a_1 \\geq 3$, $a_1+a_2+\\cdots+a_n=100$, 则$n$的最大值为\\bracket{20}.\n\\fourch{$9$}{$10$}{$11$}{$12$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -417632,7 +419568,9 @@ "id": "016320", "content": "设$\\{a_n\\}$是等差数列. 下列结论中正确的是\\bracket{20}.\n\\twoch{若$a_1+a_2>0$, 则$a_2+a_3>0$}{若$0\\sqrt{a_1 a_3}$}{若$a_1+a_3<0$, 则$a_1+a_2<0$}{若$a_1<0$, 则$(a_2-a_1)(a_2-a_3)>0$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -417652,7 +419590,9 @@ "id": "016321", "content": "设等差数列$\\{a_n\\}$的前$n$项和为$S_n$, 已知$a_3=12$, $S_{12}>0$, $S_{13}<0$.\\\\\n(1) 求公差$d$的取值范围;\\\\\n(2) 指出$S_1$、$S_2$、$S_3$、$\\cdots$、$S_{12}$中哪一个最大, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417672,7 +419612,9 @@ "id": "016322", "content": "已知等差数列$\\{a_n\\}$的公差$d>0$, 设$\\{a_n\\}$的前$n$项和为$S_n$, $a_1=1$, $S_2S_3=36$.\\\\\n(1) 求$d, S_n$;\\\\\n(2) 求$m$、$k$($m$、$k$为正整数) 的值, 使得$a_m+a_{m+1}+a_{m+2}+\\cdots+a_{m+k}=65$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417692,7 +419634,9 @@ "id": "016323", "content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, $b_n$为数列$\\{S_n\\}$的前$n$项积, 已知$\\dfrac{2}{S_n}+\\dfrac{1}{b_n}=2$.($n \\geq 1$, $n \\in \\mathbf{N}$)\\\\\n(1) 证明: 数列$\\{b_n\\}$是等差数列;\\\\\n(2) 求$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417712,7 +419656,9 @@ "id": "016324", "content": "等比数列$x, 3 x+3,6 x+6, \\cdots$的第四项等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417732,7 +419678,9 @@ "id": "016325", "content": "若等差数列$\\{a_n\\}$和等比数列$\\{b_n\\}$满足$a_1=b_1=-1$, $a_4=b_4=8$, 则$\\dfrac{a_2}{b_2}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417752,7 +419700,9 @@ "id": "016326", "content": "等比数列$\\{a_n\\}$的各项均为实数, 其前$n$项的和为$S_n$, 已知$S_3=\\dfrac{7}{4}$, $S_6=\\dfrac{63}{4}$, 则$a_8=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417772,7 +419722,9 @@ "id": "016327", "content": "已知数列$\\{a_n\\}$是以$q$为公比的等比数列. 若$b_n=-2 a_n$, 则数列$\\{b_n\\}$是\\bracket{20}.\n\\twoch{以$q$为公比的等比数列}{以$-q$为公比的等比数列}{以$2 q$为公比的等比数列}{以$-2 q$为公比的等比数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -417792,7 +419744,9 @@ "id": "016328", "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=1+\\lambda a_n$, 其中$\\lambda \\neq 0$.\\\\\n(1) 证明$\\{a_n\\}$是等比数列, 并求其通项公式;\\\\\n(2) 若$S_5=\\dfrac{31}{32}$, 求$\\lambda$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417812,7 +419766,9 @@ "id": "016329", "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=\\dfrac{3 n^2-n}{2}$($n \\geq 1$, $n \\in \\mathbf{N}$).\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 证明: 对任意的$n>1$, 都存在正整数$m$, 使得$a_1, a_n, a_m$成等比数列.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417832,7 +419788,9 @@ "id": "016330", "content": "已知$\\{a_n\\}$是首项为$1$, 公差为$2$的等差数列, $S_n$表示$\\{a_n\\}$的前$n$项和.\\\\\n(1) 求$a_n$及$S_n$;\\\\\n(2) 设$\\{b_n\\}$是首项为$2$的等比数列, 公比$q$满足$q^2-(a_4+1) q+S_4=0$, 求$\\{b_n\\}$的通项公式及其前$n$项和$T_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417852,7 +419810,9 @@ "id": "016331", "content": "已知数列$\\{a_n\\}$和$\\{b_n\\}$满足: $a_1=1$, $a_2=2$, $a_n>0$, $b_n=\\sqrt{a_n a_{n+1}}$($n$为正整数), 且$\\{b_n\\}$是以$q$为公比的等比数列.\\\\\n(1) 证明$a_{n+2}=a_n q^2$;\\\\\n(2) 若$c_n=a_{2 n-1}+2 a_{2 n}$, 证明数列$\\{c_n\\}$是等比数列;\\\\\n(3) 求和: $\\dfrac{1}{a_1}+\\dfrac{1}{a_2}+\\dfrac{1}{a_3}+\\dfrac{1}{a_4}+\\cdots+\\dfrac{1}{a_{2 n-1}}+\\dfrac{1}{a_{2 n}}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417872,7 +419832,9 @@ "id": "016332", "content": "已知数列$\\{a_n\\}$中, $a_1=3$, $a_{n+1}+a_n=3 \\cdot 2^n$, $n$为正整数.\\\\\n(1) 证明数列$\\{a_n-2^n\\}$是等比数列, 并求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 在数列$\\{a_n\\}$中, 是否存在连续三项成等差数列? 若存在, 求出所有符合条件的项; 若否存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -417892,7 +419854,9 @@ "id": "016333", "content": "记$S_n$为等比数列$\\{a_n\\}$的$n$项和. 若$a_1=\\dfrac{1}{3}$, $a_4^2=a_6$, 则$S_5=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417912,7 +419876,9 @@ "id": "016334", "content": "一直角三角形三内角的正弦值成等比数列, 其最小内角是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417932,7 +419898,9 @@ "id": "016335", "content": "等比数列$\\{a_n\\}$中, $a_{15}=10$, $a_{45}=90$, 则$a_{60}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417952,7 +419920,9 @@ "id": "016336", "content": "设数列$\\{a_n\\}$是公比为$q$($q>1$)的等比数列, 若$a_{2020}$和$a_{2021}$是方程$4 x^2+8 x+3=0$的两根, 则$a_{2021}+a_{2022}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417972,7 +419942,9 @@ "id": "016337", "content": "已知数列$\\{a_n\\}$为等比数列, 若$a_1=27$, $a_k=\\dfrac{1}{243}$, 公比$q=-\\dfrac{1}{3}$, 则这个数列的前$k$项的和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -417992,7 +419964,9 @@ "id": "016338", "content": "$\\{a_n\\}$是等比数列, $a_2=2$, $a_5=\\dfrac{1}{4}$, 则$a_1 a_2+a_2 a_3+\\cdots+a_n a_{n+1}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418012,7 +419986,9 @@ "id": "016339", "content": "我国古代数学名著《算法统宗》中有如下问题: ``远望巍巍塔七层, 红光点点倍加增, 共灯三百八十一, 请问尖头几盏灯?''意思是: 一座$7$层塔共挂了$381$盏灯, 且相邻两层中的下一层灯数是上一层灯数的$2$倍, 则塔的顶层共有灯\\bracket{20}.\n\\fourch{$1$盏}{$3$盏}{$5$盏}{$9$盏}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -418032,7 +420008,9 @@ "id": "016340", "content": "某校学生在研究民间剪纸艺术时, 发现剪纸时经常会沿纸的某条对称轴形纸对折, 规格为$20 \\mathrm{dm} \\times 12 \\mathrm{dm}$的长方形纸, 对折$1$次共可以得到$10 \\mathrm{dm} \\times 12 \\mathrm{dm}, 20 \\mathrm{dm} \\times 6 \\mathrm{dm}$两种规格的图形, 它们的面积之和$S_1=240 \\mathrm{dm}^2$, 对折$2$次共可以得到$5 \\mathrm{dm} \\times 12 \\mathrm{dm}, 10 \\mathrm{dm} \\times 6 \\mathrm{dm}, 20 \\mathrm{dm} \\times 3 \\mathrm{dm}$三种规格的图形, 它们的面积之和$S_2=180 \\mathrm{dm}^2$, 以此类推, 则对折$4$次共可以得到不同规格图形的种数为\\blank{50}; 如果对折$n$次, 那么$\\displaystyle\\sum_{k=1}^n S_k=$\\blank{50}$\\mathrm{dm}^2$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418052,7 +420030,9 @@ "id": "016341", "content": "等比数列$\\{a_n\\}$的公比为$q$, 前$n$项和为$S_n$, 设甲: $q>0$, 乙: $\\{S_n\\}$是递增数列, 则\\bracket{20}.\n\\twoch{甲是乙的充分条件但不是必要条件}{甲是乙的必要条件但不是充分条件}{甲是乙的充要条件}{甲既不是乙的充分条件也不是乙的必要条件}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -418072,7 +420052,9 @@ "id": "016342", "content": "已知等比数列$\\{a_n\\}$中, $a_1=\\dfrac{1}{3}$, $q=\\dfrac{1}{3}$.\\\\\n(1) $S_n$为$\\{a_n\\}$的前$n$项和, 证明: $S_n=\\dfrac{1-a_n}{2}$;\\\\\n(2) 设$b_n=\\log _3 a_1+\\log _3 a_2+\\cdots+\\log _3 a_n$, 求数列$\\{b_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418092,7 +420074,9 @@ "id": "016343", "content": "已知公比大于$1$的等比数列$\\{a_n\\}$满足$a_2+a_4=20$, $a_3=8$.\\\\\n(1) 求$\\{a_n\\}$的通项公式;\\\\\n(2) 记$b_m$为$\\{a_n\\}$在区间$(0, m]$($m$为正整数)中的项的个数, 求数列$\\{b_m\\}$的前$100$项和$S_{100}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418112,7 +420096,9 @@ "id": "016344", "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=\\dfrac{3}{2}(a_n-1)$, $n \\geq 1$, $n \\in \\mathbf{N}$.\\\\\n(1) 求$\\{a_n\\}$的通项公式;\\\\\n(2) 若对于任意的正整数$n$, 有$k \\cdot a_n \\geq 4 n+1$成立, 求实数$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418132,7 +420118,9 @@ "id": "016345", "content": "下列数列中, 既是严格递增数列又是无穷数列的是\\bracket{20}.\n\\fourch{$1, \\dfrac{1}{2}, \\dfrac{1}{3}, \\dfrac{1}{4}, \\cdots$}{$-1,-2,-3,-4, \\cdots$}{$-1,-\\dfrac{1}{2},-\\dfrac{1}{4},-\\dfrac{1}{8}, \\cdots$}{$1, \\sqrt{2}, \\sqrt{3}, \\cdots, \\sqrt{n}$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -418152,7 +420140,9 @@ "id": "016346", "content": "已知一个数列的前$4$项为$1, \\sqrt{2}-1, \\sqrt{3}-\\sqrt{2}, 2-\\sqrt{3}$, 则它的一个通项公式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418172,7 +420162,9 @@ "id": "016347", "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=n^2+1$($n \\geq 1$, $n \\in \\mathbf{N}$), 则$a_8$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418192,7 +420184,9 @@ "id": "016348", "content": "设数列$\\{a_n\\}$的前$n$项和为$S_n$, $S_n=\\dfrac{a_1(3^n-1)}{2}$($n \\geq 1$, $n \\in \\mathbf{N}$), 且$a_4=54$, 则$a_1$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418212,7 +420206,9 @@ "id": "016349", "content": "已知数列$\\{a_n\\}$满足$a_n=n^2$($n \\geq 1$, $n \\in \\mathbf{N}$), 若对于一切$n$, $b_n$中的第$a_n$项恒等于$\\{a_n\\}$中的第$b_n$项, 则$\\dfrac{\\lg (b_1 b_4 b_9 b_{16})}{\\lg (b_1 b_2 b_3 b_4)}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418232,7 +420228,9 @@ "id": "016350", "content": "根据下列数列的前五项或前六项, 写出数列的一个通项公式.\\\\\n(1) $\\dfrac{2}{3}, \\dfrac{4}{15}, \\dfrac{6}{35}, \\dfrac{8}{63}, \\dfrac{10}{99}$;\\\\\n(2) $\\dfrac{3}{2}, 1, \\dfrac{7}{10}, \\dfrac{9}{17}, \\dfrac{11}{26}$;\\\\\n(3) $0,1,0,1,0$;\\\\\n(4) $1,2,3,1,2,3$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418252,7 +420250,9 @@ "id": "016351", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=n^2-n-30$.\\\\\n(1) 写出这个数列的前三项, 并指出$60$是此数列的第几项?\\\\\n(2) 该数列前$n$项和$S_n$是否存在最值? 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418272,7 +420272,9 @@ "id": "016352", "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n$满足$\\log _2(S_n+1)=n+1$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418292,7 +420294,9 @@ "id": "016353", "content": "已知在正项数列$\\{a_n\\}$中, $S_n$表示前$n$项和且$2 \\sqrt{S_n}=a_n+1$, 求$a_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418312,7 +420316,9 @@ "id": "016354", "content": "已知数列$\\{a_n\\}$中, $a_1+3 a_2+5 a_3+\\cdots+(2 n-1) a_n=(2 n-3) \\cdot 2^{n+1}$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418332,7 +420338,9 @@ "id": "016355", "content": "已知数列$\\{a_n\\}$满足前$n$项和$S_n=n^2+1$, 数列$\\{b_n\\}$满足$b_n=\\dfrac{2}{a_n+1}$, 且前$n$项和为$T_n$, 设$c_n=T_{2 n+1}-T_n$.\\\\\n(1) 求数列$\\{b_n\\}$的通项公式;\\\\\n(2) 判断数列$\\{c_n\\}$的单调性;\\\\\n(3) 当$n \\geq 2$时, $T_{2 n+1}-T_n<\\dfrac{1}{5}-\\dfrac{7}{12} \\log _a(a-1)$恒成立, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418352,7 +420360,9 @@ "id": "016356", "content": "数列$\\{a_n\\}: 2,-6,12,-20,30, \\cdots$的一个通项公式是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418372,7 +420382,9 @@ "id": "016357", "content": "设$a_n=-n^2+10 n+11$, 则数列$\\{a_n\\}$从首项到第\\blank{50}项的和最大.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418392,7 +420404,9 @@ "id": "016358", "content": "已知数列$\\{a_n\\}$中, $a_1=\\dfrac{1}{2}$, $a_{n+1}=1-\\dfrac{1}{a_n}$, 则$a_{2022}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418412,7 +420426,9 @@ "id": "016359", "content": "已知$\\{a_n\\}$是递增数列, 且对任意$n \\in \\mathbf{N}$, $n\\ge 1$都有$a_n=n^2+\\lambda n$恒成立, 则实数$\\lambda$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418432,7 +420448,9 @@ "id": "016360", "content": "已知数列$\\{a_n\\}$的首项$a_1=1$, 且$a_n=2 a_{n-1}+1$($n \\geq 2$, $n \\in \\mathbf{N}$), 则$a_5=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418452,7 +420470,9 @@ "id": "016361", "content": "已知数列$\\{a_n\\}$中, $a_1=1$, $a_2=3$, $a_n=a_{n-1}+\\dfrac{1}{a_{n-2}}$($n \\geq 3$, $n \\in \\mathbf{N}$), 则$a_5=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418472,7 +420492,9 @@ "id": "016362", "content": "数列$\\{a_n\\}: 1,-\\dfrac{5}{8}, \\dfrac{7}{15},-\\dfrac{9}{24}, \\cdots$的一个通项公式是\\bracket{20}.\n\\twoch{$a_n=(-1)^{n+1} \\dfrac{2 n-1}{n^2+n}$}{$a_n=(-1)^{n-1} \\dfrac{2 n+1}{n^3+3 n}$}{$a_n=(-1)^{n+1} \\dfrac{2 n-1}{n^2+2 n}$}{$a_n=(-1)^{n+1} \\dfrac{2 n+1}{n^2+2 n}$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -418492,7 +420514,9 @@ "id": "016363", "content": "已知数列$\\{a_n\\}$的通项$a_n=\\dfrac{n a}{n b+c}$($a$、$b$、$c$为正实数), 则$a_n$与$a_{n+1}$的大小关系是\\bracket{20}.\n\\fourch{$a_n>a_{n+1}$}{$a_nS_{2021}$, 则数列$\\{a_n\\}$严格递增}{若$T_{2022}>T_{2021}$, 则数列$\\{a_n\\}$严格递增}{若数列$\\{S_n\\}$严格递增, 则$a_{2022} \\geq a_{2021}$}{若数列$\\{T_n\\}$严格递增, 则$a_{2022} \\geq a_{2021}$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -418532,7 +420558,9 @@ "id": "016365", "content": "数列$\\{a_n\\}$的前$n$项和$S_n$满足关系式$\\lg (S_n-1)=n$($n \\geq 1$, $n \\in \\mathbf{N}$), 求$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418552,7 +420580,9 @@ "id": "016366", "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n$, $a_1=2$, 且对于任意正整数$n$, 总有$4S_{n+1}=1+3S_n$.\\\\\n(1) 求数列$\\{a_n\\}$的通项公式$a_n$及前项和$S_n$;\\\\\n(2) 指出数列$\\{a_n\\}$中的最大项、最小项及其相应的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418572,7 +420602,9 @@ "id": "016367", "content": "已知数列$\\{a_n\\}$与$\\{b_n\\}$满足$b_{n+1} a_n+b_n a_{n+1}=(-2)^n+1$, $b_n=\\dfrac{3+(-1)^{n-1}}{2}$, $n \\geq 1$, $n \\in \\mathbf{N}$, 且$a_1=2$.\\\\\n(1) 求$a_2, a_3$的值;\\\\\n(2) 设$c_n=a_{2 n+1}-a_{2 n-1}$($n$为正整数), 证明$\\{c_n\\}$是等比数列;\\\\\n(3) 设$S_n$为$\\{a_n\\}$的前$n$项和, 证明$\\dfrac{S_1}{a_1}+\\dfrac{S_2}{a_2}+\\cdots+\\dfrac{S_{2 n-1}}{a_{2 n-1}}+\\dfrac{S_{2 n}}{a_{2 n}} \\leq n-\\dfrac{1}{3}$($n \\geq 1$, $n \\in \\mathbf{N}$).", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418592,7 +420624,9 @@ "id": "016368", "content": "用数学归纳法证明``$2^{n+1} \\geq n^2+n+2$($n \\in \\mathbf{N}$, $n \\geq 1$)''时, 第一步应证明\\blank{150}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418612,7 +420646,9 @@ "id": "016369", "content": "观察下列等式: $1^3=1$, $1^3+2^3=9$, $1^3+2^3+3^3=36$, $1^3+2^3+3^3+4^3=100$, $\\cdots$. 猜想: $1^3+2^3+3^3+\\cdots+n^3=$\\blank{50}.($n$为正整数)", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418632,7 +420668,9 @@ "id": "016370", "content": "用数学归纳法证明: $(n+1)(n+2) \\cdots(n+n)=2^n \\cdot 1 \\cdot 3 \\cdot \\cdots \\cdot(2 n-1)$时, 从``$k$到$k+1$''左边需增乘的代数式是\\bracket{20}.\n\\fourch{$2 k+1$}{$\\dfrac{2 k+1}{k+1}$}{$2(2 k+1)$}{$\\dfrac{2(2 k+1)}{k+1}$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -418652,7 +420690,9 @@ "id": "016371", "content": "如果命题$P(n)$对$n=k$成立, 则它对$n=k+2$也成立, 又若$P(n)$对$n=2$成立, 则下列结论中正确的是\\bracket{20}.\n\\twoch{$P(n)$对所有自然数成立}{$P(n)$对所有正偶数成立}{$P(n)$对所有正奇数成立}{$P(n)$对所有大于$1$的自然数成立}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -418672,7 +420712,9 @@ "id": "016372", "content": "某个命题与整数$n$有关, 如果当$n=k$时命题成立, 可推出$n=k+1$时命题成立, 现已知当$n=3$时命题不成立, 则命题在\\bracket{20}.\n\\fourch{$n=4$时成立}{$n=4$时不成立}{$n=2$时成立}{$n=2$时不成立}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -418692,7 +420734,9 @@ "id": "016373", "content": "用数学归纳法证明: $1-\\dfrac{1}{2}+\\dfrac{1}{3}-\\dfrac{1}{4}+\\cdots+\\dfrac{1}{2 n-1}-\\dfrac{1}{2 n}=\\dfrac{1}{n+1}+\\dfrac{1}{n+2}+\\cdots+\\dfrac{1}{2 n}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418712,7 +420756,9 @@ "id": "016374", "content": "用数学归纳法证明下述整除问题. 求证: $11^{n+2}+12^{2 n+1}$($n$为正整数) 被$133$整除.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418732,7 +420778,9 @@ "id": "016375", "content": "用数学归纳法证明不等式: $1+\\dfrac{1}{\\sqrt{2}}+\\dfrac{1}{\\sqrt{3}}+\\cdots+\\dfrac{1}{\\sqrt{n}}<2 \\sqrt{n}$($n$为正整数).", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418752,7 +420800,9 @@ "id": "016376", "content": "在数列$\\{a_n\\},\\{b_n\\}$中, $a_1=2$, $b_1=4$, 且$a_n, b_n, a_{n+1}$成等差数列, $b_n, a_{n+1}, b_{n+1}$成等比数列, 求$a_2, a_3, a_4$及$b_2, b_3, b_4$, 由此猜测$\\{a_n\\},\\{b_n\\}$的通项公式, 并证明你的结论.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418772,7 +420822,9 @@ "id": "016377", "content": "设函数$f(x)=x-\\sin x$, 数列$\\{a_n\\}$满足$a_{n+1}=f(a_n)$.\\\\\n(1) 若$a_1=2$, 试比较$a_2$与$a_3$的大小;\\\\\n(2) 若$0n^2$. 验证不等式成立所取的第一个值应该是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418852,7 +420910,9 @@ "id": "016381", "content": "设$f(k)=\\dfrac{1}{k+1}+\\dfrac{1}{k+2}+\\dfrac{1}{k+3}+\\cdots+\\dfrac{1}{2 k}$($k \\geq 1$, $k \\in \\mathbf{N}$), 则$f(k+1)=f(k)+$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418872,7 +420932,9 @@ "id": "016382", "content": "设$f(n)>0$且$f(n_1+n_2)=f(n_1) f(n_2)$, $f(2)=4$($n \\geq 1$, $n \\in \\mathbf{N}$), 则$f(n)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418892,7 +420954,9 @@ "id": "016383", "content": "$k$为正偶数, $P(k)$表示等式$1-\\dfrac{1}{2}+\\dfrac{1}{3}-\\dfrac{1}{4}+\\cdots+\\dfrac{1}{k-1}-\\dfrac{1}{k}=2(\\dfrac{1}{k+2}+\\dfrac{1}{k+4}+\\cdots+\\dfrac{1}{2 k})$, 则$P(4)$表示的等式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -418912,7 +420976,9 @@ "id": "016384", "content": "用数学归纳法证明$1^2+2^2+\\cdots+(n-1)^2+n^2+(n-1)^2+\\cdots+2^2+1^2=\\dfrac{n(2 n^2+1)}{3}$时, 由$n=$$k$的假设到证明$n=k+1$时, 等式左边应添加的式子是\\bracket{20}.\n\\fourch{$(k+1)^2+2 k^2$}{$(k+1)^2+k^2$}{$(k+1)^2$}{$\\dfrac{1}{3}(k+1)[2(k+1)^2+1]$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -418932,7 +420998,9 @@ "id": "016385", "content": "某学生在证明等差数列前$n$项和公式时, 证法如下:\\\\\n\\textcircled{1} 当$n=1$时, $S_1=a_1$显然成立;\\\\\n\\textcircled{2} 假设$n=k$($k \\in \\mathbf{N}$, $k \\geq 1$)时, 公式成立, 即$S_k=k a_1+\\dfrac{k(k-1)}{2} d$, 当$n=k+1$时,\n\\begin{align*}\nS_{k+1}&=a_{1}+a_{2}+\\cdots+a_{k}+a_{k+1}\\\\\n&=a_{1}+(a_{1}+d)+(a_{1}+2 d)+\\cdots+[a_{1}+(k-1) d]+(a_{1}+k d)\\\\\n&=(k+1) a_{1}+(d+2 d+\\cdots+k d)=(k+1) a_{1}+\\dfrac{k(k+1)}{2} d\\\\\n&=(k+1) a_{1}+\\dfrac{(k+1)[(k+1)-1]}{2} d.\n\\end{align*}\n所以$n=k+1$时, 公式成立. 以上证明中错误的是\\bracket{20}.\n\\twoch{当$n$取第一个值$1$时, 证明不对}{归纳假设写法不对}{从$n=k$到$n=k+1$的推理中未用归纳假设}{从$n=k$到$n=k+1$的推理有错误}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -418952,7 +421020,9 @@ "id": "016386", "content": "某个命题与正整数$n$有关, 若$n=k$($k$为正整数)时命题成立, 可推出$n=k+1$时, 命题也成立, 现已知当$n=5$时, 该命题不成立, 那么可推出\\bracket{20}.\n\\twoch{当$n=6$时该命题不成立}{当$n=6$时该命题成立}{当$n=4$时该命题不成立}{当$n=4$时该命题成立}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -418972,7 +421042,9 @@ "id": "016387", "content": "用数学归纳法证明: $1+a+a^2+\\cdots+a^{n+1}=\\dfrac{1-a^{n+2}}{1-a}$($a \\neq 0$, $a \\neq 1$, $n$为正整数).", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -418992,7 +421064,9 @@ "id": "016388", "content": "数列$\\{a_n\\}$满足$S_n=2 n-a_n$.\\\\\n(1) 计算$a_1, a_2, a_3, a_4$并由此猜想通项$a_n$的表达式;\\\\\n(2) 用数学归纳法证明 (1) 中的猜想.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -419012,7 +421086,9 @@ "id": "016389", "content": "数列$\\{a_n\\}$和$\\{b_n\\}$分别是等比数列和等差数列, 它们的前四项之和分别是$120$和$60$, 而第二项与第四项之和分别是$90$和$34$, 集合$A=\\{a_1^2, a_2^2, \\cdots, a_n^2, \\cdots\\}$, $B=\\{b_1, b_2, \\cdots, b_n, \\cdots\\}$. 求证: $A \\subset B$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -419032,7 +421108,9 @@ "id": "016390", "content": "等差数列$\\{a_n\\}$中, 公差$d=1$, $a_4+a_{17}=8$, 则$a_2+a_4+a_6+\\cdots+a_{20}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -419052,7 +421130,9 @@ "id": "016391", "content": "数列$\\{a_n\\}$的前$n$项的和$S_n=n^2+1$, 则此数列的通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -419072,7 +421152,9 @@ "id": "016392", "content": "若数列$\\{a_n\\}$满足: $a_1=1$, $a_{n+1}=2 a_n$($n \\geq 1$, $n \\in \\mathbf{N}$), 则前$6$项的和$S_6=$\\blank{50}. (用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -419092,7 +421174,9 @@ "id": "016393", "content": "数列$\\{a_n\\}$的前$n$项的和为$S_n=\\dfrac{1}{3} a_n-2$, 则$\\displaystyle \\lim_{n \\to \\infty}(a_1+a_2+\\cdots+a_n)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -419112,7 +421196,9 @@ "id": "016394", "content": "在等比数列$\\{a_n\\}$中, $a_5 a_6=\\dfrac{3 \\pi}{2}$, 则$\\sin (2 a_4 a_7)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -419132,7 +421218,9 @@ "id": "016395", "content": "已知等差数列$\\{a_n\\}$前$9$项的和为$27$, $a_{10}=8$, 则$a_{100}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -419152,7 +421240,9 @@ "id": "016396", "content": "设等比数列$\\{a_n\\}$的公比$q=\\dfrac{1}{2}$, 前$n$项和为$S_n$, 则$\\dfrac{S_4}{a_4}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -419172,7 +421262,9 @@ "id": "016397", "content": "设数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_1=-1$, $a_{n+1}=S_{n+1} S_n$, 则$S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -419192,7 +421284,9 @@ "id": "016398", "content": "在等比数列$\\{a_n\\}$中, $a_n>0$, 且$a_1 \\cdot a_2 \\cdot \\cdots \\cdot a_7 \\cdot a_8=16$, 则$a_4+a_5$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -419212,7 +421306,9 @@ "id": "016399", "content": "某企业投资$72$万元兴建一座环保建材厂, 第$1$年各种经营成本为$12$万元, 以后每年的经营成本增加$4$万元, 每年销售环保建材的收人为$50$万元, 则该厂获取的纯利润达到最大值时是在第\\blank{50}年.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -419232,7 +421328,9 @@ "id": "016400", "content": "设$\\{a_n\\}$是公差为$d$的等差数列, $S_n$是$\\{a_n\\}$的前$n$项和, $\\{b_n\\}$是公比为$q$的等比数列, $T_n$是$\\{b_n\\}$的前$n$项积. 用类比的方法, 将等差数列$\\{a_n\\}$前$n$项和$S_n$的公式推广到等比数列$\\{b_n\\}$中有\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -419252,7 +421350,9 @@ "id": "016401", "content": "如果$f(n)=1+\\dfrac{1}{2}+\\dfrac{1}{3}+\\cdots+\\dfrac{1}{n}+\\dfrac{1}{n+1}+\\cdots+\\dfrac{1}{2^n}$($n \\geq 1$, $n \\in \\mathbf{N}$), 那么$f(k+1)-f(k)$共有\\blank{50}项.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -419272,7 +421372,9 @@ "id": "016402", "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=3^n+k$($k$为常数), 那么下述结论中正确的是\\bracket{20}.\n\\twoch{$k$为任意实数时, $\\{a_n\\}$是等比数列}{$k=-1$时, $\\{a_n\\}$是等比数列}{$k=0$时, $\\{a_n\\}$是等比数列}{$\\{a_n\\}$不可能是等比数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -419292,7 +421394,9 @@ "id": "016403", "content": "已知等差数列$\\{a_n\\}$的公差为$d$, 前$n$项和为$S_n$, 则``$d>0$''是``$S_4+S_6>$$2S_5$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -419312,7 +421416,9 @@ "id": "016404", "content": "若$a$、$b$、$c$成等差数列, 则函数$f(x)=a^2+b x+c$的图像与$x$轴的交点的个数是\\bracket{20}.\n\\fourch{0 个}{1 个}{2 个}{不确定}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -419332,7 +421438,9 @@ "id": "016405", "content": "设$S_n=\\dfrac{1}{2}+\\dfrac{1}{6}+\\dfrac{1}{12}+\\cdots+\\dfrac{1}{n(n+1)}$, 且$S_n \\cdot S_{n+1}=\\dfrac{3}{4}$, 则$n$的值为\\bracket{20}.\n\\fourch{9}{8}{7}{6}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -419352,7 +421460,9 @@ "id": "016406", "content": "已知等差数列$\\{a_n\\}$中, $a_4=14$, 前$10$项和$S_{10}=185$.\\\\\n(1) 求$a_n$;\\\\\n(2) 将$\\{a_n\\}$中的第$2$项, 第$4$项, $\\cdots$, 第$2^n$项按原来的顺序排成一个新数列, 求此数列的前$n$项和$G_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -419372,7 +421482,9 @@ "id": "016407", "content": "设$S_n$为数列$\\{a_n\\}$的前$n$项的和, 且$S_n=\\dfrac{3}{2}(a_n-1)$($n \\geq 1$, $n \\in \\mathbf{N}$), 数列$\\{b_n\\}$的通项公式$b_n=4 n+5$($n \\geq 1$, $n \\in \\mathbf{N}$).\\\\\n(1) 求证: 数列$\\{a_n\\}$是等比数列;\\\\\n(2) 若$d \\in\\{a_1, a_2, a_3, \\cdots\\} \\cap\\{b_1, b_2, b_3, \\cdots\\}$, 则称$d$为数列$\\{a_n\\}$和$\\{b_n\\}$的公共项, 按它们在原数列中的先后顺序排成一个新的数列$\\{d_n\\}$, 求数列$\\{d_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -419392,7 +421504,9 @@ "id": "016408", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且$a_2 a_n=S_2+S_n$对一切正整数$n$都成立.\\\\\n(1) 求$a_1, a_2$的值;\\\\\n(2) 设$a_1>0$, 数列$\\{\\lg \\dfrac{10 a_1}{a_n}\\}$的前$n$项和为$T_n$, 当$n$为何值时, $T_n$最大? 并求出$T_n$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -419412,7 +421526,9 @@ "id": "016409", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n$, 且满足$a_1=\\dfrac{1}{2}$, $a_n+2S_n S_{n-1}=0$($n \\in \\mathbf{N}$).\\\\\n(1) 判断$\\{\\dfrac{1}{S_n}\\}$是否为等差数列? 并证明你的结论;\\\\\n(2) 求$S_n$和$a_n$;\\\\\n(3) 求证: $S_1^2+S_2^2+\\cdots+S_n^2 \\leq \\dfrac{1}{2}-\\dfrac{1}{4 n}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -419432,7 +421548,9 @@ "id": "016410", "content": "定义$R_p$数列$\\{a_n\\}$: 对实数$p$, 满足: \\textcircled{1} $a_1+p \\geq 0$, $a_2+p=0$; \\textcircled{2} 对于任意正整数$n$, $a_{4 n-1}=latex]\n\\draw [->] (-0.5,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (1,0.2) -- (1,0) node [below] {$1$} coordinate (1);\n\\draw (0,1) node [left] {$A$} coordinate (A);\n\\draw (1,1) node [right] {$B$} coordinate (B);\n\\draw (0,-1) node [left] {$P$} coordinate (P);\n\\path [name path = AB, draw] (A)--(B);\n\\path [name path = OB, draw] (O)--(B);\n\\path [name path = PM, draw] (P)--(0.8,1.5);\n\\path [name intersections = {of = AB and PM, by = M}];\n\\path [name intersections = {of = OB and PM, by = N}];\n\\draw (M) node [above left] {$M$} -- (O);\n\\draw (N) node [below right] {$N$} coordinate (N);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\triangle OMN$的面积$S$关于$l$的斜率$k$的函数关系式;\\\\\n(2) 当$k$为何值时, $S$, 取得最大值, 并求此最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420132,7 +422318,9 @@ "id": "016445", "content": "已知定点$P(6,4)$与定直线$l_1: y=4 x$, 过$P$点的直线$l$与$l_1$交于第一象限$Q$点, 与$x$轴正半轴交于点$M$, 求使$\\triangle OQM$面积最小的直线$l$方程.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw [->] (0,0) -- (16,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,8) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\filldraw (6,4) node [right] {$P$} coordinate (P) circle (0.06);\n\\draw (0,0) -- (2,8) node [above] {$l_1$} coordinate (l_1);\n\\draw (1.8,7.2) node [right] {$Q$} coordinate (Q) (11.25,0) node [below] {$M$} coordinate (M);\n\\draw ($(Q)!-0.1!(M)$) -- ($(Q)!1.1!(M)$);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420152,7 +422340,9 @@ "id": "016446", "content": "直线$y=1$的斜率是\\blank{50}, 倾斜角是\\blank{50}, 一个法向量可以是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420172,7 +422362,9 @@ "id": "016447", "content": "已知直线$l$在$x$轴上的截距为$-3$, 倾斜角为$\\alpha$且$\\cos \\alpha=\\dfrac{3}{5}$, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420192,7 +422384,9 @@ "id": "016448", "content": "边长为$3$的等边三角形$OAB$的顶点$O$为坐标原点, 顶点$B$在$y$轴的正半轴上, 则$AB$所在直线的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420212,7 +422406,9 @@ "id": "016449", "content": "若$\\alpha \\in[\\dfrac{\\pi}{6}, \\dfrac{\\pi}{2})$, 则直线$2 x \\cos \\alpha+3 y+1=0$的倾斜角的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420232,7 +422428,9 @@ "id": "016450", "content": "已知直线$l$过$P(-1,-1)$且平分平行四边形$ABCD$的面积, 若此平行四边形的两个顶点为$B(1,4)$、$D(5,0)$, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420252,7 +422450,9 @@ "id": "016451", "content": "已知方程$y=a|x|, y=x+a$($a>0$)所确定的图像有两个交点, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420272,7 +422472,9 @@ "id": "016452", "content": "已知直线$l$过圆$x^2+(y-3)^2=4$的圆心, 且与直线$x+y+1=0$平行, 则$l$的方程是\\bracket{20}.\n\\fourch{$x+y-2=0$}{$x-y-2=0$}{$x+y-3=0$}{$x-y+3=0$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -420292,7 +422494,9 @@ "id": "016453", "content": "已知函数$f(x)=a \\sin x-b \\cos x$($a b \\neq 0$), 若$f(\\dfrac{\\pi}{4}-x)=f(\\dfrac{\\pi}{4}+x)$, 则直线$a x-b y+c=0$的倾斜角为\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{4}$}{$\\dfrac{\\pi}{3}$}{$\\dfrac{2 \\pi}{3}$}{$\\dfrac{3 \\pi}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -420312,7 +422516,9 @@ "id": "016454", "content": "入射光线在直线$l_1: 2 x-y-3=0$上, 经过$x$轴反射到直线$l_2$上, 再经过$y$轴反射到直线$l_3$上, 则直线$l_3$的方程为\\bracket{20}.\n\\fourch{$x-2 y+3=0$}{$2 x-y+3=0$}{$2 x+y-3=0$}{$2 x-y+6=0$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -420332,7 +422538,9 @@ "id": "016455", "content": "直线$l$与直线$x-3 y+10=0$, $2 x+y-8=0$分别交于$M$、$N$两点, 且点$(0,1)$平分线段$MN$, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420352,7 +422560,9 @@ "id": "016456", "content": "过点$P(2,1)$作直线$l$, 分别与$x$、$y$轴的正半轴交于点$A$、$B$.\\\\\n(1) 当$\\triangle ABO$的面积$S$最小时, 求直线$l$的方程;\\\\\n(2) 当$|PA| \\cdot|PB|$最小时, 直线$l$的方程;\\\\\n(3) 若点$P$恰为线段$AB$的三等分点, 求此直线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420372,7 +422582,9 @@ "id": "016457", "content": "求证: 不论$\\lambda$取什么实数, 直线$(2 \\lambda-1) x+(\\lambda+3) y-(\\lambda-11)=0$都经过一个定点, 并求这个定点的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420392,7 +422604,9 @@ "id": "016458", "content": "证明: 直线$(m+2) x-(m+1) y-2(3+2 m)=0$与点$P(-2,2)$的距离$d \\leq 4 \\sqrt{2}$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420412,7 +422626,9 @@ "id": "016459", "content": "``$a=1$''是``直线$l_1: a x+2 y-1=0$与直线$l_2: x+(a+1) y+4=0$平行''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充分必要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -420432,7 +422648,9 @@ "id": "016460", "content": "``$m=\\dfrac{1}{2}$''是``直线$(m+2) x+3 m y+1=0$与直线$(m-2) x+(m+2) y-3=0$相互垂直''的 \\bracket{20}.\n\\twoch{充分必要条件}{充分而不必要条件}{必要而不充分条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -420452,7 +422670,9 @@ "id": "016461", "content": "直线$x=-2$与直线$\\sqrt{3} x-y+1=0$的夹角是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420472,7 +422692,9 @@ "id": "016462", "content": "圆$C: x^2+y^2-2 x-4 y+4=0$的圆心到直线$3 x+4 y+4=0$的距离\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420492,7 +422714,9 @@ "id": "016463", "content": "已知平行直线$l_1: 2 x+y-1=0$, $l_2: 4 x+2 y+1=0$, 则$l_1, l_2$的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420512,7 +422736,9 @@ "id": "016464", "content": "已知两条直线$l_1: x+m^2 y+6=0$, $l_2: (m-2) x+3 m y+2 m=0$, 当$m$取何值时, $l_1$、$l_2$\\\\\n(1) 相交;\\\\\n(2) 平行;\\\\\n(3) 重合.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420532,7 +422758,9 @@ "id": "016465", "content": "已知等腰直角三角形的斜边所在直线方程是$3 x-y+2=0$, 直角顶点$C(\\dfrac{14}{5}, \\dfrac{2}{5})$, 求两条直角边所在的直线方程和此三角形面积.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420552,7 +422780,9 @@ "id": "016466", "content": "一束光线经过点$A(-2,1)$由直线$l: x-3 y+2=0$反射后, 经过点$B(3,5)$射出, 求反射光线所在直线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420572,7 +422802,9 @@ "id": "016467", "content": "已知点$A(-4,5)$、$B(2,1)$, 试在$x$轴上求一点$M$, 使$|MA|+|MB|$最小. .", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420592,7 +422824,9 @@ "id": "016468", "content": "求直线$l_1: 3 x-2 y-6=0$关于直线$l_2: 2 x-3 y+1=0$对称的直线$l_3$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420612,7 +422846,9 @@ "id": "016469", "content": "已知$\\triangle ABC$的顶点$A(3,-1)$, 边$AB$上的中线所在直线方程为$6 x+10 y-59=0$, 角$B$的平分线所在直线方程为$x-4 y+10=0$, 求$BC$边所在直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420632,7 +422868,9 @@ "id": "016470", "content": "在第一象限内, 直线$y=x$上有两点$B$、$C(C$点在$B$点上方), 且$|BC|=\\sqrt{2}$, 在$x$轴上有点$A(2,0)$, 当点$B$位于何处时, $\\angle BAC$取得最大值?", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420652,7 +422890,9 @@ "id": "016471", "content": "过点$(-1,3)$且垂直于直线$x-2 y+3=0$的直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420672,7 +422912,9 @@ "id": "016472", "content": "若直线$x-m y-4=0$与直线$y=-2 x+4$的夹角为$\\arccos \\dfrac{2 \\sqrt{5}}{5}$, 则实数$m$为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420692,7 +422934,9 @@ "id": "016473", "content": "已知直线$l_1: x+a y=1$, $l_2: a x+y=1$, 若$l_1\\parallel l_2$, 则$l_1$与$l_2$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420712,7 +422956,9 @@ "id": "016474", "content": "已知点$P(a, b)$在直线$l: 12 x+5 y-33=0$上, 则$\\sqrt{a^2+b^2}$的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420732,7 +422978,9 @@ "id": "016475", "content": "已知$\\triangle ABC$的三个顶点$A(1,0)$、$B(-1, \\sqrt{2})$、$C(3, \\dfrac{\\sqrt{2}}{2})$, 则$\\angle BAC$的平分线所在直线的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420752,7 +423000,9 @@ "id": "016476", "content": "已知两条平行直线$l_1$与$l_2$分别过点$P_1(1,0)$与点$P_2(0,5)$, $l_1$、$l_2$之间的距离为$d$, 则$d$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420772,7 +423022,9 @@ "id": "016477", "content": "实数$k=5$是直线$(k-3) x+(4-k) y=-1$与直线$2(k-3) x-2 y+(2-k)=0$平行的\\bracket{20}.\n\\twoch{充要条件}{充分非必要条件}{必要非充分条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -420792,7 +423044,9 @@ "id": "016478", "content": "过点$P(1,2)$引直线, 使$A(2,3), B(4,-5)$到它的距离相等, 则这条直线方程为\\bracket{20}.\n\\twoch{$4 x+y-6=0$}{$x+4 y-6=0$}{$2 x+3 y-7=0$或$x+4 y-6=0$}{$3 x+2 y-7=0$或$4 x+y-6=0$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -420812,7 +423066,9 @@ "id": "016479", "content": "下列命题中:\\\\\n\\textcircled{1} 两条平行直线$3 x-2 y+5=0$与$6 x-4 y+8=0$间的距离是$d=\\dfrac{|5-4|}{\\sqrt{3^2+(-2)^2}}$;\\\\\n\\textcircled{2} 已知直线$l$经过点$P(1,1)$且与$l_1: y=\\sqrt{3} x+1$和$l_2: y=\\sqrt{3} x+3$分别交于两点$A$、$B$, 若$|AB|=\\sqrt{2}$, 则直线$l$的方程是$(2-\\sqrt{3}) x-y+\\sqrt{3}-1=0$;\\\\\n\\textcircled{3} 若$0 \\leq \\theta \\leq \\dfrac{\\pi}{2}$, 当点$(1, \\cos \\theta)$到直线$x \\sin \\theta+y \\cos \\theta-1=0$的距离是$\\dfrac{1}{4}$时, 这条直线的斜率为$-\\dfrac{\\sqrt{3}}{3}$;\\\\\n\\textcircled{4} 若直线$a x+b y=2$经过点$M(\\cos \\alpha, \\sin \\alpha)$, 则$a^2+b^2 \\geq 4$.\\\\\n其中正确命题的个数是\\bracket{20}.\n\\fourch{0}{1}{2}{3}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -420832,7 +423088,9 @@ "id": "016480", "content": "已知直线$l: 5 x+2 y+3=0$.\\\\\n(1) 求直线$l_1: 3 x+7 y-13=0$与直线$l$的夹角;\\\\\n(2) 设直线$l_2$经过点$P(2,1)$, 且与$l$的夹角等于$45{^ \\circ}$, 求直线$l_2$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420852,7 +423110,9 @@ "id": "016481", "content": "已知三条直线$l_1: 4 x+y=4$, $l_2: m x+y=0$, $l_3: 2 x-3 m y=4$.\\\\\n(1) 若$l_1 \\perp l_3$, 求$m$的值;\\\\\n(2) 若三条直线不能围成三角形, 求$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420872,7 +423132,9 @@ "id": "016482", "content": "已知平面上的线段$l$及点$P$, 在$l$上任取一点$Q$, 线段$PQ$长度的最小值称为点$P$到线段$l$的距离, 记作$d(P, l)$.\\\\\n(1) 求点$P(1,1)$到线段$l: x-y-3=0$($3 \\leq x \\leq 5$)的距离$d(P, l)$;\\\\\n(2) 求点$P(t, 0)$到线段$l: x-y-3=0$($1 \\leq x \\leq 4$)的距离$d(P, l)$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -420892,7 +423154,9 @@ "id": "016483", "content": "直线$l: a x+3 m y+2 a=0$($m \\neq 0$)过点$(1,-1)$, 则直线$l$的倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420912,7 +423176,9 @@ "id": "016484", "content": "已知直线$l_1: k x+(1-k) y-3=0$, $l_2: (k-1) x+(2 k+3) y-2=0$, 若$l_1 \\perp l_2$, 则$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420932,7 +423198,9 @@ "id": "016485", "content": "已知直线$l$的方程为$(2 a^2-7 a+3) x+(a^2-9) y+3 a^2=0$, 则当$a=$\\blank{50}时, 直线$l$与$x$轴平行.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420952,7 +423220,9 @@ "id": "016486", "content": "已知直线$l$经过点$A(-4,1)$, 且和直线$\\sqrt{3} x-y+2$的夹角为$30{^ \\circ}$, 则直线$l$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420972,7 +423242,9 @@ "id": "016487", "content": "已知点$P(-1,1)$、$Q(2,2)$, 若直线$l: x+m y+m=0$与线段$PQ$的延长线相交, 则$m$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -420992,7 +423264,9 @@ "id": "016488", "content": "已知直线$l_1: (k-3) x+(4-k) y+1=0$, 与$l_2: 2(k-3) x-2 y+3=0$平行, 则$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421012,7 +423286,9 @@ "id": "016489", "content": "点$(4, t)$到直线$4 x-3 y=1$的距离不大于$3$, 则$t$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421032,7 +423308,9 @@ "id": "016490", "content": "直线$y=\\dfrac{1}{2} x$关于直线$x=1$对称的直线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421052,7 +423330,9 @@ "id": "016491", "content": "方程$|x|+|y|=1$所表示的图形是\\blank{50}, 该图形所围成的区域的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421072,7 +423352,9 @@ "id": "016492", "content": "已知$A(2,6)$, $B$、$C$分别为直线$x-y-4=0$和$y$轴上的动点, 且$A$、$B$、$C$三点不共线, 则$\\triangle ABC$周长的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421092,7 +423374,9 @@ "id": "016493", "content": "若直线$m$被两平行线$l_1: x-y+1=0$与$l_2: x-y+3=0$所截得的线段的长为$2 \\sqrt{2}$, 则$m$的倾斜角可以是: \\textcircled{1} $15{^ \\circ}$; \\textcircled{2} $30{^ \\circ}$; \\textcircled{3} $45{^ \\circ}$; \\textcircled{4} $60{^ \\circ}$; \\textcircled{5} $75{^ \\circ}$. 其中正确答案的序号是\\blank{50}. (写出所有正确答案的序号)", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421112,7 +423396,9 @@ "id": "016494", "content": "设直线系$M: x \\cos \\theta+(y-2) \\sin \\theta=1$($0 \\leq \\theta \\leq 2 \\pi$), 对于下列四个命题:\\\\\n\\textcircled{1} $M$中所有直线均经过一个定点;\\\\\n\\textcircled{2} 存在定点$P$不在$M$中的任一条直线上;\\\\\n\\textcircled{3} 对于任意整数$n$($n \\geq 3$), 存在正$n$边形, 其所有边均在$M$中的直线上;\\\\\n\\textcircled{4} $M$中的直线所能围成的正三角形面积都相等.\\\\\n其中真命题的序号是\\blank{50}.(写出所有真命题的序号).", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421132,7 +423418,9 @@ "id": "016495", "content": "点$A(5,0)$、$B(1,-4 \\sqrt{3})$到直线的距离都是$4$, 满足条件的直线有\\bracket{20}.\n\\fourch{一条}{两条}{三条}{四条}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -421152,7 +423440,9 @@ "id": "016496", "content": "过点$A(4,1)$且在两坐标轴上的截距互为相反数的直线的方程是\\bracket{20}.\n\\twoch{$x+y=5$}{$x-y=3$}{$x+y=5$或$x-4 y=0$}{$x-y=3$或$x-4 y=0$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -421172,7 +423462,9 @@ "id": "016497", "content": "经过点$(5,10)$且与原点距离为$5$的直线的斜率是\\bracket{20}.\n\\fourch{$\\dfrac{3}{4}$}{$2$}{$\\dfrac{1}{2}$}{$\\dfrac{3}{4}$或不存在}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -421192,7 +423484,9 @@ "id": "016498", "content": "若动点$A(x_1, y_1)$、$B(x_2, y_2)$分别在直线$l_1: x+y-7=0$和$l_2: x+y-5=0$上移动, 则线段$AB$的中点$M$到原点的距离的最小值为\\bracket{20}.\n\\fourch{$2 \\sqrt{3}$}{$3 \\sqrt{3}$}{$3 \\sqrt{2}$}{$4 \\sqrt{2}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -421212,7 +423506,9 @@ "id": "016499", "content": "已知$\\triangle ABC$的$AB$、$AC$边上的高所在的直线方程分别为$2 x-3 y+1=0$和$x+y=0$, 点$A$的坐标为$(1,2)$, 求$BC$边所在的直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421232,7 +423528,9 @@ "id": "016500", "content": "已知直线$l$过点$A(-2,1)$, 且与直线$m: 2 x-y+1=0$的夹角为$\\arctan \\dfrac{1}{2}$, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421252,7 +423550,9 @@ "id": "016501", "content": "已知实数$x$、$y$满足$y=x^2-2 x+2$($-1 \\leq x \\leq 1$), 求$\\dfrac{y+3}{x+2}$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421272,7 +423572,9 @@ "id": "016502", "content": "已知点$A(-1,0)$、$B(1,0)$, 直线$l: a x+b y+c=0$(其中$a$、$b$、$c \\in \\mathbf{R}$), 点$P$在直线$l$上.\\\\\n(1) 若$a$、$b$、$c$是常数列, 求$|PB|$的最小值;\\\\\n(2) 若$a$、$b$、$c$成等差数列, 且$PA \\perp l$, 求$|PB|$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421292,7 +423594,9 @@ "id": "016503", "content": "已知过点$A(1,1)$, 且斜率为$-m$($m>0$)的直线$l$与$x$轴、$y$轴分别交于$P$、$Q$两点, 过$P$、$Q$作直线$l': 2 x+y=0$的垂线, 垂足分别为$R$、$S$(如图); 求四边形$PQSR$面积的最小值和此时直线$l$的方程.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (-1.5,3) coordinate (B) -- (0.75,-1.5) coordinate (C) node [right] {$l'$};\n\\filldraw (1,1) node [right] {$A$} coordinate (A) circle (0.03);\n\\draw (0,2.5) node [right] {$Q$} coordinate (Q) ({5/3},0) node [below] {$P$} coordinate (P);\n\\draw ($(B)!(Q)!(C)$) node [left] {$S$} coordinate (S);\n\\draw ($(B)!(P)!(C)$) node [below] {$R$} coordinate (R);\n\\draw ($(Q)!-0.2!(P)$) -- ($(Q)!1.2!(P)$) node [right] {$l$};\n\\draw (Q)--(S) (P)--(R);\n\\draw pic [draw,scale = 0.3] {right angle = R--S--Q};\n\\draw pic [draw,scale = 0.3] {right angle = P--R--S};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421312,7 +423616,9 @@ "id": "016504", "content": "圆心在直线$x-2 y=0$上的圆$C$与$y$轴的正半轴相切, 圆$C$截$x$轴所得弦的长为$2 \\sqrt{3}$, 则圆$C$的标准方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421332,7 +423638,9 @@ "id": "016505", "content": "若$P$、$Q$是圆$x^2+y^2-2 x+4 y+4=0$上的动点, 则$|PQ|$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421352,7 +423660,9 @@ "id": "016506", "content": "设直线$l: y=k x+b$($k>0$)与圆$x^2+y^2=1$和圆$(x-4)^2+y^2=1$均相切, 则$l$的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421372,7 +423682,9 @@ "id": "016507", "content": "若点$P(1,1)$为圆$x^2+y^2-6 x=0$的弦$MN$的中点, 则弦$MN$所在直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421392,7 +423704,9 @@ "id": "016508", "content": "若圆$C_1: x^2+y^2=1$和圆$C_2: x^2+y^2-6 x-8 y-k=0$没有公共点, 则实数$k$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421412,7 +423726,9 @@ "id": "016509", "content": "已知圆心在直线$2 x-y-7=0$上的圆$C$与$y$轴交于两点$A(0,-4)$、$B(0,-2)$, 求圆$C$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421432,7 +423748,9 @@ "id": "016510", "content": "求经过下列各点的圆$x^2+y^2=4$的切线方程:\\\\\n(1) 过点$A(1, \\sqrt{3})$;\\\\\n(2) 过点$B(2,3)$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421452,7 +423770,9 @@ "id": "016511", "content": "已知直线$l: (2 m+1) x+(m+1) y=7 m+4$, 圆$C: (x-1)^2+(y-2)^2=25$.\\\\\n(1) 证明: $m \\in \\mathbf{R}$时, $l$与圆$C$恒相交;\\\\\n(2) 求相交弦长的最小值及相应的$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421472,7 +423792,9 @@ "id": "016512", "content": "已知圆$C$过点$P(1,1)$, 且与圆$M: (x+2)^2+(y+2)^2=r^2$($r>0$)关于直线$x+y+2=0$对称.\\\\\n(1) 求圆$C$的方程;\\\\\n(2) 设$Q$为圆$C$上的一个动点, 求$\\overrightarrow{PQ} \\cdot \\overrightarrow{MQ}$的最小值;\\\\\n(3) 过点$P$作两条相异直线分别与圆$C$相交于$A$、$B$, 且直线$PA$和直线$PB$的倾斜角互补, $O$为坐标原点, 试判断直线$OP$和$AB$是否平行? 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421492,7 +423814,9 @@ "id": "016513", "content": "已知圆系方程$x^2+y^2+2 k x+(4 k+10) y+5 k^2+20 k=0$($k \\in \\mathbf{R}$), 是否存在直线$l$, 使该方程表示的任何一圆截得的弦长是$4 \\sqrt{5}$, 若存在, 求出$l$的方程, 若不存在, 举一反例.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421512,7 +423836,9 @@ "id": "016514", "content": "已知直线$l$过点$(-1,0)$且与直线$2 x-y=0$垂直, 则圆$x^2+y^2-4 x+8 y=0$与直线$l$相交所得的弦长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421532,7 +423858,9 @@ "id": "016515", "content": "已知圆$C$的圆心坐标是$(0, m)$, 半径长是$r$. 若直线$2 x-y+3=0$与圆相切于点$A(-2,-1)$, 则圆$C$的标准方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421552,7 +423880,9 @@ "id": "016516", "content": "已知集合$A=\\{(x, y) | y=k x,\\ x \\in \\mathbf{R}\\}$, $B=\\{(x, y) | y=6-\\sqrt{9-x^2},\\ x \\in \\mathbf{R}\\}$, 且集合$A$与$B$的交集中有且只有两个元素, 则$k$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421572,7 +423902,9 @@ "id": "016517", "content": "在平面直角坐标系$xOy$中, $A$为直线$l: y=2 x$上在第一象限内的点, $B(5,0)$, 以$AB$为直径的圆$C$与直线$l$交于另一点$D$. 若$\\overrightarrow{AB} \\cdot \\overrightarrow{CD}=0$, 则点$A$的横坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421592,7 +423924,9 @@ "id": "016518", "content": "已知半径为$1$的圆经过点$(3,4)$, 则其圆心到原点的距离的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421612,7 +423946,9 @@ "id": "016519", "content": "已知点$M$为直线$x+y-3=0$上的动点, 过点$M$引圆$x^2+y^2=1$的两条切线, 切点分别为$A$、$B$, 则点$P(0,-1)$到直线$AB$的距离的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421632,7 +423968,9 @@ "id": "016520", "content": "直线$y=\\dfrac{\\sqrt{3}}{3} x$与圆$(x-1)^2+y^2=1$的位置关系是\\bracket{20}.\n\\fourch{相交且过圆心}{相交但不过圆心}{相切}{相离}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -421652,7 +423990,9 @@ "id": "016521", "content": "在坐标平面内, 与点$A(1,2)$的距离为$1$, 且与点$B(3,1)$的距离为$2$的直线共有\\bracket{20}.\n\\fourch{1 条}{2 条}{3 条}{4 条}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -421672,7 +424012,9 @@ "id": "016522", "content": "已知点$P$在圆$(x-5)^2+(y-5)^2=16$上, 点$A(4,0)$、$B(0,2)$, 则下列表述中错误的是\\bracket{20}.\n\\twoch{点$P$到直线$AB$的距离小于$10$}{点$P$到直线$AB$的距离大于$2$}{当$\\angle PBA$最小时, $|PB|=3 \\sqrt{2}$}{当$\\angle PBA$最大时, $|PB|=3 \\sqrt{2}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -421692,7 +424034,9 @@ "id": "016523", "content": "已知圆$C: (x-\\sqrt{2})^2+y^2=1$, 直线$l: \\sqrt{2} x-y+m=0$, 试问$m$为何值时, 直线$l$与圆$C$相切、相交、相离?", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421712,7 +424056,9 @@ "id": "016524", "content": "已知圆满足: \\textcircled{1} 截$y$轴所得弦长为$2$; \\textcircled{2} 被$x$轴分成两段圆弧, 其弧长的比为$3: 1$; \\textcircled{3} 圆心到直线$l: x-2 y=0$的距离为$\\dfrac{\\sqrt{5}}{5}$, 求该圆的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421732,7 +424078,9 @@ "id": "016525", "content": "已知点$P(2,2)$, 圆$C: x^2+y^2-8 y=0$, 过点$P$的动直线$l$与圆$C$交于$A$、$B$两点, 线段$AB$的中点为$M$, $O$为坐标原点.\\\\\n(1) 求$M$的轨迹方程;\\\\\n(2) 当$|OP|=|OM|$时, 求$l$的方程及$\\triangle POM$的面积.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421752,7 +424100,9 @@ "id": "016526", "content": "设$P$是椭圆$\\dfrac{x^2}{5}+\\dfrac{y^2}{3}=1$上的动点, 则$P$到该椭圆的两个焦点的距离之和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421772,7 +424122,9 @@ "id": "016527", "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{4}=1$的一个焦点为$(2,0)$, 则$C$的离心率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421792,7 +424144,9 @@ "id": "016528", "content": "已知椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$的焦点分别为$A$、$B$, 一条直线经过点$A$与椭圆交于$P$、$Q$两点, 联结$PB, QB$所得$\\triangle PQB$的周长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421812,7 +424166,9 @@ "id": "016529", "content": "椭圆$(1-m) x^2-m y^2=1$的长轴长为\\bracket{20}.\n\\fourch{$2 \\sqrt{\\dfrac{1}{1-m}}$}{$2 \\sqrt{-\\dfrac{1}{m}}$}{$2 \\sqrt{\\dfrac{1}{m-1}}$}{$2 \\sqrt{\\dfrac{1}{m}}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -421832,7 +424188,9 @@ "id": "016530", "content": "若椭圆的对称轴为坐标轴, 短轴的一个端点与两焦点组成一个正三角形, 焦点到椭圆上点的最短距离为$\\sqrt{3}$, 则椭圆方程为\\bracket{20}.\n\\twoch{$\\dfrac{x^2}{12}+\\dfrac{y^2}{9}=1$}{$\\dfrac{x^2}{9}+\\dfrac{y^2}{12}=1$}{$\\dfrac{x^2}{12}+\\dfrac{y^2}{9}=1$或$\\dfrac{x^2}{9}+\\dfrac{y^2}{12}=1$}{以上都不对}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -421852,7 +424210,9 @@ "id": "016531", "content": "根据下列条件求椭圆的标准方程.\\\\\n(1) 两焦点为$F_1(-4,0)$、$F_2(4,0)$, 点$P$在椭圆上, $\\triangle PF_1F_2$面积的最大值为$12$;\\\\\n(2) 与椭圆$x^2+\\dfrac{y^2}{81}=1$有相同的焦点, 且经过点$P(3,-3)$;\\\\\n(3) 经过两点$P_1(\\sqrt{6}, 1)$, $P_2(-\\sqrt{3},-\\sqrt{2})$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421872,7 +424232,9 @@ "id": "016532", "content": "设$A_1, A_2$分别是椭圆$\\Gamma: \\dfrac{x^2}{a^2}+y^2=1$($a>1$)的左、右顶点, 点$B$为椭圆的上顶点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse ({sqrt(2)} and 1);\n\\filldraw (1,0) node [below] {$F_2$} coordinate (F_2) circle (0.03);\n\\draw ({sqrt(2)},0) node [below right] {$A_2$} coordinate (A_2);\n\\draw ({-sqrt(2)},0) node [below left] {$A_1$} coordinate (A_1);\n\\draw (0,1) node [above right] {$B$} coordinate (B);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\overrightarrow{A_1B} \\cdot \\overrightarrow{A_2B}=-4$, 求椭圆$\\Gamma$的方程;\\\\\n(2) 设$a=\\sqrt{2}$, $F_2$是椭圆的右焦点, 点$Q$是椭圆第二象限部分上一点, 若线段$F_2Q$的中点$M$在$y$轴上, 求$\\triangle F_2BQ$的面积.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421892,7 +424254,9 @@ "id": "016533", "content": "点$A$、$B$分别是椭圆$\\dfrac{x^2}{36}+\\dfrac{y^2}{20}=1$长轴的左、右端点, 点$F$是椭圆的右焦点, 点$P$在椭圆上, 且位于$x$轴上方, $PA \\perp PF$.\\\\\n(1) 求点$P$的坐标;\\\\\n(2) 设$M$是椭圆长轴$AB$上的一点, $M$到直线$AP$的距离等于$|MB|$, 求椭圆上的点到点$M$的距离$d$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421912,7 +424276,9 @@ "id": "016534", "content": "已知椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的右焦点为$F$, 上顶点为$B$, 离心率为$\\dfrac{2 \\sqrt{5}}{5}$, 且$|BF|=\\sqrt{5}$.\\\\\n(1) 求椭圆的方程;\\\\\n(2) 直线$l$与椭圆有唯一的公共点$M$, 与$y$轴的正半轴交于点$N$, 过$N$与$BF$垂直的直线交$x$轴于点$P$. 若$MP\\parallel BF$, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -421932,7 +424298,9 @@ "id": "016535", "content": "已知椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), 把圆$x^2+y^2=\\dfrac{a^2 b^2}{a^2+b^2}$称为该椭圆的协同圆. 设椭圆$C: \\dfrac{x^2}{4}+\\dfrac{y^2}{2}=1$的协同圆为圆$O$($O$为坐标系原点), 直线$l$是圆$O$的任意一条切线, 且交椭圆$C$于$A$、$B$两点, 则$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421952,7 +424320,9 @@ "id": "016536", "content": "方程$y^2-(\\lg a) x^2=\\dfrac{1}{3}-a$表示焦点在$x$轴上的椭圆, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421972,7 +424342,9 @@ "id": "016537", "content": "椭圆$x^2+4 y^2=12$的焦点为$F_1$、$F_2$, 点$P$在椭圆上, 若线段$PF_1$的中点在$y$轴上, 则$|PF_1|:|PF_2|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -421992,7 +424364,9 @@ "id": "016538", "content": "已知椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{5}=1$的左焦点为$F$, 点$P$在椭圆上且在$x$轴的上方, 若线段$PF$的中点在以原点$O$为圆心, $|OF|$为半径的圆上, 则直线$PF$的斜率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422012,7 +424386,9 @@ "id": "016539", "content": "设椭圆$\\dfrac{x^2}{2}+y^2=1$的左右焦点为$F_1$、$F_2$, 点$P$在该椭圆上, 则使得$\\triangle PF_1F_2$为等腰三角形的点$P$的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422032,7 +424408,9 @@ "id": "016540", "content": "动点$M(x, y)$到两定点$A_1(0,5), A_2(0,-5)$的距离之和为$10$, 则点$M$的轨迹方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422052,7 +424430,9 @@ "id": "016541", "content": "已知椭圆$C$的方程为$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$), 右焦点为$F(\\sqrt{2}, 0)$, 且离心率为$\\dfrac{\\sqrt{6}}{3}$. 则椭圆$C$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422072,7 +424452,9 @@ "id": "016542", "content": "椭圆$\\dfrac{x^2}{36}+\\dfrac{y^2}{16}=1$的弦的中点为$P(3,2)$, 则使弦所在的直线方程为\\bracket{20}.\n\\fourch{$3 x+2 y-12=0$}{$2 x+3 y-12=0$}{$4 x+9 y-30=0$}{$9 x+4 y-35=0$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -422092,7 +424474,9 @@ "id": "016543", "content": "设$B$是椭圆$C: \\dfrac{x^2}{5}+y^2=1$的上顶点, 点$P$在$C$上, 则$|PB|$的最大值为\\bracket{20}.\n\\fourch{$\\dfrac{5}{2}$}{$\\sqrt{6}$}{$\\sqrt{5}$}{$2$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -422112,7 +424496,9 @@ "id": "016544", "content": "过点$P(-\\sqrt{3}, 1)$作倾斜角为$\\alpha$的直线$l$交椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}{16}=1$与$A$、$B$两点, 若弦$AB$的中点在$x$轴或$y$轴上, 则倾斜角$\\alpha$的值为\\bracket{20}.\n\\fourch{$0$}{$0$或$\\dfrac{5 \\pi}{6}$}{$\\dfrac{\\pi}{2}$或$\\dfrac{5 \\pi}{6}$}{$0$或$\\dfrac{\\pi}{2}$或$\\dfrac{5 \\pi}{6}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -422132,7 +424518,9 @@ "id": "016545", "content": "设椭圆$\\dfrac{x^2}{m^2+1}+\\dfrac{y^2}{m^2}=1$($m>0$)的两个焦点分别为$F_1$、$F_2, M$是椭圆上任意一点, $\\triangle F_1MF_2$的周长为$2+2 \\sqrt{2}$.\\\\\n(1) 求椭圆的方程;\\\\\n(2) 过椭圆在$y$轴负半轴上的顶点$B$及椭圆右焦点$F_2$作一直线交椭圆于另一点$N$, 求$\\angle F_1NB$的大小(结果求反三角函数值表示).", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422152,7 +424540,9 @@ "id": "016546", "content": "已知椭圆$x^2+2 y^2=1$, 过原点的两条直线$l_1$和$l_2$分别与椭圆交于$A$、$B$和$C$、$D$, 记得到的平行四边形$ACBD$的面积为$S$.\n\\\\\n(1) 设$A(x_1, y_1)$、$C(x_2, y_2)$, 用$A, C$的坐标表示点$C$到直线$l_1$的距离, 并证明$S=2|x_1 y_2-x_2 y_1|$;\\\\\n(2) 设$l_1, l_2$的斜率之积为$-\\dfrac{1}{2}$, 求面积$S$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422172,7 +424562,9 @@ "id": "016547", "content": "在平面直角坐标系$xOy$中, 已知椭圆$\\Gamma: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的长轴长为$6$, 且经过点$Q(\\dfrac{3}{2}, \\sqrt{3}), A$为左顶点, $B$为下顶点, 椭圆上的点$P$在第一象限, $PA$交$y$轴于点$C, PB$交$x$轴于点$D$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\path [->,draw, name path = xaxis] (-4,0) -- (4,0) node [below] {$x$};\n\\path [->,draw, name path = yaxis] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse (3 and 2);\n\\draw (-3,0) node [below left] {$A$} coordinate (A);\n\\draw (0,-2) node [below right] {$B$} coordinate (B);\n\\draw (50:3 and 2) node [above] {$P$} coordinate (P);\n\\path [draw,name path = AP] (A)--(P);\n\\path [draw,name path = BP] (B)--(P);\n\\draw (A)--(B);\n\\draw [name intersections = {of = AP and yaxis, by = C}];\n\\draw (C) node [above left] {$C$};\n\\draw [name intersections = {of = BP and xaxis, by = D}];\n\\draw (D) node [below] {$D$};\n\\draw (C)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求椭圆$\\Gamma$的标准方程;\\\\\n(2) 若$\\overrightarrow{OB}+2 \\overrightarrow{OC}=\\overrightarrow{0}$, 求线段$AP$的长;\\\\\n(3) 试问: 四边形$ABDC$的面积是否为定值? 若是, 求出该定值; 若不是, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422192,7 +424584,9 @@ "id": "016548", "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的一条渐近线为$2 x+y=0$, 一个焦点为$(\\sqrt{5}, 0)$, 则双曲线的标准方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422212,7 +424606,9 @@ "id": "016549", "content": "已知双曲线$\\dfrac{x^2}{a^2}-y^2=1$($a>0$)的离心率是$\\sqrt{5}$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422232,7 +424628,9 @@ "id": "016550", "content": "已知$F$是双曲线$C: \\dfrac{x^2}{4}-\\dfrac{y^2}{5}=1$的一个焦点, 点$P$在$C$上, $O$为坐标原点, 若$|OP|=|OF|$, 则$\\triangle OPF$的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422252,7 +424650,9 @@ "id": "016551", "content": "已知双曲线$C_1$、$C_2$的顶点重合, $C_1$的方程为$\\dfrac{x^2}{4}-y^2=1$, 若$C_2$的一条渐近线的斜率是$C_1$的一条渐近线的斜率的$2$倍, 则$C_2$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422272,7 +424672,9 @@ "id": "016552", "content": "设$P$是双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{9}=1$上一点, 双曲线的一条渐近线方程为$3 x-2 y=0$, $F_1$、$F_2$分别是双曲线的左、右焦点, 若$|PF_1|=3$, 则$|PF_2|=$\\bracket{20}.\n\\fourch{$1$或$5$}{$6$}{$7$}{$9$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -422292,7 +424694,9 @@ "id": "016553", "content": "求由下列条件所确定的双曲线的标准方程.\\\\\n(1) 实轴在$y$轴上, 且实轴长是虚轴长的$2$倍, 焦距是$10$;\\\\\n(2) 与椭圆$\\dfrac{x^2}{12}+\\dfrac{y^2}{3}=1$有公共焦点, 一条渐近线方程为$y=\\dfrac{\\sqrt{5}}{2} x$;\\\\\n(3) 过点$A(2,2)$和$B(4,2 \\sqrt{13})$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422312,7 +424716,9 @@ "id": "016554", "content": "双曲线$\\Gamma: \\dfrac{x^2}{16}-\\dfrac{y^2}{9}=1$的左、右焦点分别为$F_1$、$F_2$, 直线$l$经过$F_2$且与$\\Gamma$的两条渐近线中的一条平行, 与另一条相交且交点在第一象限.\\\\\n(1) 设$P$为$\\Gamma$右支上的任意一点, 求$|PF_1|$的最小值;\\\\\n(2) 设$O$为坐标原点, 求$O$到$l$的距离, 并求$l$与$\\Gamma$的交点坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422332,7 +424738,9 @@ "id": "016555", "content": "双曲线$x^2-\\dfrac{y^2}{b^2}=1$($b>0$)的左、右焦点分别为$F_1$、$F_2$, 直线$l$过$F_2$且与双曲线交于$A$、$B$两点.\\\\\n(1) 若$l$的倾斜角为$\\dfrac{\\pi}{2}$, $\\triangle F_1AB$是等边三角形, 求双曲线的渐近线方程;\\\\\n(2) 设$b=\\sqrt{3}$, 若$l$的斜率存在, 且$|AB|=4$, 求$l$的斜率.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422352,7 +424760,9 @@ "id": "016556", "content": "设双曲线$C: \\dfrac{x^2}{a^2}-y^2=1$($a>0$)与直线$l: x+y=1$相交于两个不同的点$A$、$B$.\\\\\n(1) 求$a$的取值范围;\\\\\n(2) 设直线$l$与$y$轴的交点为$P$, 且$\\overrightarrow{PA}=\\dfrac{5}{12} \\overrightarrow{PB}$, 求$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422372,7 +424782,9 @@ "id": "016557", "content": "已知双曲线$\\Gamma: \\dfrac{x^2}{a^2}-y^2=1$($a>0$), 若对于双曲线右支上任意两点$P_1(x_1, y_1)$、$P_2(x_2, y_2)$均有$x_1 x_2-y_1 y_2>0$, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422392,7 +424804,9 @@ "id": "016558", "content": "双曲线$x^2-y^2=16$的两条渐近线的夹角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422412,7 +424826,9 @@ "id": "016559", "content": "双曲线$\\dfrac{x^2}{m^2-4}-\\dfrac{y^2}{m+1}=1$的焦点在$y$轴上, 则$m$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422432,7 +424848,9 @@ "id": "016560", "content": "在平面直角坐标系$xOy$中, 若双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的右焦点$F(c, 0)$到一条渐近线的距离为$\\dfrac{\\sqrt{3}}{2} c$, 则其离心率的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422452,7 +424870,9 @@ "id": "016561", "content": "已知点$F_1(-\\sqrt{2}, 0)$、$F_2(\\sqrt{2}, 0)$, 动点$P$满足$|PF_2|-|PF_1|=2$. 当点$P$的纵坐标是$\\dfrac{1}{2}$时, 点$P$到坐标原点的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422472,7 +424892,9 @@ "id": "016562", "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($b>a>0$)的左、右焦点分别为$F_1$、$F_2$, 过右焦点作平行于一条渐近线的直线交双曲线于点$A$, 若$\\triangle AF_1F_2$的内切圆半径为$\\dfrac{b}{4}$, 则双曲线的离心率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422492,7 +424914,9 @@ "id": "016563", "content": "过双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的右顶点作$x$轴的垂线与$C$的一条渐近线相交于点$A$. 若以$C$的右焦点为圆心, 以$2$为半径的圆经过$A$、$O$两点 ($O$为坐标原点), 则双曲线$C$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422512,7 +424936,9 @@ "id": "016564", "content": "下列关于双曲线$\\Gamma: \\dfrac{x^2}{6}-\\dfrac{y^2}{3}=1$的判断中, 正确的是\\bracket{20}.\n\\twoch{渐近线方程为$x \\pm 2 y=0$}{实轴长为 12}{焦点坐标为$(\\pm 3,0)$}{顶点坐标为$(\\pm 6,0)$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -422532,7 +424958,9 @@ "id": "016565", "content": "设$F_1$、$F_2$是双曲线$C: x^2-\\dfrac{y^2}{3}=1$的两个焦点, $O$为坐标原点, 点$P$在$C$上且$|OP|=2$, 则$\\triangle PF_1F_2$的面积为\\bracket{20}.\n\\fourch{$\\dfrac{7}{2}$}{$3$}{$\\dfrac{5}{2}$}{2}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -422552,7 +424980,9 @@ "id": "016566", "content": "设点$M$、$N$均在双曲线$C: \\dfrac{x^2}{4}-\\dfrac{y^2}{3}=1$上运动, $F_1$、$F_2$是双曲线$C$的左、右焦点, 则$| \\overrightarrow{MF_1}+$$\\overrightarrow{MF_2}-2 \\overrightarrow{MN} |$的最小值为\\bracket{20}.\n\\fourch{$2 \\sqrt{3}$}{$4$}{$2 \\sqrt{7}$}{以上都不对}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -422572,7 +425002,9 @@ "id": "016567", "content": "已知$a \\in \\mathbf{R}$, 双曲线$\\Gamma: \\dfrac{x^2}{a^2}-y^2=1$.\\\\\n(1) 若点$(2,1)$在双曲线上, 求$\\Gamma$的焦点坐标;\\\\\n(2) 若$a=1$, 直线$y=k x+1$与$\\Gamma$相交于$A$、$B$两点, 且线段$AB$中点的横坐标为$1$, 求实数$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422592,7 +425024,9 @@ "id": "016568", "content": "设双曲线$C: \\dfrac{x^2}{2}-\\dfrac{y^2}{3}=1$, $F_1$、$F_2$为其左右两个焦点.\\\\\n(1) 设$O$为坐标原点, $M$为双曲线$C$右支上任意一点, 求$\\overrightarrow{OM} \\cdot \\overrightarrow{F_1M}$的取值范围;\\\\\n(2) 若动点$P$与双曲线$C$的两个焦点$F_1$、$F_2$的距离之和为定值, 且$\\cos \\angle F_1PF_2$的最小值为$-\\dfrac{1}{9}$, 求动点$P$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422612,7 +425046,9 @@ "id": "016569", "content": "已知双曲线$C$的中心在原点, $D(1,0)$是它的一个顶点, $\\overrightarrow {d}=(1, \\sqrt{2})$是它的一条渐近线的一个方向向量.\\\\\n(1) 求双曲线$C$的方程;\\\\\n(2) 若过点$(-3,0)$任意作一条直线与双曲线$C$交于$A$、$B$两点($A$、$B$都不同于点$D$), 求$\\overrightarrow{DA} \\cdot \\overrightarrow{DB}$的值;\\\\\n(3) 对于双曲线$\\Gamma: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$, $a \\neq b$), $E$为它的右顶点, $M$、$N$为双曲线$\\Gamma$上的两点($M$、$N$都不同于点$E$), 且$EM \\perp EN$, 求证: 直线$MN$与$x$轴的交点是一个定点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422632,7 +425068,9 @@ "id": "016570", "content": "若抛物线$y^2=2 p x$的焦点与双曲线$x^2-y^2=2$的右焦点重合, 则$p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422652,7 +425090,9 @@ "id": "016571", "content": "抛物线$x=8 y^2$的准线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422672,7 +425112,9 @@ "id": "016572", "content": "抛物线$y^2=2 p x$($p>0$)上的动点$Q$到焦点的距离的最小值为$1$, 则$p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422692,7 +425134,9 @@ "id": "016573", "content": "斜率为$\\sqrt{3}$的直线过抛物线$C: y^2=4 x$的焦点, 且与$C$交于$A$、$B$两点, 则$|AB|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422712,7 +425156,9 @@ "id": "016574", "content": "设$O$为坐标原点, 直线$x=2$与抛物线$C: y^2=2px$($p>0$)交于$D$、$E$两点, 若$OD \\perp OE$, 则$C$的焦点坐标为\\bracket{20}.\n\\fourch{$(\\dfrac{1}{4}, 0)$}{$(\\dfrac{1}{2}, 0)$}{$(1,0)$}{$(2,0)$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -422732,7 +425178,9 @@ "id": "016575", "content": "求满足下列条件的抛物线的方程:\\\\\n(1) 焦点坐标为$(0,-2)$, 顶点在原点;\\\\\n(2) 与椭圆$4 x^2+5 y^2=20$有相同焦点, 顶点在原点;\\\\\n(3) 对称轴是$x$轴, 顶点在原点, 抛物线上的点$M(-3, m)$到焦点的距离是$5$;\\\\\n(4) 以原点为顶点, 坐标轴为对称轴, 且过直线$y=x$与圆$x^2+y^2+6 x=0$的交点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422752,7 +425200,9 @@ "id": "016576", "content": "已知抛物线$C: y^2=2 p x$($p>0$)的焦点$F$到准线的距离为$2$.\\\\\n(1) 求$C$的方程;\\\\\n(2) 已知$O$为坐标原点, 点$P$在$C$上, 点$Q$满足$\\overrightarrow{PQ}=9 \\overrightarrow{QF}$, 求直线$OQ$斜率的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422772,7 +425222,9 @@ "id": "016577", "content": "已知动点$M$到直线$x+2=0$的距离比到点$F(1,0)$的距离大$1$.\\\\\n(1) 求动点$M$所在的曲线$C$的方程;\\\\\n(2) 已知点$P(1,2)$, $A$、$B$是曲线$C$上的两个动点, 如果直线$PA$的斜率与直线$PB$的斜率互为相反数, 证明直线$AB$的斜率为定值, 并求出这个定值;\\\\\n(3) 已知点$P(1,2)$, $A$、$B$是曲线$C$上的两个动点, 如果直线$PA$的斜率与直线$PB$的斜率之和为$2$, 证明: 直线$AB$过定点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422792,7 +425244,9 @@ "id": "016578", "content": "已知抛物线方程$y^2=4 x$, $F$为焦点, $P$为抛物线准线上一点, $Q$为线段$PF$与抛物线的交点, 定义: $d(P)=\\dfrac{|PF|}{|FQ|}$.\\\\\n(1) 当$P(-1,-\\dfrac{8}{3})$时, 求$d(P)$;\\\\\n(2) 证明: 存在常数$a$, 使得$2 d(P)=|PF|+a$;\\\\\n(3) $P_1$、$P_2$、$P_3$为抛物线准线上三点, 且$|P_1P_2|=|P_2P_3|$, 判断$d(P_1)+d(P_3)$与$2 d(P_2)$的关系.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -422812,7 +425266,9 @@ "id": "016579", "content": "设抛物线的顶点在原点, 准线方程为$x=-2$, 则抛物线的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422832,7 +425288,9 @@ "id": "016580", "content": "抛物线以原点为顶点, 在坐标轴为对称轴, 且焦点在直线$x-2 y-4=0$上, 则抛物线的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422852,7 +425310,9 @@ "id": "016581", "content": "已知抛物线$C: y^2=4 x$的焦点为$F$, 直线$y=2 x-4$与$C$交于$A$、$B$两点, 则$\\cos \\angle AFB=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422872,7 +425332,9 @@ "id": "016582", "content": "如图是抛物线拱桥, 当水面离桥顶$2 \\mathrm{m}$时, 水面宽$4 \\mathrm{m}$, 若水面下降$1 \\mathrm{m}$, 则此时水面宽为\\blank{50}$\\mathrm{m}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\fill [pattern = bricks, domain = -3:3, samples = 100] plot (\\x,{-\\x*\\x/2}) --++ (1,0) --++ (0,5.5) --++ (-8,0) --++ (0,-5.5) -- cycle;\n\\draw [domain = -3:3, samples = 100] plot (\\x,{-\\x*\\x/2}) --++ (1,0) --++ (0,5.5) --++ (-8,0) --++ (0,-5.5) -- cycle;\n\\draw (-2,-2) -- (2,-2);\n\\draw (-2,-2) --++ (0,-3) (2,-2) --++ (0,-3);\n\\draw [<->] (-2,-4.5) -- (2,-4.5) node [midway, fill = white] {$4\\mathrm{m}$};\n\\draw [<->] (0,0) --++ (0,-2) node [midway, fill = white] {$2\\mathrm{m}$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422892,7 +425354,9 @@ "id": "016583", "content": "已知抛物线$x^2+m y=0$上的点到定点$(0,4)$和到定直线$y=-4$的距离相等, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422912,7 +425376,9 @@ "id": "016584", "content": "若抛物线$y^2=8 x$的焦点$F$与双曲线$\\dfrac{x^2}{3}-\\dfrac{y^2}{n}=1$的一个焦点重合, 则$n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -422932,7 +425398,9 @@ "id": "016585", "content": "已知$A$、$B$为平面内两定点, 过该平面内动点$M$作直线$AB$的垂线, 垂足为$N$. 若$\\overrightarrow{MN}^2=$$\\lambda \\overrightarrow{AN} \\cdot \\overrightarrow{NB}$, 其中$\\lambda$为常数, 则动点$M$的轨迹不可能是 \\bracket{20}.\n\\fourch{圆}{椭圆}{抛物线}{双曲线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -422952,7 +425420,9 @@ "id": "016586", "content": "设抛物线的顶点为$O$, 焦点为$F$, 准线为$l . P$是抛物线上异于$O$的一点, 过$P$作$PQ \\perp l$于$Q$, 则线段$FQ$的垂直平分线\\bracket{20}.\n\\fourch{经过点$O$}{经过点$P$}{平行于直线$OP$}{垂直于直线$OP$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -422972,7 +425442,9 @@ "id": "016587", "content": "设抛物线$y^2=8 x$的准线与$x$轴交于点$Q$, 若过点${Q}$的直线$l$与抛物线有公共点, 则直线$l$的斜率的取值范围是\\bracket{20}.\n\\fourch{$[-\\dfrac{1}{2}, \\dfrac{1}{2}]$}{$[-2,2]$}{$[-1,1]$}{$[-4,4]$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -422992,7 +425464,9 @@ "id": "016588", "content": "已知直线$l: y=k x-4$和抛物线$C: y^2=8 x$有且只有一个公共点, 求实数$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423012,7 +425486,9 @@ "id": "016589", "content": "动圆$C$过定点$(1,0)$, 且与直线$x=-1$相切, 设圆心$C$的轨迹$\\Gamma$方程为$F(x, y)=0$.\\\\\n(1) 求圆心$C$的轨迹方程$\\Gamma$;\\\\\n(2) 曲线$\\Gamma$上一定点$P(x_0, 2)$, 方向向量$\\overrightarrow {d}=(1,-1)$的直线$l$(不过$P$点) 与曲线$\\Gamma$交与$A$、$B$两点, 设直线$PA$、$PB$斜率分别为$k_{PA}$、$k_{PB}$, 证明: $k_{PA}+k_{PB}=0$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423032,7 +425508,9 @@ "id": "016590", "content": "已知斜率为$k$的直线$l$经过抛物线$C: y^2=4 x$的焦点$F$, 且与抛物线$C$交于不同的两点$A(x_1, y_1)$、$B(x_2, y_2)$.\n(1) 若点$A$和$B$到抛物线准线的距离分别为$\\dfrac{3}{2}$和$3$, 求$|AB|$;\\\\\n(2) 若$|AF|+|AB|=2|BF|$, 求$k$的值;\\\\\n(3) 点$M(t, 0)$, $t>0$, 对任意确定的实数$k$, 若$\\triangle AMB$是以$AB$为斜边的直角三角形, 判断符合条件的点$M$有几个, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423052,7 +425530,9 @@ "id": "016591", "content": "已知$AB$是双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$左支上过焦点$F_1$的弦, $|AB|=m$, $F_2$为右焦点, 则$\\triangle ABF_2$的周长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423072,7 +425552,9 @@ "id": "016592", "content": "过抛物线$y^2=4 x$的焦点作倾斜角为$\\dfrac{\\pi}{4}$的直线交抛物线于$A$、$B$两点, 则$|AB|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423092,7 +425574,9 @@ "id": "016593", "content": "直线$y=k x+1$与焦点在$x$轴上的椭圆$\\dfrac{x^2}{5}+\\dfrac{y^2}{m}=1$总有公共点, 则$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423112,7 +425596,9 @@ "id": "016594", "content": "设双曲线$x^2-y^2=6$的左右顶点分别为$A_1$、$A_2, P$为双曲线右支上一点, 且位于第一象限, 直线$PA_1$、$PA_2$的斜率分别为$k_1$、$k_2$, 则$k_1 \\cdot k_2$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423132,7 +425618,9 @@ "id": "016595", "content": "已知点$M(-1,1)$和抛物线$C: y^2=4 x$, 过$C$的焦点且斜率为$k$的直线与$C$交于$A$、$B$两点, 若$\\angle AMB=90{^\\circ}$, 则$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423152,7 +425640,9 @@ "id": "016596", "content": "过点$(0,3)$的直线$l$与下列曲线只有一个公共点, 求直线$l$的方程:\\\\\n(1) 椭圆$\\dfrac{x^2}{4}+y^2=1$;\\\\\n(2) 双曲线$\\dfrac{x^2}{4}-\\dfrac{y^2}{3}=1$;\\\\\n(3) 抛物线$x^2=-y$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423172,7 +425662,9 @@ "id": "016597", "content": "已知直线$l: y=a x+1$和双曲线$C: 3 x^2-y^2=1$交于$A$、$B$两点.\\\\\n(1) 当$a$为何值时, 以$AB$为直径的圆过原点;\\\\\n(2) 是否存在实数$a$, 使得$A$、$B$两点关于直线$y=2 x$对称.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423192,7 +425684,9 @@ "id": "016598", "content": "已知双曲线$\\Gamma: x^2-\\dfrac{y^2}{b^2}=1$($b>0$), 直线$l: y=k x+m$($k m \\neq 0$), $l$与$\\Gamma$交于$P$、$Q$两点, $P'$为$P$关于$y$轴的对称点, 直线$P'Q$与$y$轴交于点$N(0, n)$.\\\\\n(1) 若点$(2,0)$是$\\Gamma$的一个焦点, 求$\\Gamma$的渐近线方程;\\\\\n(2) 若$b=1$, 点$P$的坐标为$(-1,0)$, 且$\\overrightarrow{NP'}=\\dfrac{3}{2} \\overrightarrow{P' Q}$, 求$k$的值;\\\\\n(3) 若$m=2$, 求$n$关于$b$的表达式.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423212,7 +425706,9 @@ "id": "016599", "content": "已知抛物线$C: y^2=2 p x$($p>0$), 直线$l$交此抛物线于不同的两个点$A(x_1, y_1)$、$B(x_2, y_2)$, 记$N(p, 0)$, 如果直线$l$过点$M(-p, 0)$, 设线段$AB$的中点为点为$Q$, 问是否存在一条直线和一个定点, 使得点$Q$到它们的距离相等? 若存在, 求出这条直线和这个定点; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423232,7 +425728,9 @@ "id": "016600", "content": "过点$M(0,-\\dfrac{1}{2})$的直线与双曲线$x^2-y^2=\\lambda$($\\lambda \\neq 0$)恒有公共点, 则$\\lambda$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423252,7 +425750,9 @@ "id": "016601", "content": "抛物线$x^2=y$上的点到直线$2 x-y-4=0$的距离最小的点的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423272,7 +425772,9 @@ "id": "016602", "content": "已知点$P$是椭圆$\\dfrac{x^2}{45}+\\dfrac{y^2}{20}=1$上第三象限内的一点, 且与两焦点的连线互相垂直, 若$P$到直线$4 x-3 y-2 m+1=0$的距离不大于$3$, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423292,7 +425794,9 @@ "id": "016603", "content": "抛物线$y^2=2 p x$($p>0$)的焦点作直线交抛物线于$E(x_1, y_1)$、$F(x_2, y_2)$, 若$x_1+x_2=3 p$, 则弦$EF$的长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423312,7 +425816,9 @@ "id": "016604", "content": "已知抛物线$y^2=2 p x$($p>0$)的焦点为$F$, $A$、$B$为此抛物线上的异于坐标原点$O$的两个不同的点, 满足$\\overrightarrow{FA}+\\overrightarrow{FB}+\\overrightarrow{FO}=\\overrightarrow{0}$, 则$|\\overrightarrow{FA}|+|\\overrightarrow{FB}|+|\\overrightarrow{FO}|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423332,7 +425838,9 @@ "id": "016605", "content": "过点$M(-2,0)$的直线$l$与椭圆$x^2+2 y^2=2$交于点$P_1$、$P_2$, 线段$P_1P_2$的中点为$P$, 设直线$l$的斜率为$k_1$($k_1 \\neq 0$), 直线$OP$的斜率为$k_2$, 则$k_1 k_2$的值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423352,7 +425860,9 @@ "id": "016606", "content": "过点$(2,4)$作直线与抛物线$y^2=8 x$只有一个公共点, 这样的直线共有\\bracket{20}.\n\\fourch{1 条}{2 条}{3 条}{4 条}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -423372,7 +425882,9 @@ "id": "016607", "content": "过点$(\\sqrt{7}, 5)$与双曲线$\\dfrac{x^2}{7}-\\dfrac{y^2}{25}=1$有且只有一个公共点的直线有\\bracket{20}.\n\\fourch{1 条}{2 条}{3 条}{4 条}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -423392,7 +425904,9 @@ "id": "016608", "content": "$F_1(-1,0)$、$F_2(1,0)$是椭圆的两焦点, 过$F_1$的直线$l$交椭圆于$M$、$N$, 若$\\triangle MF_2N$的周长为$8$, 则椭圆方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$}{$\\dfrac{y^2}{4}+\\dfrac{x^2}{3}=1$}{$\\dfrac{x^2}{16}+\\dfrac{y^2}{15}=1$}{$\\dfrac{y^2}{16}+\\dfrac{x^2}{15}=1$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -423412,7 +425926,9 @@ "id": "016609", "content": "已知双曲线$C_1: x^2-\\dfrac{y^2}{4}=1$.\\\\\n(1) 求与双曲线$C_1$有相同的焦点, 且过点$P(4, \\sqrt{3})$的双曲线$C_2$的标准方程;\\\\\n(2) 直线$l: y=x+m$分别交双曲线$C_1$的两条渐近线于$A$、$B$两点. 当$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=3$时, 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423432,7 +425948,9 @@ "id": "016610", "content": "椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)过点$M(2,0)$, 且右焦点为$F(1,0)$, 过$F$的直线$l$与椭圆$C$相交于$A$、$B$两点. 设点$P(4,3)$, 记$PA$、$PB$的斜率分别为$k_1$和$k_2$.\\\\\n(1) 求椭圆$C$的方程;\\\\\n(2) 如果直线$l$的斜率等于$-1$, 求出$k_1 \\cdot k_2$的值;\\\\\n(3) 探讨$k_1+k_2$是否为定值? 如果是, 求出该定值; 如果不是, 求出$k_1+k_2$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423452,7 +425970,9 @@ "id": "016611", "content": "设抛物线$C: y^2=2 p x$($p>0$)的焦点为$F$, 经过点$F$的动直线$l$交抛物线$C$于$A(x_1, y_1)$、$B(x_2, y_2)$两点, 且$y_1 y_2=-4$.\\\\\n(1) 求抛物线$C$的方程;\\\\\n(2) 若直线$2 x+3 y=0$平分线段$AB$, 求直线$l$的倾斜角;\\\\\n(3) 若点$M$是抛物线$C$的准线上的一点, 直线$MF, MA, MB$, 的斜率分别为$k_0, k_1, k_2$, 求证: 当$k_0=1$时, $k_1+k_2$为定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423472,7 +425992,9 @@ "id": "016612", "content": "如果曲线$C$上任意一点的坐标都是方程$F(x, y)=0$的解, 则下列命题中正确的是\\bracket{20}.\n\\onech{曲线$C$的方程是$F(x, y)=0$}{曲线$C$上的点都在方程$F(x, y)=0$的曲线上}{方程$F(x, y)=0$的曲线是$C$}{以方程$F(x, y)=0$的解为坐标的点都在曲线$C$上}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -423492,7 +426014,9 @@ "id": "016613", "content": "方程$y=\\sqrt{4-x^2}$所对应的曲线为\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) circle (2);\n\\draw (2,0) node [below right] {$2$};\n\\draw (-2,0) node [below left] {$-2$};\n\\draw (0,2) node [above left] {$2$};\n\\draw (0,-2) node [below left] {$-2$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (2,0) arc (0:180:2);\n\\draw (2,0) node [below right] {$2$};\n\\draw (-2,0) node [below left] {$-2$};\n\\draw (0,2) node [above left] {$2$};\n\\draw (0,-2) node [below left] {$-2$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,2) arc (90:-90:2);\n\\draw (2,0) node [below right] {$2$};\n\\draw (-2,0) node [below left] {$-2$};\n\\draw (0,2) node [above left] {$2$};\n\\draw (0,-2) node [below left] {$-2$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (2,0) arc (0:90:2);\n\\draw (2,0) node [below right] {$2$};\n\\draw (-2,0) node [below left] {$-2$};\n\\draw (0,2) node [above left] {$2$};\n\\draw (0,-2) node [below left] {$-2$};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -423512,7 +426036,9 @@ "id": "016614", "content": "已知椭圆$\\dfrac{x^2}{2}+y^2=1$, 作垂直于$x$轴的垂线交椭圆于$A$、$B$两点, 作垂直于$y$轴的垂线交椭圆于$C$、$D$两点, 且$AB=CD$, 两垂线相交于点$P$, 则点$P$的轨迹是\\bracket{20}.\n\\fourch{椭圆}{双曲线}{圆}{抛物线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -423532,7 +426058,9 @@ "id": "016615", "content": "若点$M(\\dfrac{m}{2},-m)$在方程$x^2+(y-1)^2=10$所表示的曲线上, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423552,7 +426080,9 @@ "id": "016616", "content": "直角坐标平面$xOy$中, 若定点$A(1,2)$与动点$P(x, y)$满足$\\overrightarrow{OP} \\cdot \\overrightarrow{OA}=4$, 则点$P$的轨迹方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423572,7 +426102,9 @@ "id": "016617", "content": "已知曲线$C: m x^2+n y^2=1$.\\\\\n\\textcircled{1} 若$m>n>0$, 则$C$是椭圆, 其焦点在$y$轴上;\\\\\n\\textcircled{2} 若$m=n>0$, 则$C$是圆, 其半径为$\\sqrt{n}$;\\\\\n\\textcircled{3} 若$m n<0$, 则$C$是双曲线, 其渐近线为$y=\\pm \\sqrt{-\\dfrac{m}{n}}x$;\\\\\n\\textcircled{4} 若$m=0, n>0$, 则$C$是两条直线.\\\\\n则以上命题中真命题的序号有\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423592,7 +426124,9 @@ "id": "016618", "content": "数学中有许多形状优美、寓意美好的曲线, 曲线$C: x^2+y^2=1+|x| y$就是其中之一(如图). 给出下列三个结论:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -90:90, samples = 100] plot (\\x:{1/sqrt(1-sin(\\x)*cos(\\x))}) plot ({180-\\x}:{1/sqrt(1-sin(\\x)*cos(\\x))});\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 曲线$C$恰好经过$6$个整点 (即横、纵坐标均为整数的点);\\\\\n\\textcircled{2} 曲线$C$上任意一点到原点的距离都不超过$\\sqrt{2}$;\\\\\n\\textcircled{3} 曲线$C$所围成的``心形''区域的面积小于$3$.\\\\\n其中, 所有正确结论的序号是\\bracket{20}.\n\\fourch{\\textcircled{1}}{\\textcircled{2}}{\\textcircled{1}\\textcircled{2}}{\\textcircled{1}\\textcircled{2}\\textcircled{3}}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -423612,7 +426146,9 @@ "id": "016619", "content": "一动点$P$到$F(-4,0)$的距离与它到定直线$l: x=4$的距离相等, 求此动点$P$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423632,7 +426168,9 @@ "id": "016620", "content": "一动点到$F(-4,0)$的距离比它到$y$轴的距离大$4$, 求此动点的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423652,7 +426190,9 @@ "id": "016621", "content": "已知动点$P$到定点$F(1,0)$和直线$x=3$的距离之和等于$4$, 求点$P$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423672,7 +426212,9 @@ "id": "016622", "content": "已知直线$l$过定点$(0,3)$, 且是曲线$y^2=4 x$的动弦$P_1P_2$的中垂线, 求直线$l$与动弦$P_1P_2$的交点$M$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423692,7 +426234,9 @@ "id": "016623", "content": "已知向量$\\overrightarrow {a}=(x, \\sqrt{3} y)$, $\\overrightarrow {b}=(1,0)$, 且$(\\overrightarrow {a}+\\sqrt{3} \\overrightarrow {b}) \\perp(\\overrightarrow {a}-\\sqrt{3} \\overrightarrow {b})$, 则点$Q(x, y)$的轨迹$C$的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423712,7 +426256,9 @@ "id": "016624", "content": "已知$A(\\sqrt{3}, 0)$, $Q$为圆$x^2+y^2=4$上的动点, 作$AQ$的垂直平分线交线段$OQ$于$T$, 则点$T$的轨迹方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423732,7 +426278,9 @@ "id": "016625", "content": "曲线$4 x^2+9 y^2=36$关于点$Q(-3,1)$对称的曲线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423752,7 +426300,9 @@ "id": "016626", "content": "已知动点$P$在圆$x^2+y^2=4$上运动, 点$Q(4,0)$、点$M$在线段$PQ$上, 且$\\overrightarrow{PQ}=2 \\overrightarrow{MQ}$, 则点$M$的轨迹方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423772,7 +426322,9 @@ "id": "016627", "content": "已知点$A(-2,0)$、$B(3,0)$, 动点$P(x, y)$满足$\\overrightarrow{PA} \\cdot \\overrightarrow{PB}=x^2$, 则点$P$的轨迹是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423792,7 +426344,9 @@ "id": "016628", "content": "设过点$P(x, y)$的直线分别与$x$轴的正半轴和$y$轴的正半轴交于$A$、$B$两点, 点$Q$与点$P$关于$y$Y 轴对称, $O$为坐标原点, 若$\\overrightarrow{BP}=2 \\overrightarrow{PA}$, 且$\\overrightarrow{OQ} \\cdot \\overrightarrow{AB}=1$, 则$P$点的轨迹方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423812,7 +426366,9 @@ "id": "016629", "content": "曲线$F(x, y)=0$关于直线$x-y-2=0$对称的曲线方程为\\bracket{20}.\n\\fourch{$F(y+2, x)=0$}{$F(x-2, y)=0$}{$F(y+2, x-2)=0$}{$F(y-2, x+2)=0$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -423832,7 +426388,9 @@ "id": "016630", "content": "方程$|x|-1=\\sqrt{1-y^2}$所表示的曲线是\\bracket{20}.\n\\fourch{一个圆}{两个圆}{半个圆}{两个半圆}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -423852,7 +426410,9 @@ "id": "016631", "content": "已知$a$、$b \\in \\mathbf{R}$, $ab>0$, 函数$f(x)=a x^2+b$($x \\in \\mathbf{R}$). 若$f(s-t), f(s), f(s+t)$成等比数列, 则平面上点$(s, t)$的轨迹是\\bracket{20}.\n\\fourch{直线和圆}{直线和椭圆}{直线和双曲线}{直线和抛物线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -423872,7 +426432,9 @@ "id": "016632", "content": "已知一条长为$6$的线段两端点$A$、$B$分别在$x$、$y$轴上滑动, 点$M$在线段$AB$上, 且$AM: MB=1: 2$, 求动点$M$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423892,7 +426454,9 @@ "id": "016633", "content": "已知抛物线$y^2=4 p x$($p>0$), $O$为顶点, $A$、$B$为抛物线上的两动点, 且满足$OA \\perp OB$, 如果$OM \\perp AB$于点$M$, 求点$M$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423912,7 +426476,9 @@ "id": "016634", "content": "已知曲线$C: x^2+\\dfrac{y^2}{a}=1$, 直线$l: k x-y-k=0$, $O$为坐标原点.\\\\\n(1) 讨论曲线$C$所表示的轨迹形状;\\\\\n(2) 当$k=1$时, 直线$l$与曲线$C$相交于两点$M$、$N$, 若$|MN|=\\sqrt{2}$, 求曲线$C$的方程;\\\\\n(3) 当$a=-1$时, 直线$l$与曲线$C$相交于两点$M$、$N$, 试问在曲线$C$上是否存在点$Q$, 使得$\\overrightarrow{OM}+\\overrightarrow{ON}=\\lambda \\overrightarrow{OQ}$? 若存在, 求实数$\\lambda$的取值范围; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -423932,7 +426498,9 @@ "id": "016635", "content": "方程$\\begin{cases}x=\\cos ^2\\alpha, \\\\ y=\\sin ^2 \\alpha\\end{cases}$($\\alpha \\in \\mathbf{R}$)的图形为\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) circle (1);\n\\draw (1,0) node [below right] {$1$};\n\\draw (-1,0) node [below left] {$-1$};\n\\draw (0,1) node [above left] {$1$};\n\\draw (0,-1) node [below left] {$-1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0) arc (0:90:1);\n\\draw (1,0) node [below] {$1$};\n\\draw (-1,0) node [below left] {$-1$};\n\\draw (0,1) node [left] {$1$};\n\\draw (0,-1) node [below left] {$-1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,2) -- (2,-1);\n\\draw (1,0) node [below] {$1$};\n\\draw (-1,0) node [below left] {$-1$};\n\\draw (0,1) node [left] {$1$};\n\\draw (0,-1) node [below left] {$-1$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,1) -- (1,0);\n\\draw (1,0) node [below] {$1$};\n\\draw (-1,0) node [below left] {$-1$};\n\\draw (0,1) node [left] {$1$};\n\\draw (0,-1) node [below left] {$-1$};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -423952,7 +426520,9 @@ "id": "016636", "content": "点$P$的直角坐标为$(1,-\\sqrt{3})$, 则点$P$的一个极坐标为\\bracket{20}.\n\\fourch{$(2, \\dfrac{\\pi}{3})$}{$(2, \\dfrac{4 \\pi}{3})$}{$(2,-\\dfrac{\\pi}{3})$}{$(-2,-\\dfrac{4 \\pi}{3})$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -423972,7 +426542,9 @@ "id": "016637", "content": "将参数方程$\\begin{cases}x=1+2 \\cos \\theta, \\\\ y=2 \\sin \\theta\\end{cases}$($\\theta$为参数)化为普通方程, 所得方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -423992,7 +426564,9 @@ "id": "016638", "content": "把极坐标方程$\\rho=2 \\sin (\\dfrac{\\pi}{3}+\\theta)$化为直角坐标方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424012,7 +426586,9 @@ "id": "016639", "content": "已知$x$、$y$满足$(x-1)^2+(y-2)^2=4$, 则$u=2 x-3 y$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424032,7 +426608,9 @@ "id": "016640", "content": "把下列参数方程化为普通方程, 并说明曲线的类型.\\\\\n(1) $\\begin{cases}x=3+2 \\cos \\theta, \\\\ y=\\cos 2 \\theta\\end{cases}$($\\theta$为参数);\\\\\n(2) $\\begin{cases}x=\\dfrac{m+1}{m+2}, \\\\ y=\\dfrac{2 m}{m+2}\\end{cases}$($m$为参数).", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424052,7 +426630,9 @@ "id": "016641", "content": "极坐标方程$\\dfrac{1}{\\rho}(5 \\sqrt{3} \\cos \\theta-5 \\sin \\theta)=1$表示什么曲线?", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424072,7 +426652,9 @@ "id": "016642", "content": "在直角坐标系$xOy$中, 圆$C$的圆心为$C(2,1)$, 半径为$1$.\\\\\n(1) 写出圆$C$的一个参数方程;\\\\\n(2) 过点$F(4,1)$作$\\odot C$的两条切线, 以坐标原点为极点, $x$轴正半轴为极轴建立极坐标系, 求这两条切线的极坐标方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424092,7 +426674,9 @@ "id": "016643", "content": "已知曲线$C_1$、$C_2$的参数方程分别为$C_1: \\begin{cases}x=4 \\cos ^2 \\theta, \\\\ y=4 \\sin ^2 \\theta\\end{cases}$($\\theta$为参数), $C_2: \\begin{cases}x=t+\\dfrac{1}{t}, \\\\ y=t-\\dfrac{1}{t}\\end{cases}$($t$为参数).\\\\\n(1) 将$C_1$、$C_2$的参数方程化为普通方程;\\\\\n(2) 以坐标原点为极点, $x$轴正半轴为极轴建立极坐标系. 设$C_1$、$C_2$的交点为$P$, 求圆心在极轴上, 且经过极点和$P$的圆的极坐标方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424112,7 +426696,9 @@ "id": "016644", "content": "有平面直角坐标系$xOy$中, 已知椭圆$C_1: \\dfrac{x^2}{36}+\\dfrac{y^2}{4}=1$和$C_2: x^2+\\dfrac{y^2}{9}=1$, $P$为$C_1$上的动点, $Q$为$C_2$上的动点, $\\omega$为$\\overrightarrow{OP} \\cdot \\overrightarrow{OQ}$的最大值, 记$\\Omega=\\{(P, Q) | P$在$C_1$上, $Q$在$\\mathrm{C}_2$上, 且$\\overrightarrow{\\mathrm{OP}} \\cdot \\overrightarrow{\\mathrm{OQ}}=\\omega\\}$, 则$\\Omega$中\\bracket{20}.\n\\fourch{元素个数为 2}{元素个数为 4}{元素个数为 8}{含有无穷个元素}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -424132,7 +426718,9 @@ "id": "016645", "content": "参数方程$\\begin{cases}x=t \\cos \\theta, \\\\ y=2+t \\sin \\theta\\end{cases}$($t$为参数, $0 \\leq \\theta \\leq \\pi$)表示的曲线为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424152,7 +426740,9 @@ "id": "016646", "content": "对任意实数$k$, 直线$y=k x+b$与椭圆: $\\begin{cases}x=\\sqrt{3}+2 \\cos \\theta,\\\\ y=1+4 \\sin \\theta\\end{cases}$($0 \\leq \\theta \\leq 2 \\pi$)恒有公共点, 则$b$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424172,7 +426762,9 @@ "id": "016647", "content": "已知点$P$在椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}{9}=1$上, 则$2 x-y$的最大值为\\blank{50} , 最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424192,7 +426784,9 @@ "id": "016648", "content": "在极坐标系中, 直线$\\rho \\cos \\theta+\\rho \\sin \\theta=a$($a>0$)与圆$\\rho=2 \\cos \\theta$相切, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424212,7 +426806,9 @@ "id": "016649", "content": "在平面直角坐标系中, 已知$A(1,0)$、$B(0,-1)$, $P$是曲线$y=\\sqrt{1-x^2}$上一个动点, 则$\\overrightarrow{BP} \\cdot\\overrightarrow{BA}$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424232,7 +426828,9 @@ "id": "016650", "content": "直线$\\begin{cases}x=t \\sin 20{^ \\circ}+3, \\\\ y=-t \\cos 20{^ \\circ}\\end{cases}$($t$为参数$)$的倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424252,7 +426850,9 @@ "id": "016651", "content": "给出下列方程: \\textcircled{1} $\\rho=2 \\cos \\theta$; \\textcircled{2} $\\rho=-3 \\sin \\theta$; \\textcircled{3} $\\rho=1$; \\textcircled{4} $\\rho \\sin \\theta=-2$; \\textcircled{5} $\\rho \\cos \\theta=1$; \\textcircled{6} $\\rho=\\sin \\theta-\\sqrt{3} \\cos \\theta$. 其中表示圆的共有\\bracket{20}.\n\\fourch{2 个}{3 个}{4 个}{5 个}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -424272,7 +426872,9 @@ "id": "016652", "content": "动点$P(3+2 \\cos \\theta, \\cos 2 \\theta)$的轨迹的焦点坐标为\\bracket{20}.\n\\fourch{$(3,-\\dfrac{3}{2})$}{$(3 \\pm \\sqrt{5}, 0)$}{$(\\pm \\sqrt{5}, 0)$}{$(3,-\\dfrac{1}{2})$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -424292,7 +426894,9 @@ "id": "016653", "content": "在极坐标系下, 点$M(m, \\dfrac{\\pi}{3})$($m>0$)中曲线$\\rho \\cos (\\theta-\\dfrac{\\pi}{3})=2$上各点距离的最小值为\\bracket{20}.\n\\fourch{$|m-2|$}{$m-2$}{$m+2$}{不存在}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -424312,7 +426916,9 @@ "id": "016654", "content": "已知点$P(x, y)$是圆$x^2+y^2=2 y$上的动点.\\\\\n(1) 求$2 x+y$的取值范围;\\\\\n(2) 若$x+y+a \\geq 0$恒成立, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424332,7 +426938,9 @@ "id": "016655", "content": "在直角坐标系$xOy$中, 曲线$C$的参数方程为$\\begin{cases}x=2-t-t^2, \\\\ y=2-3 t+t^2\\end{cases}$($t$为参数且$t \\neq 1$), $C$与坐标轴交于$A$、$B$两点.\\\\\n(1) 求$|AB|$;\\\\\n(2) 以坐标原点为极点, $x$轴正半轴为极轴建立极坐标系, 求直线$AB$的极坐标方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424352,7 +426960,9 @@ "id": "016656", "content": "在直角坐标系$xOy$中, 以坐标原点为极点, $x$轴正半轴为极轴建立极坐标系, 曲线$C$的极坐标方程为$\\rho=2 \\sqrt{2} \\cos \\theta$.\\\\\n(1) 将$C$的极坐标方程化为直角坐标方程;\\\\\n(2) 设点$A$的直角坐标为$(1,0), M$为$C$上的动点, 点$P$满足$\\overrightarrow{AP}=\\sqrt{2} \\overrightarrow{AM}$, 写出$P$的轨迹$C_1$的参数方程, 并判断$C$与$C_1$是否有公共点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424372,7 +426982,9 @@ "id": "016657", "content": "已知$a \\in \\mathbf{R}$, 方程$a^2 x^2+(a+2) y^2+4 x+8 y+5 a=0$表示圆, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424392,7 +427004,9 @@ "id": "016658", "content": "已知$A$、$B$为定点且$|AB|=4$, 动点$P$满足$|PA|-|PB|=3$, 则$|PA|$的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424412,7 +427026,9 @@ "id": "016659", "content": "已知椭圆$\\dfrac{x^2}{16}+\\dfrac{y^2}{9}=1$的左、右焦点分别为$F_1$、$F_2$, 点$P$在椭圆上, 若$F_1$、$F_2$、$P$是一个直角三角形的三个顶点, 则点$P$到$x$轴的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424432,7 +427048,9 @@ "id": "016660", "content": "以椭圆$\\dfrac{x^2}{169}+\\dfrac{y^2}{144}=1$的右焦点为圆心, 且与双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$的渐近线相切的圆的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424452,7 +427070,9 @@ "id": "016661", "content": "若曲线$2 x=\\sqrt{4+y^2}$与直线$y=m(x+1)$有公共点, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424472,7 +427092,9 @@ "id": "016662", "content": "设集合$M=\\{(x, y) | x^2+(y-a)^2=1,\\ x \\in \\mathbf{R},\\ y \\in \\mathbf{R}\\}$, $N=\\{(x, y) | x^2-y=0,\\ x \\in \\mathbf{R},\\ y \\in \\mathbf{R}\\}$, 若集合$M \\cap N$中有$2$个元素, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424492,7 +427114,9 @@ "id": "016663", "content": "已知直线$y=x+1$与椭圆$m x^2+n y^2=1$($m>n>0$)相交于$A$、$B$两点, 若弦$AB$的中点的横坐标等于$-\\dfrac{1}{3}$, 则双曲线$\\dfrac{x^2}{m^2}-\\dfrac{y^2}{n^2}=1$的两条渐近线的夹角的正切值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424512,7 +427136,9 @@ "id": "016664", "content": "在极坐标系中, 直线$l$的方程为$\\rho \\sin (\\dfrac{\\pi}{6}-\\theta)=2$, 曲线$C$的方程为$\\rho=4 \\cos \\theta$, 则直线$l$被曲线$C$截得的弦长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424532,7 +427158,9 @@ "id": "016665", "content": "给出以下四个命题:\\\\\n\\textcircled{1} 椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{7}=1$的焦距为$2 \\sqrt{2}$;\\\\\n\\textcircled{2} 双曲线$8 m x^2-m y^2=8$的一个焦点是$(0,3)$, 则$m=-1$;\\\\\n\\textcircled{3} 设双曲线与椭圆$3 x^2+4 y^2=48$有相同的焦点, 且实轴长等于$2$, 则此双曲线的标准方程为$x^2-\\dfrac{y^2}{3}=1$;\\\\\n\\textcircled{4} 已知椭圆$\\dfrac{x^2}{m}+\\dfrac{y^2}{n}=1$与双曲线$\\dfrac{x^2}{p}-\\dfrac{y^2}{q}=1$($m , n ,p , q \\in (0,+\\infty)$)有共同的焦点$F_1$、$F_2$, $M$是椭圆和双曲线的一个交点, 则$|MF_1| \\cdot|MF_2|=m-p$.\\\\\n其中正确命题的序号为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424552,7 +427180,9 @@ "id": "016666", "content": "过抛物线$y^2=2 p x$($p>0$)的焦点$F$作倾斜角为$45{^\\circ}$的直线交抛物线于$A$、$B$两点, 若线段$AB$的长为$8$, 则$p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424572,7 +427202,9 @@ "id": "016667", "content": "椭圆$\\dfrac{x^2}{9}+\\dfrac{y^2}{2}=1$的焦点为$F_1$、$F_2$, 点$P$在椭圆上, 若$|PF_1|=4$, 则$\\angle F_1PF_2$的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424592,7 +427224,9 @@ "id": "016668", "content": "已知直线$l_1$: $4x-3y+6=0$和直线$l_2$: $x=-1$, 抛物线$y^2=4x$上一动点$P$到直线$l_1$和直线$l_2$的距离之和的最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424612,7 +427246,9 @@ "id": "016669", "content": "若直线$l$过点$(3,0)$且与双曲线$4 x^2-9 y^2=36$只有一个公共点, 则这样的直线有\\bracket{20}.\n\\fourch{1 条}{2 条}{3 条}{4 条}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -424632,7 +427268,9 @@ "id": "016670", "content": "设抛物线$C: y^2=4 x$的焦点为$F$, 过点$(-2,0)$且斜率为$\\dfrac{2}{3}$的直线与$C$交于$M$、$N$两点, 则$\\overrightarrow{FM} \\cdot \\overrightarrow{FN}=$\\bracket{20}.\n\\fourch{$5$}{$6$}{$7$}{$8$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -424652,7 +427290,9 @@ "id": "016671", "content": "直线$\\begin{cases}x=1+t \\sin \\alpha, \\\\ y=t \\sin \\alpha\\end{cases}$($t$为参数) 与圆$x^2+y^2=4 x+2 y$的位置关系是\\bracket{20}.\n\\fourch{相切}{相交}{相离}{与$\\alpha$的值有关}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -424672,7 +427312,9 @@ "id": "016672", "content": "以$(a_1, 0)$、$(a_2, 0)$为圆心的两圆均过$(1,0)$, 与$y$轴正半轴分别交于$(0, y_1)$、$(0, y_2)$, 且满足$\\ln y_1+\\ln y_2=0$, 则点$(\\dfrac{1}{a_1}, \\dfrac{1}{a_2})$的轨迹是\\bracket{20}.\n\\fourch{直线}{圆}{椭圆}{双曲线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -424692,7 +427334,9 @@ "id": "016673", "content": "已知直线$y=x+b$与双曲线$2 x^2-y^2=2$交于$A$、$B$两点, 若以$AB$为直径的圆过原点, 求实数$b$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424712,7 +427356,9 @@ "id": "016674", "content": "在直角坐标系$xOy$中, 曲线$C_1$的方程为$y=k|x|+2$, 以坐标原点为极点, $x$轴正半轴为极轴建立极坐标系, 曲线$C_2$的极坐标方程为$\\rho^2+2 \\rho \\cos \\theta-3=0$.\\\\\n(1) 求$C_2$的直角坐标方程;\\\\\n(2) 若$C_1$与$C_2$有且仅有三个公共点, 求$C_1$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424732,7 +427378,9 @@ "id": "016675", "content": "设椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的右顶点为$A$, 上顶点为$B$. 已知椭圆的离心率为$\\dfrac{\\sqrt{5}}{3}$, $|AB|=\\sqrt{13}$.\\\\\n(1) 求椭圆的方程;\\\\\n(2) 设直线$l: y=k x$($k<0$)与椭圆交于$P$、$Q$两点, $l$与直线$AB$交于点$M$, 且点$P$、$M$均在第四象限. 若$\\triangle BPM$的面积是$\\triangle BPQ$面积的$2$倍, 求$k$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424752,7 +427400,9 @@ "id": "016676", "content": "已知动圆过定点$P(1,0)$, 且与直线$l: x=-1$相切, 点$C$在直线$l$上.\\\\\n(1) 求动圆圆心$M$的轨迹方程;\\\\\n(2) 设过定点$P$且斜率为$-\\sqrt{3}$的直线与(1)中的轨迹交于$A$、$B$两点.\\\\\n(I) 问$\\triangle ABC$能否为正三角形? 若能, 求出点$C$坐标; 若不能, 说明理由;\\\\\n(II) 当$\\triangle ABC$为钝角三角形时, 求点$C$纵坐标的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424772,7 +427422,9 @@ "id": "016677", "content": "已知双曲线$C$的中心是原点, 右焦点为$F(\\sqrt{3}, 0)$, 一条渐近线$m: x+$$\\sqrt{2} y=0$, 设过点$A(-3 \\sqrt{2}, 0)$的直线$l$的方向向量$\\overrightarrow {e}=(1, k)$.\\\\\n(1) 求双曲线$C$的方程;\\\\\n(2) 若过原点的直线$a\\parallel l$, 且$a$与$l$的距离为$\\sqrt{6}$, 求$k$的值;\\\\\n(3) 证明: 当$k>\\dfrac{\\sqrt{2}}{2}$时, 在双曲线$C$的右支上不存在点$Q$, 使之到直线$l$的距离为$\\sqrt{6}$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424792,7 +427444,9 @@ "id": "016678", "content": "点$Q$在直线$b$上, 直线$b$在平面$\\beta$上, 则$Q, b, \\beta$之间的关系用符号表示为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -424812,7 +427466,9 @@ "id": "016679", "content": "下列条件中, 能够确定一个平面的是\\bracket{20}.\n\\fourch{两个点}{三个点}{一条直线和一个点}{两条相交直线}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -424832,7 +427488,9 @@ "id": "016680", "content": "在下列命题中, 不是公理的是\\bracket{20}.\n\\onech{平行于同一个平面的两个平面相互平行}{过不在同一条直线上的三点, 有且只有一个平面}{如果一条直线上的两点在一个平面上, 那么这条直线上所有的点都在此平面上}{如果两个不重合的平面有一个公共点, 那么它们有且只有一条过该点的公共直线}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -424852,7 +427510,9 @@ "id": "016681", "content": "下列命题中正确的个数为\\bracket{20}.\\\\\n\\textcircled{1} 经过三点确定一个平面;\\\\\n\\textcircled{2} 梯形可以确定一个平面;\\\\\n\\textcircled{3} 两两相交的三条直线最多可以确定三个平面;\\\\\n\\textcircled{4} 如果两个平面有三个公共点, 则这两个平面重合.\n\\fourch{0}{1}{2}{3}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -424872,7 +427532,9 @@ "id": "016682", "content": "如图, 若一个水平放置的图形的斜二测直观图是一个底角为$45{^ \\circ}$且腰和上底均为$1$的等腰梯形, 则原平面图形的面积是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) --++ (45:1) --++ (1,0) --++ (-45:1) --cycle;\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{2+\\sqrt{2}}{2}$}{$\\dfrac{1+\\sqrt{2}}{2}$}{$2+\\sqrt{2}$}{$1+\\sqrt{2}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -424892,7 +427554,9 @@ "id": "016683", "content": "已知三条直线$l_1$、$l_2$、$l_3$相交于同一点$P$, 与另一条直线$a$分别相交于$A$、$B$、$C$, 且$P \\notin a$, 求证: 四条直线$l_1$、$l_2$、$l_3$、$a$在同一平面上.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424912,7 +427576,9 @@ "id": "016684", "content": "求证: 如果两条平行直线与第三条直线均相交, 那么这三条直线共面.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424932,7 +427598,9 @@ "id": "016685", "content": "求证: 如果三条平行直线与第四条直线均相交, 那么这四条直线共面.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424952,7 +427620,9 @@ "id": "016686", "content": "在空间四边形$ABCD$中 ($A$、$B$、$C$、$D$四点不共面), $E$、$F$、$G$、$H$分别是$AB$、$BC$、$CD$、$DA$上的点, 若$EH$与$FG$交于$P$. 求证: 点$P$在直线$BD$上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (3,0) node [right] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(C)$) node [below] {$F$} coordinate (F);\n\\draw (1.4,1) node [below] {$D$} coordinate (D);\n\\draw ($(C)!{2/3}!(D)$) node [below] {$G$} coordinate (G);\n\\draw ($(B)!2!(D)$) node [right] {$P$} coordinate (P);\n\\draw (1,2.5) node [above] {$A$} coordinate (A);\n\\draw ($(A)!0.4!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(A)!{4/7}!(D)$) node [below left] {$H$} coordinate (H);\n\\draw (E)--(P)--(F)(A)--(D)--(C)--(B)--cycle(B)--(D);\n\\draw (E) circle (0.03) (H) circle (0.03) (F) circle (0.03) (G) circle (0.03); \n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424972,7 +427642,9 @@ "id": "016687", "content": "如图, 在四面体$ABCD$中作截面$PQR$, 若$PQ$、$CB$的延长线交于$M$, $RQ$、$DB$的延长线交于$N$, $RP$、$DC$的延长线交于$K$. 求证: $M$、$N$、$K$三点共线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (1.5,-0.3) node [right] {$C$} coordinate (C);\n\\draw (2,0.5) node [right] {$D$} coordinate (D);\n\\draw (1.3,2) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)--(C)--(D)--(A)(A)--(C);\n\\draw [dashed] (B)--(D);\n\\draw ($(A)!0.3!(D)$) node [right] {$R$} coordinate (R);\n\\draw ($(A)!0.48!(C)$) node [right] {$P$} coordinate (P);\n\\draw ($(A)!0.65!(B)$) node [above left] {$Q$} coordinate (Q);\n\\draw (R)--(P)--(Q);\n\\draw [dashed] (R)--(Q);\n\\path [name path = BC] (C)--($(C)!2.1!(B)$);\n\\path [name path = PQ] (P)--($(P)!3.1!(Q)$);\n\\path [name intersections = {of = BC and PQ, by = M}];\n\\draw (M) node [left] {$M$};\n\\path [name path = CD] (C)--($(D)!2.1!(C)$);\n\\path [name path = RP] (R)--($(R)!4.1!(P)$);\n\\path [name intersections = {of = CD and RP, by = K}];\n\\draw (K) node [below] {$K$};\n\\path [name path = BD] (D)--($(D)!1.4!(B)$);\n\\path [name path = RQ] (R)--($(R)!2.1!(Q)$);\n\\path [name intersections = {of = BD and RQ, by = N}];\n\\draw (N) node [below left] {$N$};\n\\draw (Q)--(M)--(B)(P)--(K)--(C)(Q)--(N);\n\\draw [dashed] (N)--(B);\n\\draw (M)--(K);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -424992,7 +427664,9 @@ "id": "016688", "content": "正方体$ABCD-A_1B_1C_1D_1$中, 棱长为$a$, $M$、$N$、$P$分别是$A_1B_1$、$AD$、$BB_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\filldraw ($(A)!0.5!(D)$) node [right] {$N$} coordinate (N) circle (0.03);\n\\filldraw ($(B)!0.5!(B_1)$) node [right] {$P$} coordinate (P) circle (0.03);\n\\filldraw ($(A_1)!0.5!(B_1)$) node [above] {$M$} coordinate (M) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n(1) 画出过$M$、$N$、$P$三点的平面与平面$ABCD$的交线和它与平面$BB_1C_1C$的交线;\\\\\n(2) 设过$M$、$N$、$P$三点的平面与$BC$交于点$K$, 求$PK$的长.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425012,7 +427686,9 @@ "id": "016689", "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$、$P$分别是$A_1B_1$、$AD$、$BB_1$的中点, 作出过$M$、$N$、$P$三点的平面$\\alpha$截正方体所得到的截面.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\filldraw ($(A)!0.5!(D)$) node [right] {$N$} coordinate (N) circle (0.03);\n\\filldraw ($(B)!0.5!(B_1)$) node [right] {$P$} coordinate (P) circle (0.03);\n\\filldraw ($(A_1)!0.5!(B_1)$) node [above] {$M$} coordinate (M) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425032,7 +427708,9 @@ "id": "016690", "content": "如图, 在空间四边形$ABCD$中, 点$M$、$N$、$P$分别在棱$AD$、$BD$、$CD$上, 点$S$在面$ABC$内, 试画出线段$SD$与过$M$、$N$、$P$三点的截面的交点$O$(写出作法).\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw ($1/3*(A)+1/3*(C)+1/3*(D)$)+(0,{2*sqrt(6)/3},0) node [above] {$A$} coordinate (A);\n\\draw (A)--(B)--(C)--(D)--cycle(A)--(C);\n\\draw [dashed] (B)--(D);\n\\filldraw ($(C)!0.5!(D)$) node [below] {$P$} coordinate (P) circle (0.03);\n\\filldraw ($(B)!0.3!(D)$) node [below] {$N$} coordinate (N) circle (0.03);\n\\filldraw ($(A)!0.4!(D)$) node [above] {$M$} coordinate (M) circle (0.03);\n\\filldraw ($1/3*(A)+1/3*(C)+1/3*(B)$) node [left] {$S$} coordinate (S) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425052,7 +427730,9 @@ "id": "016691", "content": "如果点$P$在直线$m$上, 直线$m$在平面$\\alpha$上, 则$P$、$m$、$\\alpha$之间的关系用符号表示为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425072,7 +427752,9 @@ "id": "016692", "content": "看图填空:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.5!(C)$) node [below] {$O$} coordinate (O);\n\\draw ($(A_1)!0.5!(C_1)$) node [above] {$O_1$} coordinate (O_1);\n\\draw [dashed] (A)--(C)(B)--(D)(A_1)--(C_1)(B_1)--(D_1)(O)--(O_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 直线$BD \\cap$平面$ACC_1A_1=$\\blank{50};\\\\\n(2) 平面$ACC_1A_1 \\cap$平面$ABCD=$\\blank{50};\\\\\n(3) 平面$ACB_1 \\cap$平面$DBB_1D_1=$\\blank{50};\\\\\n(4) 直线$OO_1$和平面$ACC_1A_1$的位置关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425092,7 +427774,9 @@ "id": "016693", "content": "空间三个平面最多将空间分成\\blank{50}个部分.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425112,7 +427796,9 @@ "id": "016694", "content": "若三个平面两两相交, 且三条交线互相平行, 则这三个平面把空间分成\\blank{50}个部分.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425132,7 +427818,9 @@ "id": "016695", "content": "已知等边$\\triangle ABC$边长为$a$, 那么$\\triangle ABC$的平面直观图$\\triangle A' B' C'$的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425152,7 +427840,9 @@ "id": "016696", "content": "已知点$P$、$Q$在平面$\\alpha$上, 点$M$在平面$\\beta$上, 又$\\alpha \\cap \\beta=l$,$M \\notin l$, $PQ \\cap l=R$, 过$P$、$Q$、$M$的平面为$\\gamma$, 则$\\beta \\cap \\gamma=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425172,7 +427862,9 @@ "id": "016697", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $O$是$BD_1$的中点, 直线$A_1C$交平面$AB_1D_1$于点$M$, 则下列结论中错误的是\\bracket{20}.\n\\twoch{$A_1$、$M$、$O$三点共线}{$M$、$O$、$A$、$A_1$四点共面}{$A$、$O$、$C$、$M$四点共面}{$B$、$B_1$、$O$、$M$四点共面}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425192,7 +427884,9 @@ "id": "016698", "content": "过正方体中心的平面截正方体所得的截面中, 不可能的图形为\\bracket{20}.\n\\fourch{三角形}{长方形}{对角线不相等的菱形}{六边形}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425212,7 +427906,9 @@ "id": "016699", "content": "已知平面$\\alpha$、$\\beta$、$\\gamma$两两垂直, 直线$a$、$b$、$c$满足: $a \\subset \\alpha$, $b \\subset \\beta$, $c \\subset \\gamma$, 则直线$a$、$b$、$c$不可能满足以下哪种关系\\bracket{20}.\n\\fourch{两两垂直}{两两平行}{两两相交}{两两异面}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425232,7 +427928,9 @@ "id": "016700", "content": "已知: 直线$AB$、$BC$、$CA$两两相交, 交点分别为$A$、$B$、$C$, 求证: 直线$AB$、$BC$、$CA$共面.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425252,7 +427950,9 @@ "id": "016701", "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $A_1C_1 \\cap B_1D_1=O_1$, $B_1D \\cap$截面$A_1BC_1=P$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\def\\l{3}\n\\def\\m{4}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A_1)!0.5!(C_1)$) node [above] {$O_1$} coordinate (O_1);\n\\draw (A_1)--(C_1)--(B)--cycle(B_1)--(D_1);\n\\draw [dashed] (B_1)--(D)(B)--(O_1);\n\\draw ($(B)!{2/3}!(O_1)$) node [left] {$P$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $B$、$P$、$O_1$三点共线;\\\\\n(2) 若$AB=3$, $BC=4$, $CC_1=6$, 求$DP$的长.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425272,7 +427972,9 @@ "id": "016702", "content": "在正方体$ABCD-A_1B_1C_1D_1$中.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\end{tikzpicture}\n\\end{center}\n(1) $AA_1$与$CC_1$是否在同一平面上?\\\\\n(2) 点$B$、$C_1$、$D$是否在同一平面上?\\\\\n(3) 画出平面$ACC_1A_1$与平面$BC_1D$的交线, 平面$ACD_1$与平面$BDC_1$的交线.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425292,7 +427994,9 @@ "id": "016703", "content": "在正方体$ABCD-A_1B_1C_1D_1$中与异面直线$AB, CC_1$均垂直的棱有\\blank{50}条.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425312,7 +428016,9 @@ "id": "016704", "content": "如图, 已知正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$2$, 高为$3$, 则异面直线$AA_1$与$BD_1$所成角的大小是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [dashed] (B)--(D_1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425332,7 +428038,9 @@ "id": "016705", "content": "在空间中, ``两条直线不平行'''是``两条直线异面''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425352,7 +428060,9 @@ "id": "016706", "content": "若三条直线两两垂直, 则\n\\textcircled{1} 三条直线必共点;\n\\textcircled{2} 其中必有两条是异面直线;\n\\textcircled{3} 三条直线不可能在同一平面内;\n\\textcircled{4} 其中必有两条在同一平面内.\n以上命题中不正确的序号是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}\\textcircled{3}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}{\\textcircled{2}\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{2}\\textcircled{4}}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425372,7 +428082,9 @@ "id": "016707", "content": "在直三棱柱$ABC-A_1B_1C_1$的棱所在的直线中, 与直线$BC_1$异面的直线的条数为\\bracket{20}.\n\\fourch{1}{2}{3}{4}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425392,7 +428104,9 @@ "id": "016708", "content": "如图, 已知长方体的长和宽都是$2 \\sqrt{3} \\text{cm}$, 高是$2 \\text{cm}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\def\\l{{2*sqrt(3)}}\n\\def\\m{{2*sqrt(3)}}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (A_1)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$BC$与$A_1C_1$所成的角的大小;\\\\\n(2) 求异面直线$AA_1$与$BC_1$所成的角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425412,7 +428126,9 @@ "id": "016709", "content": "如图, 已知长方体的长和宽都是$2 \\sqrt{3} \\text{cm}$, 高是$2 \\text{cm}$, 求异面直线$AC$与$BD_1$所成的角.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\def\\l{{2*sqrt(3)}}\n\\def\\m{{2*sqrt(3)}}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [dashed] (A)--(C)(B)--(D_1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425432,7 +428148,9 @@ "id": "016710", "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$分别是棱$AA_1$和$BB_1$的中点, 若$\\theta$为直线$CM$和$ND_1$所成的角, 则$\\theta$的余弦值为\\bracket{20}.(此题有图)\n\\fourch{$\\dfrac{1}{9}$}{$\\dfrac{2}{3}$}{$\\dfrac{2 \\sqrt{5}}{9}$}{$\\dfrac{4 \\sqrt{5}}{9}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425452,7 +428170,9 @@ "id": "016711", "content": "如图, 在空间四边形$ABCD$中, $AD=BC=2, E$、$F$分别为$AB$、$CD$的中点, $EF=\\sqrt{3}$, 求异面直线$AD$和$BC$所成的角.(此题有图)\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (O);\n\\draw ({1/2},0,{sqrt(3)/2}) node [below] {$F$} coordinate (F);\n\\draw ({1/4*(-1-sqrt(6))},{1/2},{1/4*(sqrt(2)-sqrt(3))}) node [left] {$E$} coordinate (E);\n\\draw (-1,0,0) node [left] {$B$} coordinate (B);\n\\draw ($(B)!2!(O)$) node [right] {$D$} coordinate (D);\n\\draw ($(D)!2!(F)$) node [below] {$C$} coordinate (C);\n\\draw ($(B)!2!(E)$) node [above] {$A$} coordinate (A);\n\\draw (B)--(C)--(D)--(A)--cycle(A)--(C);\n\\draw [dashed] (E)--(F)(B)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425472,7 +428192,9 @@ "id": "016712", "content": "在空间四边形$ABCD$中, $AD=BC$, 且异面直线$AD$和$BC$所成的角为$60{^ \\circ}$, $E$、$F$分别为$AB$、$CD$的中点, 求异面直线$EF$和$BC$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425492,7 +428214,9 @@ "id": "016713", "content": "已知正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$分别是$A_1B_1$和$B_1C_1$中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(B_1)!0.5!(C_1)$) node [above] {$N$} coordinate (N);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$M$} coordinate (M);\n\\draw (A)--(M)(C)--(N)(B_1)--(D_1);\n\\draw [dashed] (D_1)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) $AM$和$CN$是否为异面直线? 说明理由;\\\\\n(2) $D_1B$和$CC_1$是否为异面直线? 说明理由;\\\\\n(3) 求异面直线$CN$与$D_1B_1$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425512,7 +428236,9 @@ "id": "016714", "content": "如图, 在棱长为$1$的正方体$ABCD-A_1B_1C_1D_1$中, $P$为底面$ABCD$内 (包括边界)的动点, 满足直线$D_1P$与直线$CC_1$所成角的大小为$\\dfrac{\\pi}{6}$, 求线段$DP$扫过的面积的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (D) ++ ({2/3},0,{2*sqrt(2)/3}) node [left] {$P$} coordinate (P);\n\\draw [dashed] (D_1)--(P);\n\\filldraw (P) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425532,7 +428258,9 @@ "id": "016715", "content": "和两条异面直线中的一条相交的直线与另一条的位置关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425552,7 +428280,9 @@ "id": "016716", "content": "在长方体$ABCD-A_1B_1C_1D_1$中$AA_1=a$, $\\angle BAB_1=\\angle B_1A_1C_1=30{^ \\circ}$, 则异面直线$AB_1$与$A_1C_1$所成角的余弦值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425572,7 +428302,9 @@ "id": "016717", "content": "正方体上的点$P$、$Q$、$R$、$S$是其所在棱的中点, 则直线$PQ$与直线$RS$异面的是\\blank{50}.\\\\\n\\textcircled{1} \\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.5!(B)$) node [below] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [right] {$Q$} coordinate (Q);\n\\draw ($(C_1)!0.5!(D_1)$) node [above] {$S$} coordinate (S);\n\\draw ($(D_1)!0.5!(A_1)$) node [left] {$R$} coordinate (R);\n\\draw (S)--(R);\n\\draw [dashed] (P)--(Q);\n\\end{tikzpicture}\n\\textcircled{2} \\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.5!(B)$) node [below] {$Q$} coordinate (Q);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [below] {$S$} coordinate (S);\n\\draw ($(B_1)!0.5!(C_1)$) node [above] {$R$} coordinate (R);\n\\draw (S)--(R);\n\\draw (P)--(Q);\n\\end{tikzpicture}\n\\textcircled{3} \\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.5!(B)$) node [below] {$Q$} coordinate (Q);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [below] {$S$} coordinate (S);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$R$} coordinate (R);\n\\draw (S)--(R);\n\\draw (P)--(Q);\n\\end{tikzpicture}\n\\textcircled{4} \\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$Q$} coordinate (Q);\n\\draw ($(A)!0.5!(A_1)$) node [left] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(A)$) node [below] {$S$} coordinate (S);\n\\draw ($(D_1)!0.5!(C_1)$) node [above] {$R$} coordinate (R);\n\\draw [dashed] (S)--(R);\n\\draw [dashed] (P)--(Q);\n\\end{tikzpicture}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425592,7 +428324,9 @@ "id": "016718", "content": "入图是一个正方体的平面展开图, 在这个正方体中, 下列说法中正确的序号是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$E$} rectangle (4,1);\n\\draw (1,2) node [left] {$N$} rectangle (2,-1) node [right] {$F$};\n\\draw (3,0) -- (3,1) node [above] {$M$};\n\\draw (2,0) node [below right] {$B$} -- (3,1);\n\\draw (1,0) node [below left] {$A$} -- (2,-1);\n\\draw (0,0) -- (1,1) node [above left] {$D$};\n\\draw (1,2) -- (2,1) node [below left] {$C$};\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} 直线$AF$与直线$DE$相交;\\\\\n\\textcircled{2} 直线$CN$与直线$DE$平行;\\\\\n\\textcircled{3} 直线$BM$与直线$DE$是异面直线;\\\\\n\\textcircled{4} 直线$CN$与直线$BM$成$60{^\\circ}$角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425612,7 +428346,9 @@ "id": "016719", "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$分别是$CD$、$CC_1$的中点, 则异面直线$A_1M$与$DN$所成角的大小是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(C)!0.5!(D)$) node [below] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$N$} coordinate (N);\n\\draw [dashed] (A_1)--(M)(D)--(N);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425632,7 +428368,9 @@ "id": "016720", "content": "如图, 在正六棱柱的所有棱中任取两条, 则它们所在的直线是互相垂直的异面直线的概率为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (A) --++ (1,0,0) coordinate (B) --++ ({1/2},0,{sqrt(3)/2}) coordinate (C) --++ ({-1/2},0,{sqrt(3)/2}) coordinate (D) --++ (-1,0,0) coordinate (E) --++ ({-1/2},0,{-sqrt(3)/2}) coordinate (F) -- cycle;\n\\draw [dashed] (A) --++ (0,-2,0) coordinate (A1);\n\\draw [dashed] (B) --++ (0,-2,0) coordinate (B1);\n\\draw (C) --++ (0,-2,0) coordinate (C1);\n\\draw (D) --++ (0,-2,0) coordinate (D1);\n\\draw (E) --++ (0,-2,0) coordinate (E1);\n\\draw (F) --++ (0,-2,0) coordinate (F1);\n\\draw (F1) -- (E1) -- (D1) -- (C1);\n\\draw [dashed] (F1) -- (A1) -- (B1) -- (C1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425652,7 +428390,9 @@ "id": "016721", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, 下列四个结论中错误的是\\bracket{20}.\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [dashed] (A)--(C)--(D_1)--cycle;\n\\draw (B_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{直线$B_1C$与直线$AC$所成的角为$60{^ \\circ}$}{直线$B_1C$与直线$DC_1$所成的角为$45{^ \\circ}$}{直线$B_1C$与直线$AD_1$所成的角为$90{^ \\circ}$}{直线$B_1C$与直线$AB$所成的角为$90{^ \\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425672,7 +428412,9 @@ "id": "016722", "content": "在空间中, 直线$AB$平行于直线$EF$, 直线$BC$、$EF$为异面直线, 若$\\angle ABC=120{^ \\circ}$, 则异面直线$BC$、$EF$所成角的大小为\\bracket{20}.\n\\fourch{$30{^ \\circ}$}{$60{^ \\circ}$}{$120{^ \\circ}$}{$150{^ \\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425692,7 +428434,9 @@ "id": "016723", "content": "上海海关大楼的钟楼可以看作一个正四棱柱, 且钟楼的四个侧面均有时钟悬挂, 在$0$点到$12$点时针与分针的转动中 (包括$0$点, 但不包括$12$点), 相邻两面时钟的时针两两相互垂直的情况的次数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\begin{scope}[x = {(-10:0.9)}]\n\\draw (0,0) -- (-2,0) -- (-2,2) -- (0,2) -- cycle;\n\\draw (-1,1) circle (0.8);\n\\draw [->] (-1,1) --++ (-45:0.5);\n\\draw [->] (-1,1) --++ (-90:0.65);\n\\foreach \\i in {1,2,...,12} {\\draw (-1,1) ++ ({30*\\i}:0.7) --++ ({30*\\i}:0.05);};\n\\end{scope}\n\\begin{scope}[x = {(40:0.7)}]\n\\draw (0,0) -- (2,0) -- (2,2) -- (0,2) -- cycle;\n\\draw (1,1) circle (0.8);\n\\draw [->] (1,1) --++ (-45:0.5);\n\\draw [->] (1,1) --++ (-90:0.65);\n\\foreach \\i in {1,2,...,12} {\\draw (1,1) ++ ({30*\\i}:0.7) --++ ({30*\\i}:0.05);};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\fourch{0 次}{2 次}{4 次}{12 次}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425712,7 +428456,9 @@ "id": "016724", "content": "已知正方体$ABCD-A_1B_1C_1D_1$中, 点$E$是棱$A_1B_1$的中点.\\\\\n(1) 求证: $A_1B, EC_1$是异面直线;\\\\\n(2) 求异面直线$A_1B$、$EC_1$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425732,7 +428478,9 @@ "id": "016725", "content": "如图, 四棱锥$P-ABCD$中, 底面$ABCD$为矩形, $PA \\perp$底面$ABCD$. $AB=PA=1$, $AD=\\sqrt{3}$, $E$、$F$分别为棱$PD$、$PA$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(3)},0,0) node [right] {$D$} coordinate (D);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,1) node [left] {$B$} coordinate (B);\n\\draw ({sqrt(3)},0,1) node [right] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(P)$) node [below right] {$F$} coordinate (F);\n\\draw ($(D)!0.5!(P)$) node [right] {$E$} coordinate (E);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(C);\n\\draw [dashed] (B)--(A)--(D)(P)--(A)(F)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $B$、$C$、$E$、$F$四点共面;\\\\\n(2) 求异面直线$PB$与$AE$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425752,7 +428500,9 @@ "id": "016726", "content": "如图, 直三棱柱$ABC-A_1B_1C_1$的体积为$8$, 且$AB=AC=2$, $\\angle BAC=90{^ \\circ}$, $E$是$AA_1$的中点, $O$是$C_1B_1$的中点, 求异面直线$C_1E$与$BO$所成角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,0) node [right] {$A$} coordinate (A);\n\\draw (-1,0,0) node [left] {$C$} coordinate (C);\n\\draw (0,0,-1) node [right] {$B$} coordinate (B);\n\\draw (A) --++ (0,2,0) node [right] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,2,0) node [left] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)!0.5!(A)$) node [below left] {$E$} coordinate (E);\n\\draw ($(B_1)!0.5!(C_1)$) node [above] {$O$} coordinate (O);\n\\draw [dashed] (B)--(O)(B)--(C);\n\\draw (C_1)--(E)(C)--(A)--(B)(C_1)--(A_1)--(B_1)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425772,7 +428522,9 @@ "id": "016727", "content": "``直线$a$与平面$M$没有公共点''是``直线$a$与平面$M$平行''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425792,7 +428544,9 @@ "id": "016728", "content": "已知直线$a$在平面$\\beta$上, 则``直线$l \\perp a$''是``直线$l \\perp \\beta$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425812,7 +428566,9 @@ "id": "016729", "content": "下列命题中正确的是\\bracket{20}.\n\\onech{若直线$l_1\\parallel$平面$\\alpha$, 直线$l_2\\parallel$平面$\\alpha$, 则$l_1\\parallel l_2$}{若直线$l$上有两个点到平面$\\alpha$的距离相等, 则$l\\parallel \\alpha$}{直线$l$与平面$\\alpha$所成角的范围是$(0, \\dfrac{\\pi}{2})$}{若直线$l_1$平面$\\alpha$, 直线$l_2 \\perp$平面$\\alpha$, 则$l_1\\perp l_2$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -425832,7 +428588,9 @@ "id": "016730", "content": "正方体$ABCD-A_1B_1C_1D_1$的棱长为$1$, 则$A_1B_1$到平面$ABC_1D_1$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425852,7 +428610,9 @@ "id": "016731", "content": "如图, $BC$是 Rt$\\triangle ABC$的斜边, 过$A$作$\\triangle ABC$所在平面$\\alpha$的垂线$AP$, 联结$PB$、$PC$, 过$A$作$AD \\perp BC$于$D$, 联结$PD$, 那么图中直角三角形共有\\blank{50}个.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (1,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,-1) node [left] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(C)$) node [below] {$D$} coordinate (D);\n\\draw (A)--(B)--(P)--cycle;\n\\draw [dashed] (A)--(D)--(P)(B)--(C)(A)--(C)--(P);\n\\draw (A) ++ (-0.5,0,0.5) coordinate (X);\n\\draw (X) --++ (2,0,0) coordinate (Y);\n\\draw (Y) --++ (0,0,-1.5) coordinate (Z);\n\\draw (X) --++ (0,0,-1.5) coordinate (W);\n\\path [name path = WZ] (W)--(Z);\n\\path [name path = AP] (A)--(P);\n\\path [name path = BP] (B)--(P);\n\\path [name intersections = {of = AP and WZ, by = S}];\n\\path [name intersections = {of = BP and WZ, by = T}];\n\\draw (W)--(S)(T)--(Z);\n\\draw (X) ++ (0.3,0) node [above] {$\\alpha$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425872,7 +428632,9 @@ "id": "016732", "content": "如图, 在棱长为$a$的正方体$ABCD-A'B'C'D'$中, $E$为$DD'$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A'$} coordinate (A');\n\\draw (B) ++ (0,\\l,0) node [right] {$B'$} coordinate (B');\n\\draw (C) ++ (0,\\l,0) node [above right] {$C'$} coordinate (C');\n\\draw (D) ++ (0,\\l,0) node [above left] {$D'$} coordinate (D');\n\\draw (A') -- (B') -- (C') -- (D') -- cycle;\n\\draw (A) -- (A') (B) -- (B') (C) -- (C');\n\\draw [dashed] (D) -- (D');\n\\draw ($(D)!0.5!(D')$) node [left] {$E$} coordinate (E);\n\\draw [dashed] (B)--(D')(A)--(C)--(E)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BD'\\parallel$平面$AEC$;\\\\\n(2) 求直线$BD'$与平面$ADD' A'$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425892,7 +428654,9 @@ "id": "016733", "content": "在三棱锥$P-ABC$中, $PA=PB=PC=AC=2 \\sqrt{2}$, $BA=BC=2$, $O$是线段$AC$的中点, $M$是线段$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above left] {$O$} coordinate (O);\n\\draw ({-sqrt(2)},0,0) node [left] {$A$} coordinate (A);\n\\draw ($(A)!2!(O)$) node [right] {$C$} coordinate (C);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (0,{sqrt(6)},0) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(C)$) node [below] {$M$} coordinate (M);\n\\draw (A)--(B)--(C)--(P)--cycle(P)--(M);\n\\draw [dashed] (A)--(C)(B)--(O)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $PO \\perp$平面$ABC$;\\\\\n(2) 求直线$PM$与平面$PBO$所成的角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425912,7 +428676,9 @@ "id": "016734", "content": "已知边长为$6$的正方形$ABCD$所在平面外一点$P$, $PD \\perp$平面$ABCD$, $PD=8$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw (0,0,0) node [left] {$D$} coordinate (D);\n\\draw (0,0,6) node [below] {$A$} coordinate (A);\n\\draw (6,0,6) node [below] {$B$} coordinate (B);\n\\draw (6,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,8,0) node [left] {$P$} coordinate (P);\n\\draw (P)--(D)--(A)--(B)--(C)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 联结$PB$、$AC$, 证明: $PB \\perp AC$;\\\\\n(2) 求$PB$与平面$ABCD$所成的角的大小;\\\\\n(3) 求点$D$到平面$PAC$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425932,7 +428698,9 @@ "id": "016735", "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, 已知点$P$在平面$ABCD$上(点$P$不在直线$AC$上), 过点$P$在平面$ABCD$上画一条直线与平面$AA_1C_1C$平行, 应当如何画? 并说明理由.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\def\\l{4}\n\\def\\m{3}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\filldraw ($1/3*(A)+1/3*(B)+1/3*(C)$) node [left] {$P$} coordinate (P) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -425952,7 +428720,9 @@ "id": "016736", "content": "拿一张矩形的纸对折后略微展开, 坚立在桌面上, 折痕与桌面的位置关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425972,7 +428742,9 @@ "id": "016737", "content": "两条平行线中的一条平行于一个平面, 那么另一条与此平面的位置关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -425992,7 +428764,9 @@ "id": "016738", "content": "已知$a$、$b$、$l$是空间中的三条直线, 其中直线$a$、$b$在平面$\\alpha$上, 则``$l \\perp a$且$l \\perp b$''是``$l \\perp$平面$\\alpha$''的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426012,7 +428786,9 @@ "id": "016739", "content": "$PA \\perp$圆$O$所在的平面, $AB$是圆$O$的直径, $C$是圆$O$上的一点, $E$、$F$分别是点$A$在$PB$、$PC$上的射影, 给出下列结论: \\textcircled{1} $AF \\perp PB$; \\textcircled{2} $EF \\perp PB$; \\textcircled{3} $AF \\perp BC$; \\textcircled{4} $AE \\perp$平面$PBC$. 其中正确结论的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426032,7 +428808,9 @@ "id": "016740", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, 若长方体的体对角线$AC_1$与过点$A$的相邻三个面所成的角分别为$\\alpha$、$\\beta$、$\\gamma$, 则$\\sin ^2 \\alpha+\\sin ^2 \\beta+\\sin ^2 \\gamma=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426052,7 +428830,9 @@ "id": "016741", "content": "已知$\\triangle ABC$, 点$P$是平面$ABC$外一点, 点$O$是点$P$在平面$ABC$上的投影.\\\\\n(1) 当点$P$到$\\triangle ABC$的三个顶点的距离相等时, 点$O$是$\\triangle ABC$的\\blank{50};\\\\\n(2) 当点$P$到$\\triangle ABC$的三边的距离相等且$O$点在$\\triangle ABC$内时, 点$O$是$\\triangle ABC$的\\blank{50};\\\\\n(3) 当$PA \\perp PB$, $PB \\perp PC$, $PC \\perp PA$时, 点$O$是$\\triangle ABC$的\\blank{50}.\n(选填``内心''、``外心''、``垂心'').", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426072,7 +428852,9 @@ "id": "016742", "content": "如图, 已知点$A \\in$平面$\\alpha$, 点$O \\in \\alpha$, 直线$a \\subset \\alpha$, 点$P \\notin \\alpha$且$PO \\perp \\alpha$, 则``直线$a \\perp$直线$OA$''是``直线$a \\perp$直线$PA$''的\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [above left] {$O$} coordinate (O);\n\\draw (1.5,0) node [above right] {$A$} coordinate (A);\n\\draw (0,1.3) node [left] {$P$} coordinate (P);\n\\path [name path = OP] (O) -- ($(P)!2!(O)$);\n\\path [name path = PA] (A) -- ($(P)!1.8!(A)$);\n\\draw ($(O)!-0.2!(A)$) -- ($(O)!1.5!(A)$);\n\\draw ($(P)!-0.3!(O)$) -- (O);\n\\draw ($(P)!-0.2!(A)$) -- (A);\n\\path [name path = para, draw] (O) ++ (-2,-0.7) --++ (1.4,1.4) --++ (4,0) --++ (-1.4,-1.4) -- cycle;\n\\draw (O) ++ (-1.5,-0.7) node [above] {$\\alpha$};\n\\draw (A) ++ (0,-0.4) --++ (0.8,0.8) node [right] {$a$};\n\\path [name intersections = {of = OP and para, by = S}];\n\\path [name intersections = {of = PA and para, by = T}];\n\\draw [dashed] (O) -- (S) (A) -- (T);\n\\draw (S) -- ($(O)!1.5!(S)$) (T) -- ($(A)!1.5!(T)$);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分也不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426092,7 +428874,9 @@ "id": "016743", "content": "如图, 已知正方体$ABCD-A_1B_1C_1D_1$, $M$、$N$分别是$A_1D$、$D_1B$的中点, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A_1)!0.5!(D)$) node [left] {$M$} coordinate (M);\n\\draw ($(D_1)!0.5!(B)$) node [right] {$N$} coordinate (N);\n\\draw (D_1)--(B_1);\n\\draw [dashed] (A_1)--(D)--(B)--(D_1)(M)--(N);\n\\end{tikzpicture}\n\\end{center}\n\\onech{直线$A_1D$与直线$D_1B$相交, 直线$MN\\parallel$平面$ABCD$}{直线$A_1D$与直线$D_1B$平行, 直线$MN \\perp$平面$BDD_1B_1$}{直线$A_1D$与直线$D_1B$垂直, 直线$MN\\parallel$平面$ABCD$}{直线$A_1D$与直线$D_1B$异面, 直线$MN \\perp$平面$BDD_1B_1$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426112,7 +428896,9 @@ "id": "016744", "content": "设正四棱柱$ABCD-A_1B_1C_1D_1$的底面边长为$1$, 高为$2$, 平面$\\alpha$经过顶点$A$, 且与棱$AB$、$AD$、$AA_1$所在直线所成的角都相等, 则满足条件的平面$\\alpha$共有\\bracket{20}.\n\\fourch{1 个}{2 个}{3 个}{4 个}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426132,7 +428918,9 @@ "id": "016745", "content": "如图, 直三棱柱$ABC-A_1B_1C_1$的底面为直角三角形, 两直角边$AB$和$AC$的长分别为$4$和$2$, 侧棱$AA_1$的长为$5$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (0,0,-2) node [right] {$C$} coordinate (C);\n\\draw (-4,0,0) node [left] {$B$} coordinate (B);\n\\draw (A) --++ (0,5,0) node [below left] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,5,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,5,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (B)--(A)--(C)(B_1)--(A_1)--(C_1)--cycle;\n\\draw [dashed] (B)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求三棱柱$ABC-A_1B_1C_1$的体积;\\\\\n(2) 设$M$为$BC$中点, 求直线$A_1M$与平面$ABC$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426152,7 +428940,9 @@ "id": "016746", "content": "如图, 已知$S$为平行四边形$ABCD$所在平面外一点, $M$、$N$分别是线段$SA$. 线段$BD$上的点, 且满足$\\dfrac{SM}{MA}=\\dfrac{BN}{ND}$, 求证: $MN\\parallel$平面$SBC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (3,0,0) node [below] {$B$} coordinate (B);\n\\draw (0,0,-3) node [below] {$D$} coordinate (D);\n\\draw (3,0,-3) node [right] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(C)$) ++ (0,2,0) node [above] {$S$} coordinate (S);\n\\draw (S)--(A)--(B)--(C)--cycle(S)--(B);\n\\draw [dashed] (A)--(D)--(C)(S)--(D)--(B);\n\\filldraw ($(S)!0.4!(A)$) node [left] {$M$} coordinate (M) circle (0.03);\n\\filldraw ($(B)!0.4!(D)$) node [above] {$N$} coordinate (N) circle (0.03);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426172,7 +428962,9 @@ "id": "016747", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E$是棱$DD_1$的中点.\\\\\n(1) 求直线$BE$与平面$ABB_1A_1$所成的角;\\\\\n(2) 在棱$C_1D_1$上是否存在一点$F$, 使得$B_1F\\parallel$平面$A_1BE$? 若存在, 求出$F$点的位置, 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426192,7 +428984,9 @@ "id": "016748", "content": "空间不重合的两个平面的位置关系有\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426212,7 +429006,9 @@ "id": "016749", "content": "二面角的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426232,7 +429028,9 @@ "id": "016750", "content": "若$\\alpha$、$\\beta$是两个不重合的平面, 在下列条件中可判定平面$\\alpha, \\beta$平行的是\\bracket{20}.\n\\onech{$\\alpha$、$\\beta$都垂直于同一个平面}{$\\alpha$内不共线的三点到$\\beta$的距离相等}{$l$、$m$是平面$\\alpha$上的直线, 且$l\\parallel \\beta$, $m\\parallel \\beta$}{$l$、$m$是两条异面直线, 且$l\\parallel \\alpha$, $m\\parallel \\alpha$, $l\\parallel \\beta$, $m\\parallel \\beta$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426252,7 +429050,9 @@ "id": "016751", "content": "设$m$、$n$为两条直线, $\\alpha$、$\\beta$为两个平面, 则下列命题中假命题是\\bracket{20}.\n\\twoch{若$m \\perp n$, $m \\perp \\alpha$, $n \\perp \\beta$, 则$\\alpha \\perp \\beta$}{若$m\\parallel n$, $m \\perp \\alpha$, $n\\parallel \\beta$, 则$\\alpha \\perp \\beta$}{若$m \\perp n$, $m\\parallel \\alpha$, $n\\parallel \\beta$, 则$\\alpha\\parallel \\beta$}{若$m\\parallel n$, $m \\perp \\alpha$, $n \\perp \\beta$, 则$\\alpha\\parallel \\beta$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426272,7 +429072,9 @@ "id": "016752", "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 二面角$A-DD_1-C$的大小为\\blank{50};\\\\\n(2) 二面角$D_1-AB-D$的大小为\\blank{50};\\\\\n(3) 二面角$A-D_1B_1-C_1$的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426292,7 +429094,9 @@ "id": "016753", "content": "在正方体$ABCD-A_1B_1C_1D_1$中.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 平面$ACC_1A_1 \\perp$平面$BB_1D_1D$;\\\\\n(2) 求二面角$A_1-AC-B$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426312,7 +429116,9 @@ "id": "016754", "content": "在三棱柱$ABC-A_1B_1C_1$中, $E$、$F$、$G$、$H$分别是$AB$、$AC$、$A_1B_1$、$A_1C_1$的中点. 求证: 平面$EFA_1\\parallel$平面$BCHG$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (1.5,0,1) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (A) --++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(A)!0.5!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [above] {$F$} coordinate (F);\n\\draw ($(A_1)!0.5!(B_1)$) node [below left] {$G$} coordinate (G);\n\\draw ($(A_1)!0.5!(C_1)$) node [above] {$H$} coordinate (H);\n\\draw (A)--(B)--(C)(A_1)--(B_1)--(C_1)--cycle(G)--(H)(B)--(G)(A_1)--(E);\n\\draw [dashed] (A)--(C)(E)--(F)--(A_1)(H)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426332,7 +429138,9 @@ "id": "016755", "content": "在三棱柱$ABC-A_1B_1C_1$中, 若$D_1$、$D$分别为$B_1C_1$、$BC$的中点, 求证: 平面$A_1BD_1\\parallel$平面$AC_1D$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (1.5,0,1) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (A) --++ (0,2,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) --++ (0,2,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) --++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(B_1)!0.5!(C_1)$) node [above] {$D_1$} coordinate (D_1);\n\\draw ($(B)!0.5!(C)$) node [below] {$D$} coordinate (D);\n\\draw (A)--(B)--(C)(A_1)--(B_1)--(C_1)--cycle(A_1)--(B)--(D_1)--cycle(C_1)--(D);\n\\draw [dashed] (A)--(C)(D)--(A)--(C_1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426352,7 +429160,9 @@ "id": "016756", "content": "如图, 在矩形$ABCD$中, $AB=3$, $AD=4$, $PA \\perp$平面$ABCD$, $PA=\\dfrac{4}{5}$, 求二面角$A-BD-P$的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (4,0,0) node [right] {$D$} coordinate (D);\n\\draw (4,0,-3) node [right] {$C$} coordinate (C);\n\\draw (0,0,-3) node [above] {$B$} coordinate (B);\n\\draw (0,0.8,0) node [above] {$P$} coordinate (P);\n\\draw (P)--(A)--(D)--(C)--cycle(P)--(D)(C)--(B)--(P);\n\\draw [dashed] (A)--(B)(B)--(D);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426372,7 +429182,9 @@ "id": "016757", "content": "在棱长为$10$的正方体$ABCD-A_1B_1C_1D_1$中, $P$为左侧面$AD_1A_1$上一点, 已知点$P$到$A_1D_1$的距离为$3$, $P$到$AA_1$的距离为$2$, 则过点$P$且与$A_1C$平行的直线交正方体于$P$、$Q$两点, 则$Q$点所在的平面是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\def\\l{10}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [dashed] (A_1) ++ (0,-3,0) --++ (0,0,-2) node [right] {$P$} coordinate (P) --++ (0,3,0);\n\\draw [dashed] (A_1)--(C);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$AA_1B_1B$}{$BB_1C_1C$}{$CC_1D_1D$}{$ABCD$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426392,7 +429204,9 @@ "id": "016758", "content": "正四面体相邻两个面所成二面角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426412,7 +429226,9 @@ "id": "016759", "content": "类比平面内``垂直于同一条直线的两条直线互相平行''的性质, 可推出空间中有下列结论:\\\\\n\\textcircled{1} 垂直于同一条直线的两条直线互相平行;\\\\\n\\textcircled{2} 垂直于同一条直线的两个平面互相平行;\\\\\n\\textcircled{3} 垂直于同一个平面的两条直线互相平行;\\\\\n\\textcircled{4} 垂直于同一个平面的两个平面互相平行.\\\\\n其中正确的是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426432,7 +429248,9 @@ "id": "016760", "content": "已知平面$\\alpha\\parallel$平面$\\beta$, $P$是平面$\\alpha$、$\\beta$外一点, 过点$P$的直线$m$与$\\alpha$、$\\beta$分别交于$A$、$C$两点, 过点$P$的直线$n$与$\\alpha$、$\\beta$分别交于$B$、$D$两点, 且$PA=6$, $AC=9$, $PD=8$, 则$BD=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426452,7 +429270,9 @@ "id": "016761", "content": "设$\\alpha, \\beta$表示两个不同的平面, $l$表示一条直线, 且$l \\subset \\alpha$, 则$l\\parallel \\beta$是$\\alpha\\parallel \\beta$的\\blank{50}条件.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426472,7 +429292,9 @@ "id": "016762", "content": "二面角$\\alpha-l-\\beta$的大小等于$60{^\\circ}$, 其内一点$P$到$\\alpha$、$\\beta$的距离分别为$1$厘米和$2$厘米, 则点$P$到棱$l$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426492,7 +429314,9 @@ "id": "016763", "content": "设$m$、$n$是空间两条不同的直线, $\\alpha$、$\\beta$是空间两个不重合的平面, 给出下面命题:\\\\\n\\textcircled{1} 若$m \\perp \\alpha$, $m\\parallel \\beta$, 则$\\alpha \\perp \\beta$;\\\\\n\\textcircled{2} 若$m\\parallel \\alpha$, $n\\parallel \\beta$, 则$m\\parallel n$;\\\\\n\\textcircled{3} 若$m \\subset \\alpha$, $n \\subset \\alpha$, $m\\parallel \\beta$, $n\\parallel \\beta$, 则$\\alpha\\parallel \\beta$;\\\\\n\\textcircled{4} 若$m \\perp \\alpha$, $n\\parallel \\beta$, $\\alpha\\parallel \\beta$, 则$m \\perp n$.\\\\\n其中正确命题有\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426512,7 +429336,9 @@ "id": "016764", "content": "空间四边形$ABCD$中, 若$AD \\perp BC$, $AD \\perp BD$, 则\\bracket{20}.\n\\twoch{平面$ABC \\perp$平面$ADC$}{平面$ABC \\perp$平面$ADB$}{平面$ABC \\perp$平面$DBC$}{平面$ABC \\perp$平面$ADC$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426532,7 +429358,9 @@ "id": "016765", "content": "下列命题中, 错误的是\\bracket{20}.\n\\onech{一条直线与两个平行平面中的一个相交, 则必与另一个平面相交}{平行于同一平面的两个不同平面平行}{如果平面$\\alpha$不垂直于平面$\\beta$, 那么平面$\\alpha$内一定不存在直线垂直于平面$\\beta$}{若直线$l$不平行于平面$\\alpha$, 则在平面$\\alpha$内不存在与$l$平行的直线}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426552,7 +429380,9 @@ "id": "016766", "content": "设$\\alpha$、$\\beta$是两个不重合的平面, $l$、$m$是两条不重合的直线, 则``$\\alpha\\parallel \\beta$''的一个充分非必要条件是\\bracket{20}.\n\\twoch{$l \\subset \\alpha$, $m \\subset_\\alpha$且$l\\parallel \\beta$, $m\\parallel \\beta$}{$l \\subset \\alpha$, $m \\subset \\beta$, 且$l\\parallel m$}{$l \\perp \\alpha$, $m \\perp \\beta$且$l\\parallel m$}{$l\\parallel \\alpha$, $m\\parallel \\beta$, 且$l\\parallel m$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426572,7 +429402,9 @@ "id": "016767", "content": "如图, 在正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$、$G$分别是$AD$、$DD_1$、$DC$的中点, 过点$B$画出一个平面与平面$EFG$平行.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.5!(D)$) node [left] {$E$} coordinate (E);\n\\draw ($(D)!0.5!(C)$) node [above] {$G$} coordinate (G);\n\\draw ($(D)!0.5!(D_1)$) node [right] {$F$} coordinate (F);\n\\draw [dashed] (E)--(F)--(G)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426592,7 +429424,9 @@ "id": "016768", "content": "一个坡面度数为$30{^\\circ}$, 人在坡面上从点$A$沿着与坡面与水平面的交线$l$成$45{^ \\circ}$的方向行走$200$米到达点$B$, 求人离水平面的距离为多少米?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426612,7 +429446,9 @@ "id": "016769", "content": "如图, 在四面体$ABCD$中, $CB=CD$, $AD \\perp BD$, 且$E$、$F$分别是$AB$、$BD$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$A$} coordinate (A);\n\\draw (-0.5,1,2) node [left] {$C$} coordinate (C);\n\\draw (C)--(A)--(B)--cycle(C)--(D)--(A);\n\\draw [dashed] (B)--(D);\n\\draw ($(B)!0.5!(D)$) node [left] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(A)$) node [right] {$E$} coordinate (E);\n\\draw [dashed] (C)--(F)--(E);\n\\draw (C)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $EF\\parallel$平面$ACD$;\\\\\n(2) 求证: 平面$EFC \\perp$平面$BCD$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426632,7 +429468,9 @@ "id": "016770", "content": "已知圆柱的底面半径为$1$, 高为$2$, 则圆柱的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426652,7 +429490,9 @@ "id": "016771", "content": "已知一个圆锥的底面半径为$6$, 其体积为$30 \\pi$则该圆锥的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426672,7 +429512,9 @@ "id": "016772", "content": "四面体$ABCD$四个面的重心分别为$E$、$F$、$G$、$H$, 则四面体$EFGH$的表面积与四面体$ABCD$的表面积的比值是\\bracket{20}.\n\\fourch{$\\dfrac{1}{27}$}{$\\dfrac{1}{16}$}{$\\dfrac{1}{9}$}{$\\dfrac{1}{8}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426692,7 +429534,9 @@ "id": "016773", "content": "已知圆柱底面半径为$1$, 高为$2$, $AB$为上底面圆的一条直径, $C$为下底面圆周上的一个动点, 则$\\triangle ABC$的面积的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426712,7 +429556,9 @@ "id": "016774", "content": "三棱柱$ABC-A_1B_1C_1$中, 若$E$、$F$分别为$AB$、$AC$的中点, 平面$EB_1C_1$将三棱柱分成体积为$V_1, V_2$的两部分, 那么$V_1: V_2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426732,7 +429578,9 @@ "id": "016775", "content": "已知$ABCD$是边长为$1$的正方形, 正方形$ABCD$绕$AB$旋转形成一个圆柱.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$A$} coordinate (A);\n\\draw (0,2,0) node [right] {$B$} coordinate (B);\n\\draw (-2,0,0) node [left] {$D$} coordinate (D);\n\\draw (-2,2,0) node [left] {$C$} coordinate (C);\n\\draw (B) ellipse (2 and 0.5);\n\\draw (D) arc (180:360:2 and 0.5) coordinate (T);\n\\draw [dashed] (T) arc (0:180:2 and 0.5);\n\\draw (C)--(B)(C)--(D)(T)--++(0,2);\n\\draw [dashed] (D)--(A)--(B);\n\\draw (-110:2 and 0.5) node [below] {$D_1$} coordinate (D_1);\n\\draw (D_1) --++ (0,2) node [above] {$C_1$} coordinate (C_1);\n\\draw (C_1) -- (B);\n\\draw [dashed] (D_1)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 求该圆柱的表面积;\\\\\n(2) 正方形$ABCD$绕$AB$逆时针旋转$\\dfrac{\\pi}{2}$至$ABC_1D_1$, 求线段$CD_1$与平面$ABCD$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426752,7 +429600,9 @@ "id": "016776", "content": "如图, 已知正三棱柱$ABC-A_1B_1C_1$的侧面对角线$A_1B$与侧面$ACC_1A_1$成$45{^\\circ}$角, $AB=4$, 求棱柱的侧面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$C$} coordinate (C);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$A$} coordinate (A);\n\\draw ({\\l/2},0,0) node [right] {$B$} coordinate (B);\n\\draw (C) ++ (0,\\h) node [left] {$C_1$} coordinate (C_1);\n\\draw (A) ++ (0,\\h) node [below right] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\h) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) -- (A) -- (B) (C) -- (C_1) (A) -- (A_1) (B) -- (B_1) (C_1) -- (A_1) -- (B_1) (C_1) -- (B_1);\n\\draw [dashed] (C) -- (B);\n\\draw (A_1)--(B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426772,7 +429622,9 @@ "id": "016777", "content": "如图所示, 在平行六面体$ABCD-A_1B_1C_1D_1$中, 已知$AB=5$, $AD=4$, $AA_1=3$, $AB \\perp AD$, $\\angle A_1AB=\\angle A_1AD=\\dfrac{\\pi}{3}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (5,0,0) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,0,-4) node [below] {$D$} coordinate (D);\n\\draw (D) ++ (5,0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ ({3/2},{3/2*sqrt(2)},{-3/2},) node [left] {$A_1$} coordinate (A_1);\n\\draw ($(A_1)-(A)+(B)$) node [above] {$B_1$} coordinate (B_1);\n\\draw ($(A_1)-(A)+(C)$) node [right] {$C_1$} coordinate (C_1);\n\\draw ($(A_1)-(A)+(D)$) node [above] {$D_1$} coordinate (D_1);\n\\draw (A)--(B)--(C)--(C_1)--(D_1)--(A_1)--cycle(A_1)--(B_1)--(C_1)(B_1)--(B);\n\\draw [dashed] (A)--(D)--(C)(D)--(D_1);\n\\filldraw (A) ++ ({3/2},0,{-3/2}) node [right] {$O$} coordinate (O) circle (0.03);\n\\draw [dashed] (A_1)--(O);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 顶点$A_1$在底面$ABCD$上的射影$O$在$\\angle BAD$的平分线上;\\\\\n(2) 求这个平行六面体的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426792,7 +429644,9 @@ "id": "016778", "content": "从边长为$a$的正三角形的顶点, 在各边上截取长度为$x$的线段, 以这些线段为边作成有两个角是直角的四边形, 这样的四边形有三个, 把这三个四边形剪掉, 用剩下部分折成一个底为正三角形的无盖形容器.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-30:2) coordinate (C_1) (-30:1) coordinate (C);\n\\draw (-150:2) coordinate (B_1) (-150:1) coordinate (B);\n\\draw (90:2) coordinate (A_1) (90:1) coordinate (A);\n\\draw (A) --++ (30:0.5) (A) --++ (150:0.5);\n\\draw (B) --++ (-90:0.5) (B) --++ (150:0.5);\n\\draw (C) --++ (-90:0.5) (C) --++ (30:0.5);\n\\draw (A_1)--(B_1)--(C_1)--cycle;\n\\draw [dashed] (A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) coordinate (A) ({sqrt(3)},0,0) coordinate (B) ({sqrt(3)/2},0,{3/2}) coordinate (C);\n\\draw (A) --++ (0,0.5,0) coordinate (A_1) (B) --++ (0,0.5,0) coordinate (B_1) (C) --++ (0,0.5,0) coordinate (C_1);\n\\draw (A)--(A_1)(B)--(B_1)(C)--(C_1);\n\\draw (A_1)--(B_1)--(C_1)--cycle;\n\\draw (A)--(C)--(B);\n\\draw [dashed] (A)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求这容器的容积$V(x)$;\\\\\n(2) 求使$V(x)$最大时的$x$及$V(x)$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -426812,7 +429666,9 @@ "id": "016779", "content": "三棱柱$ABC-A_1B_1C_1$侧棱$AA_1$与侧面$BCC_1B$的距离为$d$, $S_{BCC_1B1}=s$, 则三棱柱$ABC$$A_1B_1C_1$的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426832,7 +429688,9 @@ "id": "016780", "content": "一个长方体共一顶点的三个面的面积分别是$\\sqrt{2}, \\sqrt{3}, \\sqrt{6}$, 这个长方体对角线的长是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426852,7 +429710,9 @@ "id": "016781", "content": "过教室的一个墙角$C_1$的三个平面两两互相垂直, 空中一点$A$到三个面的距离分别为$3,4,5$.\\\\\n(1) 点$A$到墙角$C_1$的距离为\\blank{50};\\\\\n(2) $C_1A$与三个墙面所成角的大小分别为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426872,7 +429732,9 @@ "id": "016782", "content": "如图, 一圆柱体的底面周长为$24 \\text{cm}$, 高$AB$为$16 \\text{cm}, BC$是上底面的直径. 一只昆虫从点$A$出发, 沿着圆柱的侧面爬行到点$C$, 则昆虫爬行的最短距离是\\blank{50}$\\text{cm}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.2]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw ({24/pi},0) node [right] {$B$} coordinate (B);\n\\draw (A) --++ (0,16) node [left] {$D$} coordinate (D);\n\\draw (B) --++ (0,16) node [right] {$C$} coordinate (C);\n\\draw (C)--(D);\n\\draw [dashed] (A)--(B);\n\\draw (C) arc (0:360:{12/pi} and {3/pi});\n\\draw (A) arc (180:360:{12/pi} and {3/pi});\n\\draw [dashed] (B) arc (0:180:{12/pi} and {3/pi});\n\\draw (A) parabola (C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426892,7 +429754,9 @@ "id": "016783", "content": "用边长分别为$8 \\pi$和$6 \\pi$的矩形卷成圆柱, 则圆柱的底面面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426912,7 +429776,9 @@ "id": "016784", "content": "有一无盖圆柱形容器, 其表面积为定值, 当它的底面半径与高之比为\\blank{50}时, 此容器的容量最大.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -426932,7 +429798,9 @@ "id": "016785", "content": "下列命题:\\\\\n\\textcircled{1} 有一个侧面是矩形的棱柱是直棱柱;\\\\\n\\textcircled{2} 有两个侧面是矩形的棱柱是直棱柱;\\\\\n\\textcircled{3} 有两个相邻侧面是矩形的棱柱是直棱柱;\\\\\n\\textcircled{4} 有三个侧面是矩形棱柱是直棱柱.\\\\\n其中正确命题的个数是\\bracket{20}.\n\\fourch{1}{2}{3}{4}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426952,7 +429820,9 @@ "id": "016786", "content": "如果圆柱的底面直径为$4$, 母线长为$2$, 那么圆柱的侧面展开图的面积等于\\bracket{20}.\n\\fourch{$8 \\pi$}{$4 \\pi$}{$16 \\pi$}{$8$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426972,7 +429842,9 @@ "id": "016787", "content": "魏晋时期数学家刘徽在他的著作《九章算术注》中, 称一个正方体内两个互相垂直的内切圆柱所围成的几何体为``牟合方盖''(如图), 刘徽通过计算得知正方体的内切球的体积与``牟合方盖''的体积之比应为$\\pi: 4$. 在某一球内任意取一点, 则此点取自球的一个内接正方体的``牟合方盖''的概率为 \\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, x = {(-10:1cm)}, y = {(-130:0.6cm)}, z = {(90:1cm)}]\n\\def\\l{2}\n\\draw (-1,-1,1) coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\foreach \\i in {0,15,...,345}\n{\\draw [blue!50] ({cos(\\i)},-1,{sin(\\i)}) --++ (0,2,0);\n\\draw [red!50] ({cos(\\i)},{sin(\\i)},-1) --++ (0,0,2);};\n\\foreach \\i in {-1,-0.8,-0.6,...,1.05}\n{\\draw [domain = 0:360, blue!50] plot ({cos(\\x)},\\i,{sin(\\x)});\n\\draw [domain = 0:360, red!50] plot ({cos(\\x)},{sin(\\x)},\\i);};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex, x = {(-10:1cm)}, y = {(-130:0.6cm)}, z = {(90:1cm)}]\n\\def\\l{2}\n\\draw (-1,-1,1) coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [domain = 0:360] plot ({cos(\\x)},{sin(\\x)},{-sin(\\x)});\n\\draw [domain = 0:360] plot ({cos(\\x)},{sin(\\x)},{sin(\\x)});\n\\draw (0,-1,1) -- (0,1,1) -- (0,1,-1);\n\\draw [dashed] (0,1,-1) -- (0,-1,-1) -- (0,-1,1);\n\\draw [domain = 0:360, gray] plot ({cos(\\x)},1,{sin(\\x)});\n\\draw [domain = 0:360, gray, dashed] plot ({cos(\\x)},-1,{sin(\\x)});\n\\draw [domain = 0:360, gray] plot ({cos(\\x)},{sin(\\x)},1);\n\\draw [domain = 0:360, gray, dashed] plot ({cos(\\x)},{sin(\\x)},-1);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{2}{3}$}{$\\dfrac{4 \\sqrt{3}}{9 \\pi}$}{$\\dfrac{\\sqrt{3}}{9}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -426992,7 +429864,9 @@ "id": "016788", "content": "一个长方体全面积是$20 \\text{cm}^2$, 所有棱长的和是$24 \\text{cm}$, 求长方体的对角线长.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427012,7 +429886,9 @@ "id": "016789", "content": "如图, 在三棱柱$ABC-A_1B_1C_1$中, 底面是边长为$a$的正三角形, $AA_1$与$AB$、$AC$所成的角均为$60{^\\circ}$, 且$AA_1=BA_1=CA_1$, 求该棱柱的体积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ ({\\h/2},{\\h/sqrt(3)*sqrt(2)},{\\h/sqrt(3)/2}) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ ({\\h/2},{\\h/sqrt(3)*sqrt(2)},{\\h/sqrt(3)/2}) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ ({\\h/2},{\\h/sqrt(3)*sqrt(2)},{\\h/sqrt(3)/2}) node [right] {$C_1$} coordinate (C_1);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427032,7 +429908,9 @@ "id": "016790", "content": "如图所示为一个半圆柱, $E$为半圆弧$CD$上一点, $CD=\\sqrt{5}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (A) --++ ({sqrt(5)},0) node [right] {$B$} coordinate (B);\n\\draw (A) --++ (0,{2*sqrt(5)}) node [left] {$D$} coordinate (D);\n\\draw (B) --++ (0,{2*sqrt(5)}) node [right] {$C$} coordinate (C);\n\\draw (C)--(D);\n\\draw (C) arc (0:180:{sqrt(5)/2} and {sqrt(5)/8});\n\\draw [dashed] (B) arc (0:180:{sqrt(5)/2} and {sqrt(5)/8});\n\\draw ($(C)!0.5!(D)$) ++ (120:{sqrt(5)/2} and {sqrt(5)/8}) node [above] {$E$} coordinate (E);\n\\draw (D)--(E)--(C);\n\\draw [dashed] (A)--(E)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$AD=2 \\sqrt{5}$, 求四棱锥$E-ABCD$的体积的最大值;\\\\\n(2) 有三个条件:\n\\textcircled{1} $4 \\overrightarrow{DE} \\cdot \\overrightarrow{DC}=\\overrightarrow{EC} \\cdot \\overrightarrow{DC}$;\n\\textcircled{2} 直线$AD$与$BE$所成角的正弦值为$\\dfrac{2}{3}$;\n\\textcircled{3} $\\dfrac{\\sin \\angle EAB}{\\sin \\angle EBA}=\\dfrac{\\sqrt{6}}{2}$.\n请你从中选择两个作为条件, 求直线$AD$与平面$EAB$所成角的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427052,7 +429930,9 @@ "id": "016791", "content": "已知一个圆锥的底面半径为$6$, 其体积为$30 \\pi$, 则该圆锥的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427072,7 +429952,9 @@ "id": "016792", "content": "在棱长为$1$的正方体上, 分别用过共顶点的三条棱中点的平面截该正方体, 则截去$8$个三棱锥后, 剩下的凸多面体的体积是\\bracket{20}.\n\\fourch{$\\dfrac{2}{3}$}{$\\dfrac{7}{6}$}{$\\dfrac{4}{5}$}{$\\dfrac{5}{6}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -427092,7 +429974,9 @@ "id": "016793", "content": "正四棱锥底面边长为$4$, 侧棱长为$3$, 则其体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427112,7 +429996,9 @@ "id": "016794", "content": "学生到工厂劳动实践, 利用3D打印技术制作模型. 如图, 该模型为长方体$ABCD-A_1B_1C_1D_1$挖去四棱锥$O-EFGH$后所得几何体, 其中$O$为长方体的中心, $E$、$F$、$G$、$H$分别为所在棱的中点, $AB=BC=6 \\text{cm}$, $AA_1=4 \\text{cm}$, 3D打印所用原料密度为$0.9 \\mathrm{g} / \\text{cm}^3$, 不考虑打印损耗, 制作该模型所需原料的质量为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\def\\l{6}\n\\def\\m{6}\n\\def\\n{4}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$F$} coordinate (F);\n\\draw ($(C_1)!0.5!(B_1)$) node [above] {$G$} coordinate (G);\n\\draw ($(B_1)!0.5!(B)$) node [left] {$H$} coordinate (H);\n\\draw ($(A)!0.5!(C_1)$) node [left] {$O$} coordinate (O);\n\\draw [dashed] (O)--(E)(O)--(F)(O)--(G)(O)--(H);\n\\draw (E)--(F)--(G)--(H)--cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427132,7 +430018,9 @@ "id": "016795", "content": "已知圆锥底面的半径为$10$, 母线长为$60$, 则底面圆周上一点$B$沿侧面绕两周回到点$B$的最短距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427152,7 +430040,9 @@ "id": "016796", "content": "如图, 已知四棱锥$P-ABCD$的底面$ABCD$为矩形, $PA \\perp BD$, $PA=1$, $BC=2$, $PD=\\sqrt{5}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [right] {$A$} coordinate (A);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw (0,0,1) node [right] {$B$} coordinate (B);\n\\draw (-2,0,1) node [left] {$C$} coordinate (C);\n\\draw (-2,0,0) node [left] {$D$} coordinate (D);\n\\draw (C)--(B)--(A)--(P)--(D)--cycle(P)--(B)(P)--(C);\n\\draw [dashed] (D)--(A)(A)--(C)(B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 平面$PAB \\perp$平面$PAD$, 且$PA \\perp AC$;\\\\\n(2) 若四棱锥$P-ABCD$的每个顶点都在球$O$的球面上, 且球$O$的表面积为$6 \\pi$, 求三棱锥$B-PCD$的体积和全面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427172,7 +430062,9 @@ "id": "016797", "content": "已知圆锥的侧面展开图是一个半径为$4$的半圆.\n(1) 求圆锥的表面积;\n(2) 若用平行于圆锥的底面、且与底面的距离为$\\sqrt{3}$的平面截圆锥, 将此圆锥截成一个小圆锥和圆台两部分, 求小圆锥和圆台的体积比.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427192,7 +430084,9 @@ "id": "016798", "content": "如图, 一个圆锥的底面半径为$2 \\text{cm}$, 高为$6 \\text{cm}$, 在其中有一个高为$x \\text{cm}$的内接圆柱.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.4]\n\\draw (-2,0) coordinate (A) (2,0) coordinate (B) (0,6) coordinate (P);\n\\draw (-1,0) coordinate (S) (1,0) coordinate (T);\n\\draw (A) arc (180:360:2 and 0.5);\n\\draw [dashed] (A) arc (180:0:2 and 0.5);\n\\draw (A)--(P)--(B);\n\\draw [dashed] (S) arc (180:-180:1 and 0.25);\n\\draw [dashed] (S) --++ (0,3) arc (180:-180:1 and 0.25);\n\\draw [dashed] (T) --++ (0,3);\n\\end{tikzpicture}\n\\end{center}\n(1) 试用$x$表示圆柱的侧面积;\\\\\n(2) 当$x$为何值时, 圆柱的侧面积最大?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427212,7 +430106,9 @@ "id": "016799", "content": "如图, 给出两块相同的正三角形纸片, 要求用其中一块剪拼成一个正三棱锥模型的表面, 另一块剪拼成一个正三棱柱模型的表面, 使它们的表面积都与原三角形的面积相等, 请设计一种剪拼方法, 分别用虚线标示在图中, 并作简要说明; 并试比较你剪拼的正三棱锥和正三棱柱的体积的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) --++ (0:2) --++ (120:2) --++ (240:2) -- cycle;\n\\draw (4,0) --++ (0:2) --++ (120:2) --++ (240:2) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427232,7 +430128,9 @@ "id": "016800", "content": "一个圆锥体与和它等底等高的圆柱体体积相差$30$立方厘米, 这个圆锥体的体积是\\blank{50}立方厘米.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427252,7 +430150,9 @@ "id": "016801", "content": "如果圆锥的高为$8 \\text{cm}$, 圆锥的底面半径为$6 \\text{cm}$, 那么它的侧面展开图的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427272,7 +430172,9 @@ "id": "016802", "content": "有同学制作一圆锥模型, 模型的侧面是用一个半径为$9 \\text{cm}$, 圆心角为$240{^ \\circ}$的扇形铁皮制作, 再用一块圆铁片做底, 那么这块圆铁片的半径\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427292,7 +430194,9 @@ "id": "016803", "content": "一个圆柱体和一个圆锥体的高相等, 体积也相等, 圆锥体的底面积是$12$, 圆柱体的底面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427312,7 +430216,9 @@ "id": "016804", "content": "在棱长为$a$的正方体$ABCD-A_1B_1C_1D_1$中, $P$、$Q$是对角线$A_1C$上的点, 若$PQ=\\dfrac{a}{2}$, 则三棱锥$P-BDQ$的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427332,7 +430238,9 @@ "id": "016805", "content": "如图, 在四边形$ABCD$中, $\\angle DAB=90{^\\circ}$, $\\angle ADC=135{^ \\circ}$, $AB=3$, $CD=2 \\sqrt{2}$, $AD=2$. 四边形$ABCD$绕$AD$旋转一周所成几何体的表面积为\\blank{50}; 体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0) node [right] {$B$} coordinate (B);\n\\draw (0,2) node [left] {$D$} coordinate (D);\n\\draw (2,4) node [above] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\draw [dashed] (A) -- ($(D)!1.5!(A)$) (D) -- ($(A)!2.5!(D)$);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427352,7 +430260,9 @@ "id": "016806", "content": "若一个正棱锥的底面边长与侧棱长相等, 则这个棱锥一定不是\\bracket{20}.\n\\fourch{三棱锥}{四棱锥}{五棱锥}{六棱锥}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -427372,7 +430282,9 @@ "id": "016807", "content": "一副三角板由一块有一个内角为$60{^\\circ}$的直角三角形和一块等腰直角三角形组成, 如图所示, $\\angle B=\\angle F=90{^\\circ}$, $\\angle A=60{^\\circ}$, $\\angle D=45{^\\circ}$, $BC=DE$, 现将两块三角形板拼接在一起, 得三棱锥$F-CAB$, 取$BC$中点$O$与$AC$中点$M$, 则下列判断中错误的是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$C$} coordinate (C);\n\\draw (2,0) node [right] {$B$} coordinate (B);\n\\draw (2,{-2/sqrt(3)}) node [right] {$A$} coordinate (A);\n\\draw (A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$D$} coordinate (D);\n\\draw (2,0) node [right] {$E$} coordinate (E);\n\\draw (1,1) node [above] {$F$} coordinate (F);\n\\draw (D)--(E)--(F)--cycle;\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$C(D)$} coordinate (C);\n\\draw (2,0,0) node [right] {$B(E)$} coordinate (B);\n\\draw (2,0,{2/sqrt(3)}) node [below] {$A$} coordinate (A);\n\\draw (1,1,0) node [above] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(B)$) node [above right] {$Q$} coordinate (Q);\n\\draw ($(C)!0.5!(A)$) node [below] {$M$} coordinate (M);\n\\draw (C)--(F)--(B)--(A)--cycle(F)--(A)(F)--(M);\n\\draw [dashed] (C)--(B)(F)--(Q)--(M);\n\\end{tikzpicture}\n\\end{center}\n\\twoch{直线$BC \\perp$面$OFM$}{$AC$与面$OFM$所成的角为定值}{设面$ABF \\cap$面$MOF=l$, 则有$l\\parallel AB$}{三棱锥$F-COM$体积为定值}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -427392,7 +430304,9 @@ "id": "016808", "content": "正四棱台的上、下底面的边长分别为$2$、$4$, 侧棱长为$2$, 则其体积为\\bracket{20}.\n\\fourch{$20+12 \\sqrt{3}$}{$28 \\sqrt{2}$}{$\\dfrac{56}{3}$}{$\\dfrac{28 \\sqrt{2}}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -427412,7 +430326,9 @@ "id": "016809", "content": "已知三棱锥$P-ABC$的底面是直角三角形, $\\angle ACB=90{^\\circ}$, $CB=4$, $AB=20$, $D$为$AB$中点, 且$\\triangle PDB$是正三角形, 平面$PAC \\perp$平面$ABC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (20,0,0) node [right] {$B$} coordinate (B);\n\\draw ({96/5},0,{-8*sqrt(6)/5}) node [above] {$C$} coordinate (C);\n\\draw (15,{5*sqrt(21/2)/2},{-5*sqrt(3/2)/2}) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--(P)--cycle (P)--(B);\n\\draw [dashed] (A)--(C);\n\\draw ($(A)!0.5!(B)$) node [below] {$D$} coordinate (D);\n\\draw ($(P)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw (P)--(D)--(M)--(C);\n\\draw [dashed] (C)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $PA \\perp$平面$PBC$;\\\\\n(2) 设二面角$B-AP-C$大小为$\\alpha$, 求$\\sin \\alpha$的值;\\\\\n(3) 若$M$为$PB$中点, 求三棱锥$M-BCD$的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427432,7 +430348,9 @@ "id": "016810", "content": "如图, 设底面半径为$2$的圆锥的顶点、底面中心依次为$P$、$O$, $AB$为其底面的直径, 点$C$位于底面圆周上, 且$\\angle BOC=90{^ \\circ}$, 异面直线$PA$与$CB$所成角的大小为$60{^ \\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [above right] {$O$} coordinate (O);\n\\draw (-2,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [below right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above left] {$P$} coordinate (P);\n\\draw (A)--(P)--(B);\n\\draw (A) arc (180:360:2 and 0.5);\n\\draw [dashed] (B) arc (0:180:2 and 0.5) (A)--(B)(O)--(P);\n\\draw (-110:2 and 0.5) node [below] {$C$} coordinate (C);\n\\draw [dashed] (O)--(C)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求此圆锥的体积;\\\\\n(2) 求二面角$P-BC-O$的大小(结果用反三角函数值表示).", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427452,7 +430370,9 @@ "id": "016811", "content": "如图, 在三棱锥$A-BCD$中, 平面$ABD \\perp$平面$BCD$, $AB=AD$, $O$为$BD$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-2,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$A$} coordinate (A);\n\\draw ($(B)!0.5!(D)$) node [above left] {$O$} coordinate (O);\n\\draw (1.5,0,{sqrt(3)/2}) node [below] {$C$} coordinate (C);\n\\draw ($(A)!{1/3}!(D)$) node [above right] {$E$} coordinate (E);\n\\draw (A)--(B)--(C)--(D)--cycle(A)--(C)--(E);\n\\draw [dashed] (A)--(O)--(C)(B)--(E)(B)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $OA \\perp CD$;\\\\\n(2) 若$\\triangle OCD$是边长为$1$的等边三角形, 点$E$在棱$AD$上, $DE=2EA$, 且二面角$E-BC-D$的大小为$45{^ \\circ}$, 求三棱锥$A-BCD$的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427472,7 +430392,9 @@ "id": "016812", "content": "Rt$\\triangle ABC$的三个顶点在半径为$13$的球面上, 两直角边的长分别为$6$和$8$, 则球心到平面$ABC$的距离是\\bracket{20}.\n\\fourch{$5$}{$6$}{$10$}{$12$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -427492,7 +430414,9 @@ "id": "016813", "content": "一平面截一球得到直径是$6 \\text{cm}$的圆面, 球心到这个平面的距离是$4 \\text{cm}$, 则该球的体积是\\bracket{20}.\n\\fourch{$\\dfrac{100 \\pi}{3} \\text{cm}^3$}{$\\dfrac{208 \\pi}{3} \\text{cm}^3$}{$\\dfrac{500 \\pi}{3} \\text{cm}^3$}{$\\dfrac{416 \\sqrt{3} \\pi}{3} \\text{cm}^3$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -427512,7 +430436,9 @@ "id": "016814", "content": "一个与球心距离为$1$的平面截球所得的圆面面积为$\\pi$, 则球的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427532,7 +430458,9 @@ "id": "016815", "content": "已如$A$、$B$、$C$是半径为$1$的球$O$的球面上的三个点, 且$AC \\perp BC$, $AC=BC=1$, 则三棱球$O-ABC$的体积为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{2}}{12}$}{$\\dfrac{\\sqrt{3}}{12}$}{$\\dfrac{\\sqrt{2}}{4}$}{$\\dfrac{\\sqrt{3}}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -427552,7 +430480,9 @@ "id": "016816", "content": "某地球仪上北纬$30{^ \\circ}$纬线的长度为$12 \\pi \\text{cm}$, 该地球仪的半径是\\blank{50}$\\text{cm}$, 表面积是\\blank{50}$\\text{cm}^2$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427572,7 +430502,9 @@ "id": "016817", "content": "北京靠近北纬$40{^ \\circ}$线, 求北纬$40{^ \\circ}$线的长度约等于多少$\\text{km}$? (地球半径大约为$6370 \\text{km}$)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427592,7 +430524,9 @@ "id": "016818", "content": "在半径为$13 \\text{cm}$的球面上有$A$、$B$、$C$三点, $AB=BC=AC=12 \\text{cm}$, 求球心到经过这三点的截面的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427612,7 +430546,9 @@ "id": "016819", "content": "正四面体的棱长为$a$, 则外接球的表面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427632,7 +430568,9 @@ "id": "016820", "content": "正四面体的棱长为$a$, 则内切球的表面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427652,7 +430590,9 @@ "id": "016821", "content": "在三棱锥$P-ABC$中, $PA \\perp$平面$ABC$, $\\angle BAC=120{^ \\circ}$, $AP=2 \\sqrt{2}$, $AB=AC=4$则三棱锥$P-ABC$的外接球的表面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427672,7 +430612,9 @@ "id": "016822", "content": "半径为$R$的球$O$中有一内接圆柱, 当圆柱的侧面积最大时, 求球的表面积与该圆柱的侧面积之差.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427692,7 +430634,9 @@ "id": "016823", "content": "一个透明的球形装饰品内放置了两个公共底面的圆锥, 且这两个圆锥的顶点和底面圆周周在这个球面上, 如图, 已知圆锥底面面积是这个球面面积的$\\dfrac{3}{16}$, 设球的半径为$R$, 圆锥底面半径为$r$.\n\\begin{center}\n\\begin{tikzpicture}\n\\draw (0,0) circle (2) coordinate (O);\n\\draw ({-sqrt(3)},-1) coordinate (S);\n\\draw ({sqrt(3)},-1) coordinate (T);\n\\draw (0,-2) coordinate (A);\n\\draw (0,2) coordinate (B);\n\\draw ($(A)!0.5!(O)$) coordinate (A1);\n\\draw [dashed] (A) -- (B) (S) -- (B) -- (T) -- (A) -- (S) (A1) -- (T) (O) -- (T);\n\\draw ($(A1)!0.5!(T)$) ;\n\\draw ($(O)!0.5!(T)$) ;\n\\draw (T) arc (0:-180:{sqrt(3)} and {sqrt(3)/4});\n\\draw [dashed] (T) arc (0:180:{sqrt(3)} and {sqrt(3)/4});\n\\end{tikzpicture}\n\\end{center}\n(1) 试确定$R$与$r$的关系, 并求出较大圆锥与较小圆锥的体积之比;\\\\\n(2) 求出两个圆锥的体积之和与球的体积之比.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427712,7 +430656,9 @@ "id": "016824", "content": "已知过球面上$A$、$B$、$C$三点的截面和球心的距离等于球半径的一半, 且$AB=BC=CA=2$, 则球面面积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427732,7 +430678,9 @@ "id": "016825", "content": "用平面$\\alpha$截半径为$R$的球, 如果球心到平面$\\alpha$的距离为$\\dfrac{R}{2}$, 那么截得小圆的面积与球的表面积的比值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427752,7 +430700,9 @@ "id": "016826", "content": "球的半径为$R$, 则它的外切正方体的棱长为\\blank{50} , 内接正方体的棱长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427772,7 +430722,9 @@ "id": "016827", "content": "已知正方体外接球的体积是$\\dfrac{32}{3} \\pi$, 那么正方体的棱长等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427792,7 +430744,9 @@ "id": "016828", "content": "已知球的表面积为$20 \\pi$, 球面上有$A$、$B$、$C$三点, 如果$AB=AC=BC=2 \\sqrt{3}$, 则球心到平面$ABC$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427812,7 +430766,9 @@ "id": "016829", "content": "已知$A$、$B$是球$O$的球面上两点, $\\angle AOB=90{^ \\circ}, C$为该球面上的动点, 若三棱锥$O-ABC$体积的最大值为$\\dfrac{32}{3}$, 球$O$的表面积为 \\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -427832,7 +430788,9 @@ "id": "016830", "content": "球的两个平行截面面积分别为$5 \\pi$和$8 \\pi$, 球心到这两个截面的距离之差等于$1$, 则球的直径为\\bracket{20}.\n\\fourch{$3$}{$4$}{$5$}{$6$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -427852,7 +430810,9 @@ "id": "016831", "content": "设$A$、$B$、$C$、$D$是球面上的四个点, 且在同一平面内, $AB=BC=CD=DA=3$, 球心到该平面的距离是球半径的一半, 则球的体积是\\bracket{20}.\n\\fourch{$8 \\sqrt{6} \\pi$}{$64 \\sqrt{6} \\pi$}{$24 \\sqrt{2} \\pi$}{$72 \\sqrt{2} \\pi$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -427872,7 +430832,9 @@ "id": "016832", "content": "赤道上两点$A$、$B$分别是东经$125{^ \\circ}$和西经$136{^ \\circ}$, 若$O$为球心, 则地球半径$OA$、$OB$所成角$\\angle AOB$是\\bracket{20}.\n\\fourch{$261{^ \\circ}$}{$99{^ \\circ}$}{$11{^ \\circ}$}{$349{^ \\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -427892,7 +430854,9 @@ "id": "016833", "content": "如图所示, 正四面体$ABCD$中, $E$是棱$AD$的中点, $P$是棱$AC$上一动点, $BP+PE$的最小值为$\\sqrt{14}$, 求该正四面体的外接球表面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw (1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw ($1/3*(D)+1/3*(B)+1/3*(C)$) ++ (0,{2*sqrt(6)/3},0) node [above] {$A$} coordinate (A);\n\\draw (B)--(C)--(D)--(A)--cycle(A)--(C);\n\\draw [dashed] (B)--(D);\n\\draw ($(A)!{1/3}!(C)$) node [below] {$P$} coordinate (P);\n\\draw ($(A)!0.5!(D)$) node [right] {$E$} coordinate (E);\n\\draw (B)--(P)--(E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427912,7 +430876,9 @@ "id": "016834", "content": "已知圆柱的底面直径与高都等于球的直径, 求证:\\\\\n(1) 球的表面积等于圆柱的侧面积;\\\\\n(2) 球的表面积等于圆柱全面积的$\\dfrac{2}{3}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427932,7 +430898,9 @@ "id": "016835", "content": "在四棱锥$P-ABCD$中, $BC\\parallel AD$, $AD \\perp AB$, $AB=2 \\sqrt{3}$, $AD=6$, $BC=4$, $PA=PB=PD=4 \\sqrt{3}$, 求三棱锥$P-BCD$外接球的表面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale=0.5]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (6,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,{-2*sqrt(3)}) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (4,0,0) node [right] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(D)$) ++ (0,6,0) node [above] {$P$} coordinate (P);\n\\draw (A)--(D)--(P)--cycle;\n\\draw [dashed] (A)--(B)--(C)--(D)(B)--(P)--(C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -427952,7 +430920,9 @@ "id": "016836", "content": "已知$A(1,1,-2)$、$B(1,1,1)$, 则线段$AB$的长度是\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$4$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -427972,7 +430942,9 @@ "id": "016837", "content": "向量$\\overrightarrow {m}=(8,3, a)$, $\\overrightarrow {n}=(2 b, 6,5)$, 若向量$\\overrightarrow {m}\\parallel \\overrightarrow {n}$, 则$a+b$的值为\\bracket{20}.\n\\fourch{$0$}{$\\dfrac{5}{2}$}{$\\dfrac{21}{2}$}{$8$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -427992,7 +430964,9 @@ "id": "016838", "content": "已知向量$\\overrightarrow {a}$、$\\overrightarrow {b}$、$\\overrightarrow {c}$, 满足$\\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}=\\overrightarrow{0}$, $|\\overrightarrow {a}|=3$, $|\\overrightarrow {b}|=1$, $|\\overrightarrow {c}|=4$, 则$\\overrightarrow {a} \\cdot \\overrightarrow {b}+\\overrightarrow {b} \\cdot \\overrightarrow {c}+\\overrightarrow {a} \\cdot \\overrightarrow {c}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428012,7 +430986,9 @@ "id": "016839", "content": "已知$|\\overrightarrow {a}|=2 \\sqrt{2}$, $|\\overrightarrow {b}|=\\dfrac{\\sqrt{2}}{2}$, $\\overrightarrow {a} \\cdot \\overrightarrow {b}=-\\sqrt{2}$, 则向量$\\overrightarrow {a}$、$\\overrightarrow {b}$所夹的角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428032,7 +431008,9 @@ "id": "016840", "content": "已知空间三点$A$、$B$、$C$坐标分别为$(0,0,2)$、$(2,2,0)$、$(-2,-4,-2)$, 点$P$在$xOy$平面上且$PA \\perp AB$, $PA \\perp AC$, 则$P$点坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428052,7 +431030,9 @@ "id": "016841", "content": "已知平行六面体$ABCD-A'B'C'D'$, 化简下列向量表达式, 标出化简结果的向量:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{3}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A'$} coordinate (A');\n\\draw (B) ++ (0,\\n,0) node [right] {$B'$} coordinate (B');\n\\draw (C) ++ (0,\\n,0) node [above right] {$C'$} coordinate (C');\n\\draw (D) ++ (0,\\n,0) node [above left] {$D'$} coordinate (D');\n\\draw (A') -- (B') -- (C') -- (D') -- cycle;\n\\draw (A) -- (A') (B) -- (B') (C) -- (C');\n\\draw [dashed] (D) -- (D');\n\\end{tikzpicture}\n\\end{center}\n(1) $\\overrightarrow{AB}+\\overrightarrow{BC}$;\\\\\n(2) $\\overrightarrow{AB}+\\overrightarrow{AD}+\\overrightarrow{AA'}$;\\\\\n(3) $\\overrightarrow{AB}+\\overrightarrow{AD}+\\dfrac{1}{2} \\overrightarrow{CC'}$;\\\\\n(4) $\\dfrac{1}{3}(\\overrightarrow{AB}+\\overrightarrow{AD}+\\overrightarrow{AA'})$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428072,7 +431052,9 @@ "id": "016842", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别是$BB_1$、$DC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(C)!0.5!(D)$) node [below] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(B_1)$) node [right] {$E$} coordinate (E);\n\\draw [dashed] (D)--(E)(D_1)--(F);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$AE$与$D_1F$所成的角;\\\\\n(2) 证明$AE \\perp$平面$A_1D_1F$.\\\\\n(3) 取$DE$的中点$G, CC_1$的中点$H$, 用向量求证:\\\\\n(I) $A$、$G$、$H$三点共线;\\\\\n(II) $\\overrightarrow{DE}$与$\\overrightarrow{DB}, \\overrightarrow{DD_1}$共面.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428092,7 +431074,9 @@ "id": "016843", "content": "如图, 已知长方体$ABCO-A_1B_1C_1O_1$中, $OA=2$, $AB=3$, $CC_1=2$, $E$是$BC$中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{3}\n\\def\\m{2}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$O$} coordinate (O);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (O) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (O) ++ (0,\\n,0) node [above left] {$O_1$} coordinate (O_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (O_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (O) -- (O_1);\n\\draw [dashed] (A)--(O_1);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw (B_1)--(E);\n\\draw ($(A)!{4/13}!(C)$) node [below right] {$D$} coordinate (D);\n\\draw [dashed] (O_1)--(D)(A)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求异面直线$AO_1, B_1E$所成的角;\\\\\n(2) $O_1D \\perp AC$, 垂足为$D$, 求向量$\\overrightarrow{O_1D}$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428112,7 +431096,9 @@ "id": "016844", "content": "已知空间三点$A(-2,0,2)$、$B(-1,1,2)$、$C(-3,0,4)$. 设$\\overrightarrow {a}=\\overrightarrow{AB}$, $\\overrightarrow {b}=\\overrightarrow{AC}$.\\\\\n(1) 求$\\overrightarrow {a}$和$\\overrightarrow {b}$的夹角;\\\\\n(2) 若向量$k \\overrightarrow {a}+\\overrightarrow {b}$与$k \\overrightarrow {a}-2 \\overrightarrow {b}$互相垂直, 求$k$的值;\\\\\n(3) 求$\\overrightarrow {a}$在$\\overrightarrow {b}$上的投影.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428132,7 +431118,9 @@ "id": "016845", "content": "已知向量$\\overrightarrow {a}$和$\\overrightarrow {b}$的夹角为$120{^ \\circ}$, 且$|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=5$, 则$(2 \\overrightarrow {a}-\\overrightarrow {b}) \\cdot \\overrightarrow {a}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428152,7 +431140,9 @@ "id": "016846", "content": "已知空间四边形$OABC$, 其对角线为$OB$、$AC$, $M$、$N$分别是对边$OA$、$BC$的中点, 点$G$在线段$MN$上, 且$MG=2GN$, 用向量$\\overrightarrow{OA}, \\overrightarrow{OB}, \\overrightarrow{OC}$表示向量$\\overrightarrow{OG}$: $\\overrightarrow{OG}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428172,7 +431162,9 @@ "id": "016847", "content": "已知$\\overrightarrow {a}=(8,-1,4)$, $\\overrightarrow {b}=(2,2,1)$, 则以$\\overrightarrow {a}, \\overrightarrow {b}$为邻边的平行四边形的面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428192,7 +431184,9 @@ "id": "016848", "content": "已知向量$\\overrightarrow {a} \\perp \\overrightarrow {b}$, 向量$\\overrightarrow {c}$与$\\overrightarrow {a}$、$\\overrightarrow {b}$的夹角都是$60{^ \\circ}$, 且$|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=2$, $|\\overrightarrow {c}|=3$, 则$(3 \\overrightarrow {a}-2 \\overrightarrow {b}) \\cdot$$(\\overrightarrow {b}-3 \\overrightarrow {c})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428212,7 +431206,9 @@ "id": "016849", "content": "已知: $\\overrightarrow {a}=3 \\overrightarrow {m}-2 \\overrightarrow {n}-4 \\overrightarrow {p} \\neq 0$, $\\overrightarrow {b}=(x+1) \\overrightarrow {m}+8 \\overrightarrow {n}+2 y \\overrightarrow {p}$, 且$\\overrightarrow {m}$、$\\overrightarrow {n}$、$\\overrightarrow {p}$不共面. 若$\\overrightarrow {a}\\parallel \\overrightarrow {b}$, 则$x$、$y$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428232,7 +431228,9 @@ "id": "016850", "content": "设空间两个不同的单位向量$\\overrightarrow {a}=(x_1, y_1, 0)$, $\\overrightarrow {b}=(x_2, y_2, 0)$与向量$\\overrightarrow {c}=(1,1,1)$的夹角都等于$\\dfrac{\\pi}{4}$, 则$\\overrightarrow {a}$与$\\overrightarrow {b}$夹角的大小为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428252,7 +431250,9 @@ "id": "016851", "content": "在平行六面体$ABCD-A_1B_1C_1D_1$中, $M$为$AC$与$BD$的交点, 若$\\overrightarrow{A_1B_1}=\\overrightarrow {a}$, $\\overrightarrow{A_1D_1}=\\overrightarrow {b}$, $\\overrightarrow{A_1A}=\\overrightarrow {c}$, 则下列向量中与$\\overrightarrow{B_1M}$相等的向量是\\bracket{20}.\n\\fourch{$-\\dfrac{1}{2} \\overrightarrow {a}+\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}{$\\dfrac{1}{2} \\overrightarrow {a}+\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}{$\\dfrac{1}{2} \\overrightarrow {a}-\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}{$-\\dfrac{1}{2} \\overrightarrow {a}-\\dfrac{1}{2} \\overrightarrow {b}+\\overrightarrow {c}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -428272,7 +431272,9 @@ "id": "016852", "content": "已知两个非零向量$\\overrightarrow {a}=(a_1, a_2, a_3)$, $\\overrightarrow {b}=(b_1, b_2, b_3)$, 它们平行的一个充要条件是\\bracket{20}.\n\\twoch{$\\overrightarrow {a}: |\\overrightarrow {a}|=\\overrightarrow {b}: |\\overrightarrow {b}|$}{$a_1 b_1=a_2 b_2=a_3 b_3$}{$a_1 b_1+a_2 b_2+a_3 b_3=0$}{存在非零实数$k$, 使$\\overrightarrow {a}=k \\overrightarrow {b}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -428292,7 +431294,9 @@ "id": "016853", "content": "已知向量$\\overrightarrow {a}=(2,4, x)$, $\\overrightarrow {b}=(2, y, 2)$, 若$|\\overrightarrow {a}|=6$, $\\overrightarrow {a} \\perp \\overrightarrow {b}$, 则$x+y$的值是\\bracket{20}.\n\\fourch{$-3$或$1$}{$3$或$-1$}{$-3$}{$1$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -428312,7 +431316,9 @@ "id": "016854", "content": "长方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别为$AB$、$B_1C_1$中点, 若$AB=BC=2$, $AA_1=4$, 试用向量法求:\\\\\n(1) $\\overrightarrow{A_1E}$与$\\overrightarrow{CF}$的夹角的大小;\\\\\n(2) 直线$A_1E$与$FC$所夹角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428332,7 +431338,9 @@ "id": "016855", "content": "已知$A(0,2,3)$、$B(-2,1,6)$、$C(1,-1,5)$为空间三点.\\\\\n(1) 求以$\\overrightarrow{AB}, \\overrightarrow{AC}$为边的平行四边形的面积;\\\\\n(2) 若$|\\overrightarrow {a}|=\\sqrt{3}$, 且$\\overrightarrow {a}$分别与$\\overrightarrow{AB}$、$\\overrightarrow{AC}$垂直, 求向量$\\overrightarrow {a}$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428352,7 +431360,9 @@ "id": "016856", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别为$BB_1$、$DC$的中点, 求证: $D_1F \\perp$平面$ADE$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428372,7 +431382,9 @@ "id": "016857", "content": "在棱长为$2$的正方体$ABCD-A_1B_1C_1D_1$中, $O$是底面$ABCD$的中心, $E$、$F$分别是$CC_1$、$AD$的中点, 那么异面直线$OE$和$FD_1$所成的角的余弦值等于\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{10}}{5}$}{$\\dfrac{\\sqrt{15}}{5}$}{$\\dfrac{4}{5}$}{$\\dfrac{2}{3}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -428392,7 +431404,9 @@ "id": "016858", "content": "在正三棱柱$ABC-A_1B_1C_1$中, 已知$AB=1$, $D$在棱$BB_1$上, 且$BD=1$, 若$AD$与平面$AA_1C_1C$所成的角为$\\alpha$, 则$\\alpha=$\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}{3}$}{$\\dfrac{\\pi}{4}$}{$\\arcsin \\dfrac{\\sqrt{10}}{4}$}{$\\arcsin \\dfrac{\\sqrt{6}}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -428412,7 +431426,9 @@ "id": "016859", "content": "若$PA \\perp$平面$ABC$, $\\angle ACB=90{^\\circ}$且$PA=AC=BC=a$, 则异面直线$PB$与$AC$所成角的正切值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428432,7 +431448,9 @@ "id": "016860", "content": "已知底面为矩形的四棱锥$P-ABCD$的每个顶点都在球$O$的球面上, $PA \\perp AD$, $PA=AB$, $PB=\\sqrt{2} AB$, 且$BC=2 \\sqrt{2}$. 若球$O$的体积为$\\dfrac{32 \\pi}{3}$, 则$AB=$\\blank{50}, 棱$PB$的中点到平面$PCD$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428452,7 +431470,9 @@ "id": "016861", "content": "如图, 四棱锥$P-ABCD$中, 底面$ABCD$为矩形, $PD \\perp$底面$ABCD$, $AD=PD$, $E$、$F$分别为$CD$、$PB$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [right] {$D$} coordinate (D);\n\\draw (0,0,1) node [below] {$A$} coordinate (A);\n\\draw (0,1,0) node [above] {$P$} coordinate (P);\n\\draw ({-sqrt(2)},0,0) node [left] {$C$} coordinate (C);\n\\draw ({-sqrt(2)},0,1) node [left] {$B$} coordinate (B);\n\\draw (C)--(B)--(A)--(D)--(P)--cycle(B)--(P)(A)--(P);\n\\draw [dashed] (C)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $EF \\perp$平面$PAB$;\\\\\n(2) 设$AB=\\sqrt{2} BC$, 求$AC$与平面$AEF$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428472,7 +431492,9 @@ "id": "016862", "content": "如图, 在四棱锥$O-ABCD$中, 底面$ABCD$是边长为$1$的菱形, $\\angle ABC=\\dfrac{\\pi}{4}$, $OA \\perp$底面$ABCD$, $OA=2$, $M$为$OA$的中点, $N$为$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0,0) node [above right] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$D$} coordinate (D);\n\\draw ({-sqrt(2)/2},0,{sqrt(2)/2}) node [below] {$B$} coordinate (B);\n\\draw (B) ++ (2,0,0) node [below] {$C$} coordinate (C);\n\\draw (0,4,0) node [above] {$O$} coordinate (O);\n\\draw (O)--(B)--(C)--(D)--cycle(O)--(C);\n\\draw [dashed] (B)--(A)--(D)(A)--(O);\n\\draw ($(O)!0.5!(A)$) node [left] {$M$} coordinate (M);\n\\draw ($(B)!0.5!(C)$) node [below] {$N$} coordinate (N);\n\\draw [dashed] (M)--(N)(M)--(D);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: 直线$MN\\parallel$平面$OCD$;\\\\\n(2) 求异面直线$AB$与$MD$所成角的大小;\\\\\n(3) 求点$B$到平面$OCD$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428492,7 +431514,9 @@ "id": "016863", "content": "如图, 在正四棱锥$P-ABCD$中, $PA=AB=2 \\sqrt{2}$, $E$、$F$分别为$PB$、$PD$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-sqrt(2)},0,{sqrt(2)}) node [left] {$A$} coordinate (A);\n\\draw (A) ++ ({2*sqrt(2)},0,0) node [right] {$B$} coordinate (B);\n\\draw (B) ++ (0,0,{-2*sqrt(2)}) node [right] {$C$} coordinate (C);\n\\draw (C) ++ ({-2*sqrt(2)},0,0) node [below] {$D$} coordinate (D);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw ($(P)!0.5!(B)$) node [right] {$E$} coordinate (E);\n\\draw ($(P)!0.5!(D)$) node [below] {$F$} coordinate (F);\n\\path [name path = PC] (P)--(C);\n\\path [name path = AG] (A)--($(A)!1.5!($(E)!0.5!(F)$)$);\n\\path [name intersections = {of = PC and AG, by = M}];\n\\draw (A)--(B)--(C)--(P)--cycle (P)--(B) (A)--(E)--(M) node [above right] {$M$};\n\\draw [dashed] (A)--(F) (P)--(D) (A)--(D)--(C) (F)--(M);\n\\end{tikzpicture}\n\\end{center}\n(1) 求正四棱锥$P-ABCD$的表面积;\\\\\n(2) 若平面$AEF$与棱$PC$交于点$M$, 求平面$AEMF$与平面$ABCD$所成锐二面角的大小(用反三角函数值表示).", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428512,7 +431536,9 @@ "id": "016864", "content": "如图, 四棱锥$P-ABCD$中, 底面是矩形且$AD=2$, $AB=PA=\\sqrt{2}$, $PA \\perp$底面$ABCD$, $E$是$AD$的中点, $F$在$PC$上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw ({sqrt(2)},0,0) node [right] {$B$} coordinate (B);\n\\draw (0,0,2) node [below] {$D$} coordinate (D);\n\\draw ({sqrt(2)},0,2) node [below] {$C$} coordinate (C);\n\\draw (0,{sqrt(2)},0) node [above] {$P$} coordinate (P);\n\\draw (P)--(D)--(C)--(B)--cycle(P)--(C);\n\\draw [dashed] (D)--(A)--(B)(A)--(P);\n\\filldraw ($(A)!0.5!(D)$) node [below] {$E$} coordinate (E) circle (0.03);\n\\filldraw ($(P)!0.4!(C)$) node [right] {$F$} coordinate (F) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$F$在何处时, $EF \\perp$平面$PBC$;\\\\\n(2) 在条件(1)下, $EF$是否为$PC$与$AD$的公垂线段? 若是, 求出直线$PC$与$AD$的距离; 若不是, 说明理由;\\\\\n(3) 在条件(1)下, 求直线$BD$与平面$BEF$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428532,7 +431558,9 @@ "id": "016865", "content": "若一条直线与一个正四棱柱各个面所成的角都为$\\alpha$, 则$\\cos \\alpha=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428552,7 +431580,9 @@ "id": "016866", "content": "正四面体$ABCD$的棱长为$1$, 棱$AB\\parallel$平面$\\alpha$, 则正四面体上的所有点在平面$\\alpha$内的射影构成的图形面积的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428572,7 +431602,9 @@ "id": "016867", "content": "已知正三棱柱$ABC-A_1B_1C_1$的所有棱长都相等, $D$是$A_1C_1$的中点, 则直线$AD$与平面$B_1DC$所成角的正弦值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428592,7 +431624,9 @@ "id": "016868", "content": "等边三角形$ABC$与正方形$ABDE$有一公共边$AB$, 二面角$C-AB-D$的余弦值为$\\dfrac{\\sqrt{3}}{3}$, $M$、$N$分别是$AC$、$BC$的中点, 则$EM, AN$所成角的余弦值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428612,7 +431646,9 @@ "id": "016869", "content": "在正方体$ABCD-A_1B_1C_1D_1$的侧面$AB_1$内有一动点$P$到棱$A_1B_1$与棱$BC$的距离相等, 则动点$P$所在曲线的形状为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428632,7 +431668,9 @@ "id": "016870", "content": "在三棱维$O-ABC$中, 三条棱$OA$、$OB$、$OC$两两互相垂直, 且$OA=OB=OC$, $M$是$AB$边的中点, 则$OM$与平面$ABC$所成角的大小是\\blank{50}. (用反三角函数表示)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428652,7 +431690,9 @@ "id": "016871", "content": "如图, 长方体$ABCD-A_1B_1C_1D_1$中, $AA_1=AB=2$, $AD=1$, $E$、$F$、$G$分别是$DD_1$、$AB$、$CC_1$的中点, 则异面直线$A_1E$与$GF$所成的角是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{1}\n\\def\\n{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.5!(B)$) node [below] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(C_1)$) node [right] {$G$} coordinate (G);\n\\draw ($(D)!0.5!(D_1)$) node [right] {$E$} coordinate (E);\n\\draw [dashed] (A_1)--(E)(F)--(G);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\arccos \\dfrac{\\sqrt{15}}{5}$}{$\\dfrac{\\pi}{4}$}{$\\arccos \\dfrac{\\sqrt{10}}{5}$}{$\\dfrac{\\pi}{2}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -428672,7 +431712,9 @@ "id": "016872", "content": "已知二面角$\\alpha-l-\\beta$的大小为$60{^ \\circ}$, $m$、$n$为异面直线, 且$m \\perp \\alpha$, $n \\perp \\beta$, 则$m$、$n$所成的角为\\bracket{20}.\n\\fourch{$30{^ \\circ}$}{$60{^ \\circ}$}{$90{^ \\circ}$}{$120{^ \\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -428692,7 +431734,9 @@ "id": "016873", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=2, AA_1=1$, 则$BC_1$与平面$BB_1D_1D$所成角的正弦值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt{6}}{3}$}{$\\dfrac{2 \\sqrt{6}}{5}$}{$\\dfrac{\\sqrt{15}}{5}$}{$\\dfrac{\\sqrt{10}}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -428712,7 +431756,9 @@ "id": "016874", "content": "如图, 在直棱柱$ABC-A_1B_1C_1$中, 底面是等腰直角三角形, $\\angle ACB=90{^ \\circ}$, 侧棱$AA_1=2$, $D$、$E$分别是$CC_1$与$A_1B$的中点, 点$E$在平面$ABD$上的射影是$\\triangle ABD$的重心$G$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$A$} coordinate (A);\n\\draw (0,0,2) node [below] {$B$} coordinate (B);\n\\draw (A) ++ (0,2,0) node [above] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [above] {$C_1$} coordinate (C_1);\n\\draw (A_1)--(B_1)--(C_1)--cycle(B_1)--(B)--(A)--(A_1);\n\\draw [dashed] (B)--(C)--(A)(C)--(C_1);\n\\draw ($(C)!0.5!(C_1)$) node [left] {$D$} coordinate (D);\n\\draw ($(A_1)!0.5!(B)$) node [above] {$E$} coordinate (E);\n\\draw (A_1)--(B)(E)--(A);\n\\draw [dashed] (A)--(D)--(B)(D)--(E);\n\\draw ($1/3*(A)+1/3*(B)+1/3*(D)$) node [below] {$G$} coordinate (G);\n\\draw [dashed] (E)--(G);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$A_1B$与平面$ABD$所成角的大小(结果按图所示用反三角函数值表示);\\\\\n(2) 求点$A_1$到平面$AED$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428732,7 +431778,9 @@ "id": "016875", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $AD=AA_1=1$, $AB=2$, 点$E$在棱$AB$上移动.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2]\n\\def\\l{2}\n\\def\\m{1}\n\\def\\n{1}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.6!(B)$) node [below] {$E$} coordinate (E);\n\\draw [dashed] (A_1)--(D)(D_1)--(A)(A)--(C)(E)--(C)(D_1)--(C)(D_1)--(E);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $D_1E \\perp A_1D$;\\\\\n(2) $AE$等于何值时, 二面角$D_1-EC-D$的大小为$\\dfrac{\\pi}{4}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428752,7 +431800,9 @@ "id": "016876", "content": "如图, 三棱柱$ABC-A_1B_1C_1$中, $AA_1 \\perp$底面$ABC$, $AB=AC$, $D$是$BC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{-sqrt(3)}) node [right] {$A$} coordinate (A);\n\\draw (A) ++ (0,2,0) node [above] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,2,0) node [left] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,2,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (B)--(C)--(C_1)--(B_1)--cycle(B_1)--(A_1)--(C_1);\n\\draw ($(B)!0.5!(C)$) node [below] {$D$} coordinate (D);\n\\draw [dashed] (A_1)--(D)(B_1)--(A)(B)--(A)--(C)(A)--(A_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $BC \\perp$平面$A_1AD$;\\\\\n(2) 若三棱柱$ABC-A_1B_1C_1$为正三棱柱, 则二面角$A-BB_1-C$的大小为多少;\\\\\n(3) \\textcircled{1} 若$\\angle BAC=90{^ \\circ}, BC=4$; \\textcircled{2} 三棱柱$ABC-A_1B_1C_1$的体积是$8 \\sqrt{3}$; \\textcircled{3} $AA_1=4$.\n从以上三个条件中选出两个条件求异面直线$A_1D$和$AB_1$所成角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -428772,7 +431822,9 @@ "id": "016877", "content": "正三棱锥$A-BCD$的侧棱与底面边长都为$a$, $P$为$AD$中点, $PQ \\perp BC$, 垂足为$Q$, 则$AD$与$BC$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428792,7 +431844,9 @@ "id": "016878", "content": "将菱形$ABCD$沿对角线$BD$折成空间四边形$ABC'D$, 使此空间四边形的对角线$AC'$的长等于菱形$ABCD$的对角线$AC$长的一半, 则二面角$C'-BD-A$的度数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428812,7 +431866,9 @@ "id": "016879", "content": "一个棱长都相等的正三棱锥的四个顶点恰好在一个正方体的顶点上, 则此正三棱锥的表面积与正方体的表面积之比为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428832,7 +431888,9 @@ "id": "016880", "content": "圆锥的高为$1$, 底面半径为$\\sqrt{3}$, 过顶点的截面面积最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428852,7 +431910,9 @@ "id": "016881", "content": "正四棱锥的侧棱长与底面边长都是$1$, 则侧棱与底面所成的角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428872,7 +431932,9 @@ "id": "016882", "content": "表面积相等的球和正方体的体积比为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428892,7 +431954,9 @@ "id": "016883", "content": "平行六面体各棱长均为$4$, 在由顶点$P$出发的三条棱上, 取$PA=1$, $PB=2$, $PC=3$, 则棱锥$P-ABC$的体积是该平行六面体体积的\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428912,7 +431976,9 @@ "id": "016884", "content": "已知球的两个截面面积分别为$5 \\pi$, $8 \\pi$, 而球心到这个两截面的距离之差为$1$, 那么这个球的半径为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428932,7 +431998,9 @@ "id": "016885", "content": "已知三个球的半径$R_1$、$R_2$、$R_3$满足$R_1+2R_2=3R_3$, 则它们的表面积$S_1$、$S_2$、$S_3$满足的等量关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428952,7 +432020,9 @@ "id": "016886", "content": "若等腰直角三角形的直角边长为$2$, 则以一直角边所在的直线为轴旋转一周所成的几何体体积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428972,7 +432042,9 @@ "id": "016887", "content": "已知球$O$的半径为$2$, 圆$O_1$是一小圆, $O_1O=\\sqrt{2}$, $A$、$B$是圆$O_1$上两点, 若$\\angle AO_1B=\\dfrac{\\pi}{2}$, 以$O$为圆心, $2$为半径作圆, 使点$A$、$B$在圆上, 则$A$、$B$两点间的劣弧长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -428992,7 +432064,9 @@ "id": "016888", "content": "已知三棱柱$ABC-A_1B_1C_1$的侧棱与底面边长都相等, $A_1$在底面$ABC$内的射影为$\\triangle ABC$的中心, 则$AB_1$与底面$ABC$所成角的正弦值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429012,7 +432086,9 @@ "id": "016889", "content": "若一直线上有两点到一个平面的距离都等于$1$, 则该直线与这个平面的位置关系是\\bracket{20}.\n\\twoch{直线在平面内}{直线平行平面}{直线与平面相交}{直线与平面相交或平行}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -429032,7 +432108,9 @@ "id": "016890", "content": "一个正三棱锥的侧面积为底面积的$2$倍, 底面边长为$6$, 则体积$V=$\\bracket{20}.\n\\fourch{$9 \\sqrt{5}$}{$18 \\sqrt{3}$}{$9 \\sqrt{3}$}{$3 \\sqrt{3}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -429052,7 +432130,9 @@ "id": "016891", "content": "圆锥高为$h$, 母线与底面成$60{^\\circ}$的角, 则圆锥展开图扇形的圆心角$\\theta$为\\bracket{20}.\n\\fourch{$60{^ \\circ}$}{$90{^ \\circ}$}{$120{^ \\circ}$}{$180{^ \\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -429072,7 +432152,9 @@ "id": "016892", "content": "已知正方体、等边圆柱, 球的体积相等, 分别用$S_{\\text {正}}$、$S_{\\text {柱}}$、$S_{\\text{球}}$表示其全面积, 则三个面积的数值按大小顺序排列为\\bracket{20}.\n\\fourch{$S_{\\text {正}}>S_{\\text {球}}>S_{\\text {柱}}$}{$S_{\\text {球}}>S_{\\text {正}}>S_{\\text {柱}}$}{$S_{\\text {球}}>S_{\\text {柱}}>S_{\\text {正}}$}{$S_{\\text {正}}>S_{\\text {柱}}>S_{\\text {球}}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -429092,7 +432174,9 @@ "id": "016893", "content": "在棱长为$a$的正方体$ABCD-A_1B_1C_1D_1, E$、$F$分别为$BC$与$A_1D_1$的中点.\\\\\n(1) 求直线$A_1C$与$DE$所成的角;\\\\\n(2) 求直线$AD$与平面$B_1EDF$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429112,7 +432196,9 @@ "id": "016894", "content": "在底半径为$2$母线长为$4$的圆锥中内接一个高为$\\sqrt{3}$的圆柱, 求圆柱的表面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw (-2,0) coordinate (A) (2,0) coordinate (B) (0,{2*sqrt(3)}) coordinate (P);\n\\draw (-1,0) coordinate (S) (1,0) coordinate (T);\n\\draw (A) arc (180:360:2 and 0.5);\n\\draw [dashed] (A) arc (180:0:2 and 0.5);\n\\draw (A)--(P)--(B);\n\\draw [dashed] (S) arc (180:-180:1 and 0.25);\n\\draw [dashed] (S) --++ (0,{sqrt(3)}) arc (180:-180:1 and 0.25);\n\\draw [dashed] (T) --++ (0,{sqrt(3)});\n\\draw [dashed] (B)--(0,0)--(P);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429132,7 +432218,9 @@ "id": "016895", "content": "四棱锥$P$-$ABCD$中, 底面$ABCD$正方形, 边长为$4$, $E$为$AB$中点, $PE \\perp$平面$ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$E$} coordinate (E);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,0,-1) node [below] {$A$} coordinate (A);\n\\draw (2,0,-1) node [right] {$D$} coordinate (D);\n\\draw (2,0,1) node [below] {$C$} coordinate (C);\n\\draw (2,0,0) node [right] {$F$} coordinate (F);\n\\draw (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (P)--(B)--(C)--(D)--cycle(P)--(F)(P)--(C);\n\\draw [dashed] (B)--(A)--(D)(P)--(E)(P)--(A);\n\\end{tikzpicture}\n\\end{center}\n(1) 若$\\triangle PAB$为等边三角形, 求四棱锥$P-ABCD$的体积;\\\\\n(2) 若$CD$中点为$F, PF$与平面$ABCD$所成角为$45{^ \\circ}$, 求异面直线$PC$与$AD$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429152,7 +432240,9 @@ "id": "016896", "content": "如图, 在平面四边形$ABCD$中, $AB=BC=CD=a$, $\\angle B=90{^ \\circ}$, $\\angle C=135{^ \\circ}$, 沿对角线$AC$折成直二面角.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw ({2*sqrt(3)},0) node [below] {$D$} coordinate (D);\n\\draw ($(A)!{2/3}!(D)$) ++ (0,{2*sqrt(2)/sqrt(3)}) node [above] {$C$} coordinate (C);\n\\draw ($(C)!{sqrt(2)/2}!-45:(A)$) node [left] {$B$} coordinate (B);\n\\draw (A)--(C)(A)--(B)--(C)--(D)--cycle;\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw ({2*sqrt(3)},0) node [below] {$D$} coordinate (D);\n\\draw ($(A)!{2/3}!(D)$) ++ (0,0,{-2*sqrt(2)/sqrt(3)}) node [above] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(C)$) ++ (0,{sqrt(2)},0) node [above] {$B$} coordinate (B);\n\\draw (A)--(C)(A)--(B)--(C)--(D)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 求证: $AB \\perp$平面$BCD$;\\\\\n(2) 求平面$ABD$与平面$ACD$所成的角;\\\\\n(3) 求点$C$到平面$ABD$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429172,7 +432262,9 @@ "id": "016897", "content": "如图, 已知正三棱柱$ABC-A_1B_1C_1$各棱长都为$a$, $P$为线段$A_1B$上的动点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$B$} coordinate (B);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$A$} coordinate (A);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (B) ++ (0,\\h) node [left] {$B_1$} coordinate (B_1);\n\\draw (A) ++ (0,\\h) node [below right] {$A_1$} coordinate (A_1);\n\\draw (C) ++ (0,\\h) node [right] {$C_1$} coordinate (C_1);\n\\draw (B) -- (A) -- (C) (B) -- (B_1) (A) -- (A_1) (C) -- (C_1) (B_1) -- (A_1) -- (C_1) (B_1) -- (C_1);\n\\draw [dashed] (B) -- (C);\n\\draw ($(A_1)!0.4!(B)$) node [above] {$P$} coordinate (P);\n\\draw (A_1)--(B)(P)--(A);\n\\draw [dashed] (P)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 试确定$A_1P: PB$的值, 使得$PC \\perp AB$;\\\\\n(2) 若$A_1P: PB=2: 3$, 求二面角$P-AC-B$的大小;\\\\\n(3) 在 (2) 的条件下, 求点$C_1$到面$PAC$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429192,7 +432284,9 @@ "id": "016898", "content": "某班级要从$4$名男生, $2$名女生中选派$4$人参加某次社区服务, 如果要求至少有$1$名女生, 那么不同的选派方案种数为\\bracket{20}.\n\\fourch{$14$}{$24$}{$28$}{$48$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -429212,7 +432306,9 @@ "id": "016899", "content": "$12$名同学合影, 站成前排$4$人后排$8$人, 现摄影师要从后排$8$人中抽$2$人调整到前排, 若其他人的相对顺序不变, 则不同调整方法的总数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_8^2 \\mathrm{P}_3^2$}{$\\mathrm{C}_8^2 \\mathrm{P}_6^6$}{$\\mathrm{C}_8^2 \\mathrm{P}_6^2$}{$\\mathrm{C}_8^2 \\mathrm{P}_5^2$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -429232,7 +432328,9 @@ "id": "016900", "content": "书架上层放有$6$本不同的数学书, 下层放有$5$本不同的语文书.\\\\\n(1) 从中任取一本, 有\\blank{50}种不同的取法;\\\\\n(2) 从中任取数学书与语文书各一本, 有\\blank{50}种取法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429252,7 +432350,9 @@ "id": "016901", "content": "从甲、乙等$10$名同学中挑选$4$名参加某校公益活动, 要求甲、乙中至少有$1$人参加, 则不同的挑选方法共有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429272,7 +432372,9 @@ "id": "016902", "content": "某地奥运火炬接力传递路线共分$6$段, 传递活动分别由$6$名火炬手完成. 如果第一棒火炬手只能从甲、乙、人三人中产生, 最后一棒火炬手只能从甲、乙两人中产生, 则不同的传递方案共有\\blank{50}种. (用数字作答).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429292,7 +432394,9 @@ "id": "016903", "content": "书架上放有$3$本不同的数学书, $5$本不同的语文书, $6$本不同的英语书.\n(1) 若从这些书中任取一本, 有多少种不同的取法?\\\\\n(2) 若从这些书中, 取数学书、语文书、英语书各一本, 有多少种不同的取法?\\\\\n(3) 若从这些书中取不同的科目的书两本, 有多少种不同的取法?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429312,7 +432416,9 @@ "id": "016904", "content": "由数字$0,1,2,3,4$可以组成多少个三位整数(各位上的数字允许重复)?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429332,7 +432438,9 @@ "id": "016905", "content": "由数字$0,1,2,3,4$可以组成多少个三位偶数(各位上的数字允许重复)?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429352,7 +432460,9 @@ "id": "016906", "content": "由数字$0,1,2,3,4$可以组成多少个各位数字不重复的三位整数?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429372,7 +432482,9 @@ "id": "016907", "content": "由数字$0,1,2,3,4$可以组成多少个各位数字不重复的三位偶数?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429392,7 +432504,9 @@ "id": "016908", "content": "在一块并排的$10$垄田地中, 选择$2$垄分别种植$A$、$B$两种作物, 每种种植一垄, 为有利于作物生长, 要求$A$、$B$两种作物的间隔不少于$6$垄, 不同的选法共有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429412,7 +432526,9 @@ "id": "016909", "content": "$75600$有多少个正约数? 有多少个奇约数?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429432,7 +432548,9 @@ "id": "016910", "content": "某人某天需要运动总时长大于等于$60$分种, 现有五项运动可以选择, 如下表所示, 共有\\blank{50}种运动方式组合.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline A 运动 & B 运动 & C 运动 & D 运动 & E 运动 \\\\\n\\hline $7 \\sim 8$ 点 & $8 \\sim 9$ 点 & $9 \\sim 10$ 点 & $10 \\sim 11$ 点 & $11 \\sim 12$ 点 \\\\\n\\hline 30 分钟 & 20 分钟 & 40 分钟 & 30 分钟 & 30 分钟 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429452,7 +432570,9 @@ "id": "016911", "content": "用$1,2,3,4,5$可组成\\blank{50}个三位数(各位上的数字允许重复).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429472,7 +432592,9 @@ "id": "016912", "content": "电子计算机的输人纸带每排有$8$个穿孔位置, 每个穿孔位置可穿孔或不穿孔, 则每排最多可产生\\blank{50}种不同的信息.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429492,7 +432614,9 @@ "id": "016913", "content": "乘积$(a_1+a_2+a_3)(b_1+b_2+b_3+b_4)(c_1+c_2+c_3+c_4+c_5)$展开后共有\\blank{50}项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429512,7 +432636,9 @@ "id": "016914", "content": "$210$的正约数(包括$1$和$210$)有\\blank{50}个.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429532,7 +432658,9 @@ "id": "016915", "content": "从集合$\\{0,1,2,3,4,5,6,7,8,9\\}$中任取$3$个不同元素分别作为直线方程$A x+B y+C=0$中的$A$、$B$、$C$, 则经过坐标原点的不同直线有\\blank{50}条(用数值表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429552,7 +432680,9 @@ "id": "016916", "content": "某学校为落实``双减''政策, 在每天放学后开设拓展课程供学生自愿选择, 开学第一周的安排见下表. 小明同学要在这一周内选择编程、书法、足球三门课, 不同的选课方案共\\blank{50}种.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n周一 & 周二 & 周三 & 周四 & 周五 \\\\ \\hline\n\\makecell{演讲、绘画 \\\\舞蹈、足球}& \\makecell{编程、绘画\\\\舞蹈、足球} & \\makecell{编程、书法\\\\ 舞蹈、足球} & \\makecell{书法、演讲\\\\ 舞蹈、足球} & \\makecell{书法、演讲\\\\ 舞蹈、足球} \\\\ \\hline\n\\multicolumn{5}{|l|}{注: 每位同学每天最多选一门课, 每一门课一周内最多选一次} \\\\ \\hline \n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429572,7 +432702,9 @@ "id": "016917", "content": "四张不同的贺年片分别赠送给三位朋友, 每人至少一张, 则不同的赠法种数为\\bracket{20}.\n\\fourch{$36$}{$72$}{$18$}{$24$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -429592,7 +432724,9 @@ "id": "016918", "content": "用数字$1,2,3,4,5$组成的无重复数字的四位偶数的个数为\\bracket{20}.\n\\fourch{$8$}{$24$}{$48$}{$120$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -429612,7 +432746,9 @@ "id": "016919", "content": "从$6$双不同颜色的手套中任取$4$只, 其中恰好有一双同色的取法有\\bracket{20}.\n\\fourch{$240$}{$180$}{$120$}{$60$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -429632,7 +432768,9 @@ "id": "016920", "content": "用$0,1,2,3,4,5$这六个数字.\\\\\n(1) 可以组成多少个数字不重复的三位数?\\\\\n(2) 可以组成多少个数字允许重复的三位数?\\\\\n(3) 可以组成多少个数字不允许重复的三位数的奇数?\\\\\n(4) 可以组成多少个数字不重复的小于$1000$的自然数?\\\\\n(5) 可以组成多少个大于$3000$, 小于$5421$的数字不重复的四位数?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429652,7 +432790,9 @@ "id": "016921", "content": "有四名同学参加三项不同的比赛.\\\\\n(1) 每名同学必须参加一项竞赛, 有多少种不同的结果?\\\\\n(2) 每项竞赛只许一名学生参加, 有多少种不同的结果?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429672,7 +432812,9 @@ "id": "016922", "content": "设非空集合$Q \\subseteq M$, 当$Q$中所有元素和为偶数时 (集合为单元素时和为元素本身), 称$Q$是$M$的偶子集. 若集合$M=\\{1,2,3,4,5,6,7\\}$, 求其偶子集$Q$的个数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429692,7 +432834,9 @@ "id": "016923", "content": "$n(n-1)(n-2) \\cdots 4$等于\\bracket{20}.\n\\fourch{$\\mathrm{P}_n^4$}{$\\mathrm{P}_n^{n-4}$}{$n !-4$!}{$\\mathrm{P}_n^{n-3}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -429712,7 +432856,9 @@ "id": "016924", "content": "若$\\mathrm{C}_{20}^{2 x-7}=\\mathrm{C}_{20}^x$, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429732,7 +432878,9 @@ "id": "016925", "content": "$1 !+2 !+\\cdots+10!$ 的个位数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429752,7 +432900,9 @@ "id": "016926", "content": "己知$\\mathrm{C}_n^5=252$, 则自然数$n$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429772,7 +432922,9 @@ "id": "016927", "content": "从$3$名男生和$4$名女生中抽调$4$人去清理图书, 其中至少有$1$名男生的选派方法数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429792,7 +432944,9 @@ "id": "016928", "content": "计算:\\\\\n(1) $\\dfrac{\\mathrm{C}_{100}^2+\\mathrm{C}_{100}^{99}}{\\mathrm{P}_{100}^3}$;\\\\\n(2) $\\mathrm{C}_3^3+\\mathrm{C}_4^3+\\cdots+\\mathrm{C}_{10}^3$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429812,7 +432966,9 @@ "id": "016929", "content": "证明: $\\mathrm{P}_n^m+m \\mathrm{P}_n^{m-1}=\\mathrm{P}_{n+1}^m$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429832,7 +432988,9 @@ "id": "016930", "content": "化简: $\\dfrac{\\mathrm{C}_{n+1}^m}{\\mathrm{C}_n^m}-\\dfrac{\\mathrm{C}_n^{n-m+1}}{\\mathrm{C}_n^{n-m}}$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429852,7 +433010,9 @@ "id": "016931", "content": "解下列方程或不等式:\\\\\n(1) $\\mathrm{C}_{x+2}^{x-2}+\\mathrm{C}_{x+2}^{x-3}=\\dfrac{1}{10} \\mathrm{P}_{x+3}^3$;\\\\\n(2) $\\mathrm{P}_9^x>6 \\cdot \\mathrm{P}_9^{x-2}$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429872,7 +433032,9 @@ "id": "016932", "content": "$8$人坐一排椅子, 若共有$12$张椅子, 要使$4$个空位子连在一起, 共有多少种坐法?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429892,7 +433054,9 @@ "id": "016933", "content": "马路上有编号为$1,2,3,4,5,6,7,8,9,10$的$10$只路灯, 为节约用电又看清路面, 可以把其中的$3$只关掉, 但不能同时关掉相邻的两只, 也不关掉首末两只, 求满足条件的关灯法的种数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -429912,7 +433076,9 @@ "id": "016934", "content": "平面内有$10$个点, 以其中每$2$个点为端点的线段最多有\\blank{50}条.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429932,7 +433098,9 @@ "id": "016935", "content": "平面内有$10$个点, 以其中每$2$个点为起点和终点的向量最多有\\blank{50}个?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429952,7 +433120,9 @@ "id": "016936", "content": "计算: $\\mathrm{C}_4^0+\\mathrm{C}_4^1+\\mathrm{C}_4^2+\\mathrm{C}_4^3+\\mathrm{C}_4^4=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429972,7 +433142,9 @@ "id": "016937", "content": "化简: $\\mathrm{C}_n^{m-1}+\\mathrm{C}_{n-1}^m+\\mathrm{C}_{n-1}^{m-1}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -429992,7 +433164,9 @@ "id": "016938", "content": "若$x \\in \\mathbf{N}$, $x \\ge 1$, 则$\\mathrm{C}_{16}^{x^2-x}=\\mathrm{C}_{16}^{5 x-5}$的解集是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430012,7 +433186,9 @@ "id": "016939", "content": "圆周上有$2 n$个等分点($n>1$), 以其中三个点为顶点的直角三角形的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430032,7 +433208,9 @@ "id": "016940", "content": "五个工程队承建某项工程的五个不同的子项目, 每个工程队承建$1$项, 其中甲工程队不能承建$1$号子项目, 则不同的承建方案共有\\bracket{20}.\n\\fourch{$\\mathrm{C}_4^1 \\mathrm{C}_4^4$种}{$\\mathrm{C}_4^1 \\mathrm{P}_4^4$种}{$\\mathrm{C}_4^4$种}{$\\mathrm{P}_4^4$种}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -430052,7 +433230,9 @@ "id": "016941", "content": "将$4$名教师分配到$3$所中学任教, 每所中学至少$1$名, 则不同的分配方案共有\\bracket{20}.\n\\fourch{$12$种}{$24$种}{$36$种}{$48$种}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -430072,7 +433252,9 @@ "id": "016942", "content": "一生产过程有$4$道工序, 每道工序需要安排一人照看. 现从甲、乙、丙等$6$名工人中安排$4$人分别照看一道工序, 第一道工序只能从甲、乙两工人中安排$1$人, 第四道工序只能从甲、丙两工人中安排$1$人, 则不同的安排方案共有\\bracket{20}.\n\\fourch{$24$种}{$36$种}{$48$种}{$72$种}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -430092,7 +433274,9 @@ "id": "016943", "content": "(1) 四个不同的小球放入四个不同的盒中, 一共有多少种不同的放法?\\\\\n(2) 四个不同的小球放入四个不同的盒中且恰有一个空盒的放法有多少种?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430112,7 +433296,9 @@ "id": "016944", "content": "$100$件产品中有合格品$90$件, 次品$10$件, 现从中抽取$4$件检查.\\\\\n(1) 都不是次品的取法有多少种?\\\\\n(2) 至少有$1$件次品的取法有多少种?\\\\\n(3) 不都是次品的取法有多少种?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430132,7 +433318,9 @@ "id": "016945", "content": "从$10$个不同的文艺节目中选$6$个编成一个节目单, 如果某女演员的独唱节目一定不能排在第二个节目的位置上, 则共有多少种不同的排法?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430152,7 +433340,9 @@ "id": "016946", "content": "$5$件不同礼品分送给$4$人, 每人至少一件; 而且礼品全部送出, 那么送出礼品的方法数是 \\bracket{20}.\n\\fourch{$960$}{$480$}{$240$}{$120$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -430172,7 +433362,9 @@ "id": "016947", "content": "$4$个小组, 分别从$3$个风景点中选一处进行观光旅游, 不同的选择方案的种数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_4^3$}{$\\mathrm{P}_4^3$}{$3^4$}{$4^3$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -430192,7 +433384,9 @@ "id": "016948", "content": "书架上竖排着六本书, 现将新购的$3$本书上架, 要求不调乱书架上原有的书, 那么不同的上架方式共有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430212,7 +433406,9 @@ "id": "016949", "content": "小李打算从$10$个朋友中邀请$4$人去旅游, 这$10$个朋友中, 有一对双胞胎, 对这两个朋友, 要么都邀请, 要么都不邀请, 则不同的邀请方案有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430232,7 +433428,9 @@ "id": "016950", "content": "$4$人分住两个房间, 每个房间至少住进$1$人, 则不同的安排方法数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430252,7 +433450,9 @@ "id": "016951", "content": "$7$人排成一行, 分别求符合下列要求的不同排法的种数.\\\\\n(1) 甲排中间;\\\\\n(2) 甲不排两端;\\\\\n(3) 甲、乙相邻;\\\\\n(4) 甲在乙的左边 (不一定相邻);\\\\\n(5) 甲、乙、丙连排;\\\\\n(6) 甲、乙、丙两两不相邻.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430272,7 +433472,9 @@ "id": "016952", "content": "在某电影上映当天, 一对夫妇带着他们的两个小孩一起去观看该影片, 订购的$4$张电影票恰好在同一排且连在一起. 为安全起见, 影院要求每个小孩子要有家长相邻陪坐, 则不同的坐法种数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430292,7 +433494,9 @@ "id": "016953", "content": "用$1$、$2$、$3$、$4,5,6,7,8$组成没有重复数字的八位数, 要求$1$和$2$相邻, $3$与$4$相邻, $5$与$6$相邻, 而$7$与$8$不相邻, 这样的八位数共有多少个? (用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430312,7 +433516,9 @@ "id": "016954", "content": "将$5$名运动会志愿者分配到花样滑冰、短道速滑、冰球和冰壶$4$个项目进行培训, 每名志愿者只分配到$1$个项目, 每个项目至少分配$1$名志愿者, 则不同的分配方案共有\\bracket{20}.\n\\fourch{$60$种}{$120$种}{$240$种}{$480$种}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -430332,7 +433538,9 @@ "id": "016955", "content": "把同一排$6$张座位编号为$1,2,3,4,5,6$的电影票全部分给$4$个人, 每人至少分$1$张, 至多分$2$张, 且这两张票具有连续的编号, 那么不同的分法种数是\\bracket{20}.\n\\fourch{$168$}{$96$}{$72$}{$144$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -430352,7 +433560,9 @@ "id": "016956", "content": "运动会组织方拟将$4$名志愿者全部分配到$3$个不同的奥运场馆参加接待工作(每个场馆至少分配一名志愿者), 不同的分配方案有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430372,7 +433582,9 @@ "id": "016957", "content": "假设在$100$件产品中有$3$件是次品, 从中任意抽取$5$件, 分别求满足下列条件的方法数.\\\\\n(1) 没有次品;\\\\\n(2) 恰有两件是次品;\\\\\n(3) 至少有两件是次品.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430392,7 +433604,9 @@ "id": "016958", "content": "由$1,2,3,4$四个数字组成无重复数字, 且比$2134$大的四位数的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430412,7 +433626,9 @@ "id": "016959", "content": "已知集合$A=\\{-3,-2,-1,0,1,2,3\\}$, $a$、$b \\in A$, 则满足条件$|a|<|b|$的实数$a$、$b$的不同情况有\\blank{50}种.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430432,7 +433648,9 @@ "id": "016960", "content": "设$a_1, a_2, \\cdots, a_6$为$1,2,3,4,5,6$的一个排列, 则满足$|a_1-a_2|+$$|a_3-a_4|+|a_5-a_6|=3$的不同排列的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430452,7 +433670,9 @@ "id": "016961", "content": "电视台连续播放$6$个广告, 其中含$4$个不同的商业广告和$2$个不同的公益广告, 要求首尾必须播放公益广告, 则共有\\blank{50}种不同的播放方式(结果用数值表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430472,7 +433692,9 @@ "id": "016962", "content": "从$4$名男生和$3$名女生中选出$4$人参加某个座谈会, 若这$4$人中必须既有男生又有女生, 则不同的选法共有\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430492,7 +433714,9 @@ "id": "016963", "content": "在报名的$3$名男教师和$6$名女教师中, 选取$5$人参加义务献血, 要求男、女教师都有, 则不同的选取方式的种数为\\blank{50}. (结果用数值表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430512,7 +433736,9 @@ "id": "016964", "content": "用数字$1,2,3,4,5,6,7,8,9$组成没有重复数字, 且至多有一个数字是偶数的四位数, 这样的四位数一共有\\blank{50}个. (用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430532,7 +433758,9 @@ "id": "016965", "content": "$9$名同学排成前后两排, 前排$4$人, 后排$5$人, 若其中甲乙两人必须相邻, 则这样的排法种数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430552,7 +433780,9 @@ "id": "016966", "content": "某省于$2021$年启动了中学生科技创新后备人才培养计划 (简称中学生``英才计划''), 在数学、 物理、化学、生物、计算机等学科有特长的学生人选$2021$年该省中学生``英才计划'', 他们将在大学教授的指导下进行为期一年的培养. 现有$4$名数学特长生可从$3$位数学教授中任选一位作为导师, 每位数学教授至多带$2$名数学特长生, 则不同的培养方案有\\blank{50}种. (结果用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430572,7 +433802,9 @@ "id": "016967", "content": "将$9$个人 (含甲、乙)平均分成三组, 甲、乙分在同一组, 则不同分组方法的种数为\\bracket{20}.\n\\fourch{$70$}{$140$}{$280$}{$840$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -430592,7 +433824,9 @@ "id": "016968", "content": "从$6$人中选出$4$人分别到巴黎、伦敦、悉尼、莫斯科四个城市游览, 要求每个城市有一人游览, 每人只游览一个城市, 且这$6$人中甲、乙两人不去巴黎游览, 则不同的选择方案共有\\bracket{20}.\n\\fourch{$300$种}{$240$种}{$144$种}{$96$种}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -430612,7 +433846,9 @@ "id": "016969", "content": "某市拟从$4$个重点项目和$6$个一般项目中各选$2$个项目作为本年度启动的项目, 则重点项目$A$和一般项目$B$至少有一个被选中的不同选法种数是\\bracket{20}.\n\\fourch{$15$}{$45$}{$60$}{$75$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -430632,7 +433868,9 @@ "id": "016970", "content": "某居委会从$6$个人中挑选$4$人, 安排三天值班的值班任务, 每人值班一天, 其中第一天$1$人, 第二天$1$人, 第三天$2$人, 问共有多少种不同的安排方法.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430652,7 +433890,9 @@ "id": "016971", "content": "四名志愿者参加某博览会三天的活动, 若每人参加一天, 每天至少有一人参加, 其中志愿者甲第一天不能参加, 则不同的安排方法一共有几种(结果用数值表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430672,7 +433912,9 @@ "id": "016972", "content": "如图, 一个地区分为$5$个行政区域, 现给地图着色, 要求相邻地区不得使用同一颜色, 现有$4$种颜色可供选择, 则不同的着色方法共有几种. (以数字作答)\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.6]\n\\draw (0,0) circle (1) node {$1$};\n\\draw (45:1) -- (45:2) arc (45:135:2) (135:1) -- (135:2) arc (135:225:2) (225:1) -- (225:2) arc (225:315:2) (315:1) -- (315:2) arc (315:405:2); \n\\draw (90:1.5) node {$2$} (180:1.5) node {$3$} (270:1.5) node {$4$} (0:1.5) node {$5$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430692,7 +433934,9 @@ "id": "016973", "content": "$6$本不同的书, 按下列要求各有多少种不同的选法.\\\\\n(1) 分给甲、乙、丙三人, 每人两本;\\\\\n(2) 分为三份, 每份两本;\\\\\n(3) 分为三份, 一份一本, 一份两本, 一份三本;\\\\\n(4) 分给甲、乙、丙三人, 一人一本, 一人两本, 一人三本;\\\\\n(5) 分给甲、乙、丙三人, 每人至少一本.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430712,7 +433956,9 @@ "id": "016974", "content": "在$(2-x)^{10}$的展开式中, $-\\mathrm{C}_{10}^3 2^7 x^3$是第\\blank{50}项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430732,7 +433978,9 @@ "id": "016975", "content": "二项式$(-1+\\mathrm{i})^{20}$的展开式中, 第$18$项是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430752,7 +434000,9 @@ "id": "016976", "content": "$(x^3-\\dfrac{1}{x})^4$展开式中常数项为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430772,7 +434022,9 @@ "id": "016977", "content": "$\\mathrm{C}_{10}^0-\\dfrac{\\mathrm{C}_{10}^1}{2}+\\dfrac{\\mathrm{C}_{10}^2}{4}-\\dfrac{\\mathrm{C}_{10}^3}{8}+\\cdots+\\dfrac{\\mathrm{C}_{10}^{10}}{2^{10}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430792,7 +434044,9 @@ "id": "016978", "content": "若$(x+\\dfrac{2}{x})^n$的二项展开式的各项系数之和为$729$, 则该展开式中的常数项为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430812,7 +434066,9 @@ "id": "016979", "content": "在$(x-\\dfrac{1}{x})^{10}$的二项展开式中, 常数项等于\\blank{50}(结果用数值表示).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430832,7 +434088,9 @@ "id": "016980", "content": "若$(2 x+\\dfrac{a}{x})^6$的二项展开式中的常数项为$-160$, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430852,7 +434110,9 @@ "id": "016981", "content": "$(x-2 y)(2 x-y)^5$的展开式中的$x^3 y^3$系数为\\bracket{20}.\n\\fourch{$-200$}{$-120$}{$120$}{$200$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -430872,7 +434132,9 @@ "id": "016982", "content": "已知$(\\sqrt{x}+\\dfrac{1}{2 \\sqrt[4]{x}})^n$的二项展开式中, 前三项系数成等差数列, 求二项展开式中的所有有理项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430892,7 +434154,9 @@ "id": "016983", "content": "已知$(\\sqrt[3]{x^{-4}}+x)^n$的展开式中, 第$5$、$6$、$7$三项的系数成等差数列, 求二项展开式中的常数项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430912,7 +434176,9 @@ "id": "016984", "content": "已知多项式$(x-1)^3+(x+1)^4=x^4+a_1 x^3+a_2 x^2+a_3 x+a_4$, 则$a_1=$\\blank{50}, $a_2+a_3+a_4=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430932,7 +434198,9 @@ "id": "016985", "content": "已知$(1+x)(2-x)^7=a_0+a_1 x+a_2 x^2+\\cdots+a_7 x^7+a_8 x^8$, 则$a_0=$\\blank{50}, $a_1+$$a_3+a_5+a_7=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430952,7 +434220,9 @@ "id": "016986", "content": "已知数列$\\{a_n\\}$($n$为正整数) 是首项为$a_1$, 公比为$q$的等比数列.\\\\\n(1) 求和: $a_1 \\mathrm{C}_2^0-a_2 \\mathrm{C}_2^1+a_3 \\mathrm{C}_2^2, a_1 \\mathrm{C}_3^0-a_2 \\mathrm{C}_3^1+a_3 \\mathrm{C}_3^2-a_4 \\mathrm{C}_3^3$;\\\\\n(2) 由 (1)的结果归纳概括出关于正整数$n$的一个结论, 并加以证明;\\\\\n(3) 设$q \\neq 1$, $S_n$是等比数列$\\{a_n\\}$的前$n$项和, 求:\n$S_1\\mathrm{C}_n^0-S_2\\mathrm{C}_n^1+S_3\\mathrm{C}_n^2-S_4\\mathrm{C}_n^3+\\cdots+(-1)^n S_{n+1} \\mathrm{C}_n^n$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -430972,7 +434242,9 @@ "id": "016987", "content": "$1-2 \\mathrm{C}_n^1+4 \\mathrm{C}_n^2-8 \\mathrm{C}_n^3+\\cdots+(-1)^n \\mathrm{C}_n^n \\cdot 2^n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -430992,7 +434264,9 @@ "id": "016988", "content": "已知$(1+2 x)^n$的展开式的各项系数之和为$81$, 则$n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431012,7 +434286,9 @@ "id": "016989", "content": "若在$(1+a x)^5$的展开式中$x^3$的系数为$-80$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431032,7 +434308,9 @@ "id": "016990", "content": "已知$(x \\cos \\theta+1)^5$的展开式中$x^2$的系数与$(x+\\dfrac{5}{4})^4$的展开式中$x^3$的系数相等, 则$\\cos \\theta$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431052,7 +434330,9 @@ "id": "016991", "content": "在$(1+x+\\dfrac{1}{x^{2015}})^{10}$的展开式中, $x^2$项的系数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431072,7 +434352,9 @@ "id": "016992", "content": "已知$(3 x^2+3 x-2)(x-1)^5=a_0+a_1 x+\\cdots+a_7 x^7$, 则$a_0+a_2+a_4+a_6=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431092,7 +434374,9 @@ "id": "016993", "content": "$(x-1)^{10}$的二项展开式中第$4$项是\\bracket{20}.\n\\fourch{$\\mathrm{C}_{10}^3 x^7$}{$\\mathrm{C}_{10}^4 x^6$}{$-\\mathrm{C}_{10}^3 x^7$}{$-\\mathrm{C}_{10}^4 x^6$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -431112,7 +434396,9 @@ "id": "016994", "content": "已知$(x^2-\\dfrac{\\mathrm{i}}{\\sqrt{x}})^n$的展开式中第三项与第五项的系数之比为$-\\dfrac{3}{14}$, 其中$\\mathrm{i}^2=-1$, 则展开式中常数项是\\bracket{20}.\n\\fourch{$-45 \\mathrm{i}$}{$45 \\mathrm{i}$}{$-45$}{$45$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -431132,7 +434418,9 @@ "id": "016995", "content": "$(2 x^2-n)(x-\\dfrac{2}{x})^3$的展开式的各项系数之和为$3$, 则该展开式中含$x^3$项的系数为\\bracket{20}.\n\\fourch{$2$}{$8$}{$-5$}{$-17$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -431152,7 +434440,9 @@ "id": "016996", "content": "如果$(a+\\dfrac{1}{a})^{2 n}$的展开式中, 第四项和第六项的系数相等, 求展开式中的常数项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431172,7 +434462,9 @@ "id": "016997", "content": "在$(\\sqrt{x}-\\dfrac{2}{\\sqrt[3]{x^2}})^n$的展开式中, 第三项的系数比第二项的系数大$162$, 求展开式中的一次项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431192,7 +434484,9 @@ "id": "016998", "content": "已知$S_n=2^n+\\mathrm{C}_n^12^{n-1}+\\mathrm{C}_n^22^{n-2}+\\cdots+\\mathrm{C}_n^{n-1} \\cdot 2+1$($n \\in \\mathrm{N}$, $n\\ge 1$), 求证: 当$n$为偶数时, $S_n-4 n-1$能被$64$整除.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431212,7 +434506,9 @@ "id": "016999", "content": "$(x+\\dfrac{1}{x^2})^9$的二项展开式中常数项是\\blank{50}. (用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431232,7 +434528,9 @@ "id": "017000", "content": "将序号分别为$1$、$2$、$3$、$4$、$5$的$5$张参观券全部分给$4$人, 每人至少$1$张, 如果分给同一人的$2$张参观券连号, 那么不同的分法种数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431252,7 +434550,9 @@ "id": "017001", "content": "方程$\\mathrm{C}_7^{x^2-x}=\\mathrm{C}_7^{5 x-5}$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431272,7 +434572,9 @@ "id": "017002", "content": "某地奥运火炬接力传递路线共分$6$段, 传递活动分别由$6$名火炬手完成. 如果第一棒火炬手只能从甲、乙、丙三人中产生, 最后一棒火炬手只能从甲、乙两人中产生, 则不同的传递方案共有\\blank{50}种. (用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431292,7 +434594,9 @@ "id": "017003", "content": "用$1,2,3,4,5,6$组成六位数(没有重复数字), 要求任何相邻两个数字的奇偶性不同, 且$1$和$2$相邻, 这样的六位数的个数是\\blank{50}. (用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431312,7 +434616,9 @@ "id": "017004", "content": "设$n$是正整数, 则$\\mathrm{C}_n^1+\\mathrm{C}_n^26+\\mathrm{C}_n^36^2+\\cdots+\\mathrm{C}_n^n 6^{n-1}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431332,7 +434638,9 @@ "id": "017005", "content": "从$0,1,2,3,4,5$中任取$3$个数字, 组成没有重复数字的三位数, 其中能被$5$整除的三位数共有 \\blank{50}个. (用数字作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431352,7 +434660,9 @@ "id": "017006", "content": "设函数$f(x)=\\begin{cases}x^6, & x \\geq 1, \\\\ -2 x-1, & x \\leq-1,\\end{cases}$则当$x \\leq-1$时, 则$f[f(x)]$表达式的展开式中含$x^2$项的系数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431372,7 +434682,9 @@ "id": "017007", "content": "2010 年广州亚运会组委会要从小张、小赵、小李、小罗、小王五名志愿者中选派四人分别从事翻译、导游、礼仪、司机四项不同工作, 若其中小张和小赵只能从事前两项工作, 其余三人均能从事这四项工作, 则不同的选派方案共有\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431392,7 +434704,9 @@ "id": "017008", "content": "设常数$a>0$, 若$(x+\\dfrac{a}{x})^9$的二项展开式中$x^5$的系数为 $144$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431412,7 +434726,9 @@ "id": "017009", "content": "$2$名男生和$3$名女生共$5$名同学站成一排, 若男生甲不站两端, $3$名女生中有且只有两名女生相邻, 则不同排法的种数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431432,7 +434748,9 @@ "id": "017010", "content": "观察下列等式:\n\\begin{align*}\n\\mathrm{C}_5^1+\\mathrm{C}_5^5&=2^3-2,\\\\\n\\mathrm{C}_9^1+\\mathrm{C}_9^5+\\mathrm{C}_9^9&=2^7+2^3,\\\\\n\\mathrm{C}_{13}^1+\\mathrm{C}_{13}^5+\\mathrm{C}_{13}^9+\\mathrm{C}_{13}^{13}&=2^{11}-2^5,\\\\\n\\mathrm{C}_{17}^1+\\mathrm{C}_{17}^5+\\mathrm{C}_{17}^9+\\mathrm{C}_{17}^{13}+\\mathrm{C}_{17}^{17}&=2^{15}+2^7.\n\\end{align*}\n由以上等式推测到一个一般的结论:\n对于正整数$n$, $\\mathrm{C}_{4 n+1}^1+C_{4 n+1}^5+\\mathrm{C}_{4 n+1}^9+\\cdots+\\mathrm{C}_{4 n+1}^{4 n+1}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431452,7 +434770,9 @@ "id": "017011", "content": "如果$(3 x^2-\\dfrac{2}{x^3})^n$的展开式中含有非零常数项, 则正整数$n$的最小值为\\bracket{20}.\n\\fourch{$3$}{$5$}{$6$}{$10$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -431472,7 +434792,9 @@ "id": "017012", "content": "设$(x^2+1)(2 x+1)^9=a_0+a_1(x+2)+a_2(x+2)^2+\\cdots+a_{11}(x+2)^{11}$, 则$a_0+a_1+a_2+\\cdots+a_{11}$值为\\bracket{20}.\n\\fourch{$-2$}{$-1$}{$1$}{$2$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -431492,7 +434814,9 @@ "id": "017013", "content": "某班班会准备从含甲、乙的$6$名学生中选取$4$人发言, 要求甲、乙两人至少有一人参加, 那么不同的发言顺序有\\bracket{20}.\n\\fourch{$336$ 种}{$320$ 种}{$192$ 种}{$144$ 种}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -431512,7 +434836,9 @@ "id": "017014", "content": "在平面直角坐标系中, $x$轴正半轴上有$5$个点, $y$轴正半轴上有$3$个点, 将$x$轴上的$5$个点和$y$轴上的$3$个点连成$15$条线段, 这$15$条线段在第一象限内的交点最多有\\bracket{20}.\n\\fourch{$30$ 个}{$35$ 个}{$20$ 个}{$15$ 个}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -431532,7 +434858,9 @@ "id": "017015", "content": "设常数$a>0$, $(x+\\dfrac{a}{\\sqrt{x}})^9$展开式中$x^6$的系数为$4$, 求$\\displaystyle\\lim_{n \\to \\infty} \\sum_{i=1}^n a^i$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431552,7 +434880,9 @@ "id": "017016", "content": "从$0,2,4,6,8$中任取两个数, 从$1,3,5,7,9$中任取三个数, 能组成多少个没有重复数字的五位数? 其中偶数有多少个?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431572,7 +434902,9 @@ "id": "017017", "content": "求$(3-2 x)^9$展开式中系数绝对值最大的项.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431592,7 +434924,9 @@ "id": "017018", "content": "规定$\\mathrm{C}_x^m=\\dfrac{x \\cdot(x-1) \\cdots \\cdots \\cdot(x-m+1)}{m !}$, 其中$x \\in \\mathbf{R}, m$是正整数, 且$\\mathrm{C}_x^0=$1 , 这是组合数$\\mathrm{C}_n^m$($m$、$n$是正整数, 且$m \\leq n$的一种推广).\\\\\n(1) 求$\\mathrm{C}_{-15}^3$的值;\\\\\n(2) 设$x>0$, 当$x$为何值时, $\\dfrac{\\mathrm{C}_x^3}{(\\mathrm{C}_x^1)^2}$取最小值?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431612,7 +434946,9 @@ "id": "017019", "content": "已知函数$f(x)=\\dfrac{x^2+1}{x+c}$的图像关于原点对称.\\\\\n(1) 求$f(x)$的表达式;\\\\\n(2) 对$n \\geq 2$, $n \\in \\mathbf{N}$, $x>0$, 求证: $[f(x)]^n-f(x^n) \\geq 2^n-2$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431632,7 +434968,9 @@ "id": "017020", "content": "以下各说法中, 正确的有\\blank{50}.\\\\\n\\textcircled{1} 对于随机试验, 当在同样的条件下重复进行试验时, 每次试验的所有可能结果是不知道的;\\\\\n\\textcircled{2} 连续抛掷$2$次硬币, 该试验的样本空间可以取为$\\Omega=\\{$正正, 反反, 正反$\\}$;\\\\\n\\textcircled{3} ``已知一个盒中装有$4$个白球和$5$个黑球, 从中任意取$1$个球, 该球是白球或黑球'', 此事件是必然事件;\\\\\n\\textcircled{4} ``某人射击一次, 中靶''是随机事件.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431652,7 +434990,9 @@ "id": "017021", "content": "写出下列试验的样本空间:\\\\\n(1) 同时掷三颗骰子, 记录三颗骰子出现的点数之和;\\\\\n(2) 从含有两件正品$a_1, a_2$和两件次品$b_1, b_2$的四件产品中任取两件, 观察取出产品的结果;\\\\\n(3) 用红、黄、蓝三种颜色给图中$3$个矩形随机涂色, 每个矩形只涂一种颜色, 观察涂色的情况.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) rectangle (3,1) (1,0) -- (1,1) (2,0) -- (2,1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431672,7 +435012,9 @@ "id": "017022", "content": "试验$E$: 甲、乙两人玩出拳游戏 (锤子、剪刀、布), 观察甲、乙出拳的情况.\n设事件$A$表示随机事件``甲乙平局''; 事件$B$表示随机事件``甲赢得游戏''; 事件$C$表示随机事件``乙不输''. 试用集合表示事件$A$、$B$、$C$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431692,7 +435034,9 @@ "id": "017023", "content": "在试验$E$: ``连续抛掷一枚均匀的骰子$2$次, 观察每次掷出的点数''中, 指出下列随机事件的含义:\\\\\n(1) 事件$A=\\{(1,3),(2,3),(3,3),(4,3),(5,3),(6,3)\\}$;\\\\\n(2) 事件$B=\\{(1,5),(5,1),(2,4),(4,2),(3,3)\\}$;\\\\\n(3) 事件$C=\\{(1,3),(3,1),(4,2),(2,4),(3,5),(5,3),(4,6),(6,4)\\}$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431712,7 +435056,9 @@ "id": "017024", "content": "将一个各个面上涂有颜色的正方体锯成$27$个同样大小的小正方体, 从这些小正方体中任取$1$个, 观察取到的小正方体的情况, 则事件$B$为``从小正方体中任取$1$个, 恰有两面涂有颜色'', 那么事件$B$含有\\blank{50}个样本点.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431732,7 +435078,9 @@ "id": "017025", "content": "抛掷$3$枚硬币, 试验的样本点用$(x, y, z)$表示, 集合$M$表示``既有正面朝上, 也有反面朝上'', 则$M=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431752,7 +435100,9 @@ "id": "017026", "content": "抛掷一枚质地均匀的骰子两次, 事件$M=\\{(2,6),(3,5),(4,4),(5,3),(6,2)\\}$, 则事件$M$的含义是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431772,7 +435122,9 @@ "id": "017027", "content": "已知$A=\\{-1,0,1\\}$, $B=\\{1,2\\}$, 从$A$、$B$中各取一个元素分别作点的横坐标和纵坐标, 则该试验的样本空间$\\Omega$为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431792,7 +435144,9 @@ "id": "017028", "content": "从$100$个同类产品中(其中$2$个次品)任取$3$个.\\\\\n\\textcircled{1} 三个正品; \\textcircled{2} 两个正品, 一个次品; \\textcircled{3} 一个正品, 两个次品; \\textcircled{4} 三个次品; \\textcircled{5} 至少有一个次品; \\textcircled{6} 至少有一个正品.\\\\\n其中必然事件是\\blank{50}, 不可能事件是\\blank{50}, 不确定事件是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431812,7 +435166,9 @@ "id": "017029", "content": "从$2,3,8,9$中任取两个不同数字, 分别记为$a$、$b$, 用$(a, b)$表示该试验的样本点, 则事件``$\\log _a b$为整数''可表示为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431832,7 +435188,9 @@ "id": "017030", "content": "将一枚质地均匀的骰子投两次, 得到的点数依次记为$a$、$b$, 设事件$M$为``方程$a x^2+b x+1=0$有实数解'', 则事件$M$中含有样本点的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -431852,7 +435210,9 @@ "id": "017031", "content": "下列事件中是必然事件的是\\bracket{20}.\n\\onech{从分别标有数字$1,2,3,4,5$的$5$张标签中任取一张, 得到标有数字$4$的标签}{函数$y=\\log _a^x$($a>0$且$a \\neq 1$)为增函数}{平行于同一条直线的两条直线平行}{随机选取一个实数$x$, 得$2^x<0$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -431872,7 +435232,9 @@ "id": "017032", "content": "集合$A=\\{2,3\\}$, $B=\\{1,2,4\\}$, 从$A$、$B$中各任意取一个数, 构成一个两位数, 则所有基本事件的个数为\\bracket{20}.\n\\fourch{$8$}{$9$}{$12$}{$11$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -431892,7 +435254,9 @@ "id": "017033", "content": "元旦期间, 小东和爸爸、妈妈外出旅游, 一家三口随机的成一排, 则小东恰好站在中间的站法种数为\\bracket{20}.\n\\fourch{$2$}{$3$}{$4$}{$5$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -431912,7 +435276,9 @@ "id": "017034", "content": "某商场举行购物抽奖的促销活动, 规定每个顾客从装有编号分别为$0,1,2,3$四个小球 (除编号不同外, 其他完全相同) 的抽奖箱中, 每次取出一个球记下编号后放回, 连续取两次, 若取出的两个小球的编号的和等于$6$, 则中一等奖, 等于$5$中二等奖, 等于$4$或$3$中三等奖.\\\\\n(1) 写出试验的样本空间$\\Omega$;\\\\\n(2) 设随机事件$A$为``抽中三等奖'', 随机事件$B$为``抽中奖'', 试用集合表示事件$A$和$B$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431932,7 +435298,9 @@ "id": "017035", "content": "某校夏令营有$3$名用同学$A$、$B$、$C$和$3$名女同学$X$、$Y$、$Z$其年级情况如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 一年级 & 二年级 & 三年级 \\\\\n\\hline 男同学 & A & B & C \\\\\n\\hline 女同学 & X & Y & Z \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n现从这$6$名同学中随机选出$2$人参加知识竞赛 (每人被选到的可能性相同).\\\\\n(1) 写出该试验的样本空间$\\Omega$;\\\\\n(2) 设事件$M$为选出的$2$人来自不同年级且恰有$1$名男同学和$1$名女同学, 试用集合表示$M$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431952,7 +435320,9 @@ "id": "017036", "content": "汉字是世界上最古老的文字之一, 字形结构体现着人类追求均衡对称、和谐稳定的天性. 三个汉字``土''、``口''、``木''都可以看成轴对称图形. 小敏和小慧利用``土''``口''``木''三个汉字设计了一个游戏, 规则如下: 将这三个汉字分别写在背面都相同的三张卡片上, 背面朝上, 洗匀后抽出一张, 放回洗匀后再抽出一张, 若两次抽出的汉字能构成上下结构的汉字(如``土''``土''构成``圭''), 则小敏获胜, 否则小慧获胜.\\\\\n(1) 写出该试验的样本空间$\\Omega$;\\\\\n(2) 设小敏获胜为事件$A$, 试用样本点表示$A$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -431972,7 +435342,9 @@ "id": "017037", "content": "下列问题中是古典概率的是\\bracket{20}.\n\\onech{种下一粒杨树种子, 求其能长成大树的概率}{掷一颗质地不均匀的骰子, 求掷出$1$点的概率}{在区间$[1,4]$上任取一数, 求这个数大于$1.5$的概率}{同时掷两颗质地均匀的股子, 求向上的点数之和是$5$的概率}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -431992,7 +435364,9 @@ "id": "017038", "content": "为美化环境, 从红、黄、白、紫$4$种颜色的花中任选$2$种花种在一个花坛中, 余下的$2$种花种在另一个花坛中, 则红色和紫色的花不在同一花坛的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432012,7 +435386,9 @@ "id": "017039", "content": "某三位数密码, 每位数字可在$0 \\sim 9$这$10$个数字中任选一个, 则该三位数密码中, 恰有两位数字相同的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432032,7 +435408,9 @@ "id": "017040", "content": "从$1,2,3,6$这四个数中一次随机地取$2$个数, 则所取两个数的乘积为$6$的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432052,7 +435430,9 @@ "id": "017041", "content": "在平面直角坐标系中, 从六个点: $A(0,0)$、$B(2,0)$、$C(1,1)$、$D(0,2)$、$E(2,2)$、$F(3,3)$中任取三个, 这三点能构成三角形的概率是\\blank{50}. (结果用分数表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432072,7 +435452,9 @@ "id": "017042", "content": "在$100$件产品中, 有$95$件合格品, $5$件次品, 从中任取$2$件, 计算:\\\\\n(1) 取到的$2$件都是合格品的概率;\\\\\n(2) 取到的$2$件都是次品的概率;\\\\\n(3) 取到$1$件合格品、$1$件次品的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432092,7 +435474,9 @@ "id": "017043", "content": "将一颗质地均匀的正方体骰子 (六个面的点数分别为$1,2,3,4,5,6$) 先后抛掷两次, 记第一次出现的点数为$x$, 第二次出现的点数为$y$.\\\\\n(1) 求事件``$x+y \\leq 3$''的概率;\\\\\n(2) 求事件``$|x-y|=2$''的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432112,7 +435496,9 @@ "id": "017044", "content": "用$4$个不同的球任意投人$4$个不同的盒子内, 每盒投人的球数不限, 计算:\\\\\n(1) 无空盒的概率;\\\\\n(2) 恰好有一空盒的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432132,7 +435518,9 @@ "id": "017045", "content": "从男生和女生共$36$人的班级中任意选出$2$人去完成某项任务, 这里任何人当选的机会都是相同的, 如果选出的$2$人有相同性别的概率是$\\dfrac{1}{2}$, 求这个班级中的男生, 女生各有多少人?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432152,7 +435540,9 @@ "id": "017046", "content": "设连续掷两次骰子得到的点数分别为$m$、$n$($m$、$n=1,2, \\cdots, 6$), 则直线$y=\\dfrac{m}{n} x$与圆$(x-3)^2+y^2=1$相交的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432172,7 +435562,9 @@ "id": "017047", "content": "$10$件产品中有$7$件正品, $3$件次品, 从中任取$4$件, 则恰好取到$1$件次品的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432192,7 +435584,9 @@ "id": "017048", "content": "某班委会由$4$名男生与$3$名女生组成, 现从中选出$2$人担任正副班长, 其中至少有$1$名女生当选的概率是\\blank{50}. (用分数作答)", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432212,7 +435606,9 @@ "id": "017049", "content": "小华与父母一同从重庆乘火车到广安邓小平故居参观, 火车车厢里每排有左、中、右三个座位, 小华一家三口随意坐某排的三个座位, 则小华恰好坐在中间的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432232,7 +435628,9 @@ "id": "017050", "content": "$4$张卡片上分别写有数字$1,2,3,4$, 从这$4$张卡片中随机抽取$2$张, 则取出的$2$张卡片上的数字之和为奇数的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432252,7 +435650,9 @@ "id": "017051", "content": "从$1,2,3,4,5$中任意取出两个不同的数, 则其和为$5$的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432272,7 +435672,9 @@ "id": "017052", "content": "袋子中放有大小和形状相同的小球若干个, 其中标号为$0$的小球$1$个, 标号为$1$的小球$1$个, 标号为$2$的小球$n$个. 已知从袋子中随机抽取$1$个小球, 取到标号是$2$的小球的概率是$\\dfrac{1}{2}$, 则$n$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432292,7 +435694,9 @@ "id": "017053", "content": "$4$名同学各自在周六、周日两天中任选一天参加公益活动, 则周六、周日都有同学参加公益活动的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{8}$}{$\\dfrac{3}{8}$}{$\\dfrac{5}{8}$}{$\\dfrac{7}{8}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -432312,7 +435716,9 @@ "id": "017054", "content": "从正方形四个顶点及其中心这$5$个点中, 任取$2$个点, 则这$2$个点的距离不小于该正方形边长的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{5}$}{$\\dfrac{2}{5}$}{$\\dfrac{3}{5}$}{$\\dfrac{4}{5}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -432332,7 +435738,9 @@ "id": "017055", "content": "在某地的奥运火炬传递活动中, 有编号为$1,2,3, \\cdots, 18$的$18$名火炬手. 若从中任选$3$人, 则选出的火炬手的编号能组成$3$为公差的等差数列的概率为\\bracket{20}.\n\\fourch{$\\dfrac{1}{51}$}{$\\dfrac{1}{68}$}{$\\dfrac{1}{306}$}{$\\dfrac{1}{408}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -432352,7 +435760,9 @@ "id": "017056", "content": "一袋中有$8$个大小形状相同的球, 其中$5$个黑色球, $3$个白色球.\\\\\n(1) 从袋中随机地取出两个球, 求取出的两球都是黑色球的概率;\\\\\n(2) 从袋中不放回取两次, 每次取一个球, 求取出的两球都是黑色球的概率;\\\\\n(3) 从袋中有放回取两次, 每次取一个球, 求取出的两球都是黑色球和至少有一个是黑球的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432372,7 +435782,9 @@ "id": "017057", "content": "先后抛掷两枚质地均匀的骰子.\\\\\n(1) 求点数之和为$7$的概率;\\\\\n(2) 求掷出两个$4$点的概率;\\\\\n(3) 求点数之和能被$3$整除的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432392,7 +435804,9 @@ "id": "017058", "content": "袋中有红、黄、白色球各一个, 有放回抽三次, 计算下列事件的概率;\\\\\n(1) 三次颜色各不同;\\\\\n(2) 三次颜色不全相同;\\\\\n(3) 三次取出的球无红色或无黄色.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432412,7 +435826,9 @@ "id": "017059", "content": "以下各说法中, 正确的有\\blank{50}.\\\\\n\\textcircled{1} 若$A$、$B$表示随机事件, 则$A \\cap B$与$A \\cup B$也表示事件;\\\\\n\\textcircled{2} 若事件$A$、$B$、$C$两两互斥, 则$P(A)+P(B)+P(C)=1$;\\\\\n\\textcircled{3} 事件$A$、$B$满足$P(A)+P(B)=1$, 则$A$、$B$是对立事件;\\\\\n\\textcircled{4} 若两个事件是互斥事件, 则这两个事件是对立事件.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432432,7 +435848,9 @@ "id": "017060", "content": "某射手在一次射击中射中$10$环、 $9$环、$8$环、$7$环、$7$环以下的概率分别为$0.24,0.28,0.19,0.16,0.13$. 计算这个射手在一次射击中:\\\\\n(1) 射中$10$环或$9$环的概率;\\\\\n(2) 至少射中$7$环的概率;\\\\\n(3) 射中$8$环以下的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432452,7 +435870,9 @@ "id": "017061", "content": "一盒中装有各色球$12$个, 其中$5$个红球、 $4$个黑球、 $2$个白球、 $1$个绿球, 从中随机取出$1$球, 求:\\\\\n(1) 取出$1$球是红球或黑球的概率;\\\\\n(2) 取出$1$球是红球或黑球或白球的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432472,7 +435892,9 @@ "id": "017062", "content": "一个盒子里装有三张卡片, 分别标记有数字$1,2,3$, 这三张卡片除标记的数字外完全相同, 随机有放回地抽取$3$次, 每次抽取$1$张, 将抽取的卡片上的数字依次记为$a$、$b$、$c$.\\\\\n(1) 求``抽取的卡片上的数字满足$a+b=c$''的概率;\\\\\n(2) 求``抽取的卡片上的数字$a$、$b$、$c$不完全相同''的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432492,7 +435914,9 @@ "id": "017063", "content": "甲、乙两人从$1,2,3, \\cdots, 10$中各任取一数 (不重复), 已知甲取到的数是$5$的倍数, 则甲数大于乙数的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432512,7 +435936,9 @@ "id": "017064", "content": "设某随机试验的样本空间$\\Omega=\\{0,1,2,3,4,5,6,7,8\\}$, $A=\\{2,3,4\\}$, $B=\\{3,4,5\\}$, $C=\\{5,6,7\\}$, 则:\\\\\n(1) $A \\cup B=$\\blank{50};\\\\\n(2) $\\overline {A} \\cap B=$\\blank{50};\\\\\n(3) $A \\cap(B \\cap C)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432532,7 +435958,9 @@ "id": "017065", "content": "在一次教师联欢会上, 到会的女教师比男教师多$12$人, 从这些教师中随机挑选一人表演节目, 若选中男教师的概率为$\\dfrac{9}{20}$, 则参加联欢会的教师共有\\blank{50}人.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432552,7 +435980,9 @@ "id": "017066", "content": "袋中有形状、大小都相同的$4$只球, 其中$1$只白球, $1$只红球, $2$只黄球, 从中一次随机摸出$2$只球, 则这$2$只球颜色不同的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432572,7 +436002,9 @@ "id": "017067", "content": "事件$A$、$B$互斥, 它们都不发生的概率为$\\dfrac{2}{5}$, 且$P(A)=2P(B)$, 则$P(A)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432592,7 +436024,9 @@ "id": "017068", "content": "中国乒乓球队中的甲、乙两名队员参加奥运会乒乓球女子单打比赛, 甲夺得冠军的概率为$\\dfrac{3}{7}$, 乙夺得冠军的概率为$\\dfrac{1}{4}$, 那么中国队夺得女子乒乓球单打冠军的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432612,7 +436046,9 @@ "id": "017069", "content": "某袋中有编号为$1,2,3,4,5,6$的$6$个球 (小球除编号外完全相同), 甲先从袋中摸出一个球, 记下编号后放回, 乙再从袋中摸出一个球, 记下编号, 则甲、乙两人所摸出球的编号不同的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432632,7 +436068,9 @@ "id": "017070", "content": "口袋内装有一些大小相同的红球、白球和黑球, 从中摸出$1$个球, 摸出红球的概率是$0.42$, 摸出白球的概率是$0.28$, 那么摸出黑球的概率是\\bracket{20}.\n\\fourch{$0.42$}{$0.28$}{$0.3$}{$0.7$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -432652,7 +436090,9 @@ "id": "017071", "content": "从装有$3$个红球、$2$个白球的袋中任取$3$个球, 则所取的$3$个球中至少有$1$个白球的概率是\\bracket{20}.\n\\fourch{$\\dfrac{1}{10}$}{$\\dfrac{3}{10}$}{$\\dfrac{3}{5}$}{$\\dfrac{9}{10}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -432672,7 +436112,9 @@ "id": "017072", "content": "下列四个命题中错误的有\\blank{50}.\\\\\n\\textcircled{1} 对立事件一定是互斥事件;\\\\\n\\textcircled{2} 若$A$、$B$为两个事件, 则$P(A \\cup B)=P(A)+P(B)$;\\\\\n\\textcircled{3} 若事件$A$、$B$、$C$两两互斥, 则$P(A)+P(B)+P(C)=1$;\\\\\n\\textcircled{4} 事件$A$、$B$满足$P(A)+P(B)=1$, 则$A$、$B$是对立事件.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432692,7 +436134,9 @@ "id": "017073", "content": "袋中有外形、质量完全相同的红球、黑球、黄球、绿球共$12$个, 从中任取一球, 得到红球的概率是$\\dfrac{1}{3}$, 得到黑球或黄球的概率是$\\dfrac{5}{12}$, 得到黄球或绿球的概率也是$\\dfrac{5}{12}$.\\\\\n(1) 试分别求得到黑球、黄球、绿球的概率;\\\\\n(2) 从中任取一球, 求得到的不是红球也不是绿球的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432712,7 +436156,9 @@ "id": "017074", "content": "在数学考试中, 小明的成绩在$90$分及$90$分以上的概率是$0.18$, 在$80 \\sim 89$分 (包括$80$分与$89$分, 下同) 的概率是$0.51$, 在$70 \\sim 79$分的概率是$0.15$, 在$60 \\sim 69$分的概率是$0.09,60$分以下的概率是$0.07$, 计算下列事件的概率:\\\\\n(1) 小明在数学考试中取得$80$分及$80$分以上的成绩;\\\\\n(2) 小明考试及格($60$分及$60$分以上为及格).", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432732,7 +436178,9 @@ "id": "017075", "content": "某商场有奖销售中, 购物满$100$元可得$1$张奖券, 多购多得. $1000$张奖券为一个开奖单位, 设特等奖$1$个, 一等奖$10$个, 二等奖$50$个. 设$1$张奖券中特等奖、一等奖、二等奖的事件分别为$A$、$B$、$C$, 求:\\\\\n(1) $P(A), P(B), P(C)$;\\\\\n(2) $1$张奖券的中奖概率;\\\\\n(3) $1$张奖券不中特等奖且不中一等奖的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432752,7 +436200,9 @@ "id": "017076", "content": "以下各说法中, 正确的有\\blank{50}.\\\\\n\\textcircled{1} 不可能事件与任何一个事件相互独立;\\\\\n\\textcircled{2} 必然事件与任何一个事件相互独立;\\\\\n\\textcircled{3} ``$P(A \\cap B)=P(A) P(B)$''是``事件$A$、$B$相互独立''的一个充要条件;\\\\\n\\textcircled{4} 如果两个事件相互独立, 则它们的对立事件也是相互独立的.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432772,7 +436222,9 @@ "id": "017077", "content": "判断下列事件是否为相互独立事件.\\\\\n(1) 甲组$3$名男生, $2$名女生; 乙组$2$名男生, $3$名女生, 现从甲、乙两组各选$1$名同学参加演讲比赛, ``从甲组中选出$1$名男生''与``从乙组中选出$1$名女生'';\\\\\n(2) 容器内盛有$5$个白乒乓球和$3$个黄乒乓球, ``从$8$个球中任意取出$1$个, 取出的是白球''与``从剩下的$7$个球中任意取出$1$个, 取出的还是白球''.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432792,7 +436244,9 @@ "id": "017078", "content": "分别抛掷两枚质地均匀的硬币, 设事件$A$是``第一枚为正面'', 事件$B$是``第二枚为正面'', 事件$C$是``两枚结果相同'', 则下列事件具有相互独立性的是\\blank{50}. (填序号)\\\\\n\\textcircled{1} $A$、$B$; \\textcircled{2} $A$、$C$; \\textcircled{3} $B$、$C$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432812,7 +436266,9 @@ "id": "017079", "content": "甲、乙两人破译一密码, 他们能破译的概率分别为$\\dfrac{1}{3}$和$\\dfrac{1}{4}$, 两人能否破译密码相互独立, 求两人破译时, 以下事件发生的概率:\\\\\n(1) 两人都能破译的概率;\\\\\n(2) 恰有一人能破译的概率;\\\\\n(3) 至多有一人能破译的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432832,7 +436288,9 @@ "id": "017080", "content": "计算机考试分理论考试与实际操作两部分进行, 每部分考试成绩只记``合格''与``不合格'', 两部分考试都``合格''者, 则计算机考试``合格'', 并颁发合格证书. 甲、乙、丙三人在理论考试中``合格''的概率依次为$\\dfrac{4}{5}$、$\\dfrac{3}{4}$、$\\dfrac{2}{3}$, 在实际操作考试中``合格''的概率依次为$\\dfrac{1}{2}$、$\\dfrac{2}{3}$、$\\dfrac{5}{6}$, 所有考试是否合格相互之间没有影响.\\\\\n(1) 假设甲、乙、丙三人同时进行理论与实际操作两项考试, 谁获得合格证书的可能性最大?\\\\\n(2) 这三人进行理论与实际操作两项考试后, 求恰有两人获得合格证书的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432852,7 +436310,9 @@ "id": "017081", "content": "三个元件$T_1$、$T_2$、$T_3$正常工作的概率分别为$\\dfrac{1}{2}$、$\\dfrac{3}{4}$、$\\dfrac{3}{4}$, 将它们中某两个元件并联后再和第三个元件串联接人电路, 它们是否正常工作相互独立? 在如图所示的电路中, 电路不发生故障的概率是多少?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) -- (0.5,0) rectangle (0.5,-0.1) rectangle ++ (0.8,0.2) node [midway, above] {$T_1$} (1.3,0) -- (1.8,0) -- (1.8,0.7) --++ (0.5,0) (2.3,0.6) rectangle ++ (0.8,0.2) node [midway, above] {$T_2$} (3.1,0.7) -- (3.6,0.7) -- (3.6,0) -- (4.1,0) (1.8,0) -- (1.8,-0.7) -- (2.3,-0.7) (2.3,-0.8) rectangle ++ (0.8,0.2) node [midway, above] {$T_3$} (3.1,-0.7) -- (3.6,-0.7) -- (3.6,0);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -432872,7 +436332,9 @@ "id": "017082", "content": "加工某一零件需经过三道工序, 设第一、二、三道工序的次品率分别为$\\dfrac{1}{70}$、$\\dfrac{1}{69}$、$\\dfrac{1}{68}$, 且各道工序互不影响, 则加工出来的零件的次品率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432892,7 +436354,9 @@ "id": "017083", "content": "甲、乙两人同时报考某一所大学, 甲被录取的概率为$0.6$, 乙被录取的概率为$0.7$, 两人是否被录取互不影响, 则其中至少有一人被录取的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432912,7 +436376,9 @@ "id": "017084", "content": "在甲盒内的$200$个螺杆中有$160$个是$A$型, 在乙盒内的$240$个螺母中有$180$个是$A$型. 若从甲、乙两盒内各取一个, 则能配成$A$型螺栓的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432932,7 +436398,9 @@ "id": "017085", "content": "甲、乙、丙三名同学上课后独立完成自我检测题, 甲及格的概率为$\\dfrac{4}{5}$, 乙及格的概率为$\\dfrac{2}{5}$, 丙及格的概率为$\\dfrac{2}{3}$, 则三人中至少有一人及格的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432952,7 +436420,9 @@ "id": "017086", "content": "甲、乙两队进行排球决赛, 现在的情形是甲队只要再赢一局就获冠军, 乙队需要再赢两局才能获冠军. 若每局两队获胜的概率相同, 则甲队获得冠军的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432972,7 +436442,9 @@ "id": "017087", "content": "某篮球队员在比赛中每次罚球的命中率相同, 且在两次罚球中至多命中一次的概率为$\\dfrac{16}{25}$, 则该队员每次罚球的命中率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -432992,7 +436464,9 @@ "id": "017088", "content": "坛子里放有$3$个白球, $2$个黑球, 从中不放回地摸球, 用$A_1$表示第$1$次摸到白球, $A_2$表示第$2$次摸到白球, 则$A_1$与$A_2$\\bracket{20}.\n\\fourch{是互斥事件}{是相互独立事件}{是对立事件}{不是相互独立事件}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -433012,7 +436486,9 @@ "id": "017089", "content": "一条电路上装有甲、乙两根保险丝, 甲熔断的概率为$0.85$, 乙熔断的概率为$0.74$, 甲、乙两根保险丝熔断与否相互独立, 则两根保险丝都熔断的概率为\\bracket{20}.\n\\fourch{$1$}{$0.629$}{$0$}{$0.74$或$0.85$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -433032,7 +436508,9 @@ "id": "017090", "content": "有甲、乙两批种子, 发芽率分别为$0.8$和$0.9$, 在两批种子中各取一粒, 则恰有一粒种子能发芽的概率是\\bracket{20}.\n\\fourch{$0.26$}{$0.08$}{$0.18$}{$0.72$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -433052,7 +436530,9 @@ "id": "017091", "content": "少进人某商场的的名顾客购买甲种商品的概率为$0.5$, 购买乙种商品的概率为$0.6$, 且购买甲种商品间购买Z种商品相互独立, 各顾客之间购买商品也是相互独立的. 求:\\\\\n(1) 进人商场的$1$名顾客, 甲、乙两种商品都购买的概率;\\\\\n(2) 进人商场的$1$名顾客购买甲、乙两种商品中的一种的概率;\\\\\n(3) 进人商场的$1$名顾客至少购买甲、乙两种商品中的一种的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433072,7 +436552,9 @@ "id": "017092", "content": "为应对金融危机, 刺激消费, 某市给市民发放面额为$100$元的旅游消费券, 由抽样调查预计老、中、青三类市民持有这种消费券到某旅游景点的消费额及其概率如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline & 200 元 & 300 元 & 400 元 & 500 元 \\\\\n\\hline 老年 & 0.4 & 0.3 & 0.2 & 0.1 \\\\\n\\hline 中年 & 0.3 & 0.4 & 0.2 & 0.1 \\\\\n\\hline 青年 & 0.3 & 0.3 & 0.2 & 0.2 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n某天恰好有持有这种消费券的老年人中年人、青年人各一人到该旅游景点.\\\\\n(1) 求这三人恰有两人的消费额不少于$300$元的概率;\\\\\n(2) 求这三人的消费总额大于或等于$1300$元的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433092,7 +436574,9 @@ "id": "017093", "content": "同时转动如图所示的两个质地均匀的转盘, 记转盘甲得到的数为$x$, 转盘乙得到的数为$y$(若指针停在边界上则重新转), $x, y$构成数对$(x, y)$, 则所有数对$(x, y)$中, 求满足$x y=4$的概率.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0) circle (2);\n\\draw (-2,0) -- (2,0) (0,-2) -- (0,2);\n\\draw (45:1) node {$2$};\n\\draw (-45:1) node {$3$};\n\\draw (-135:1) node {$4$};\n\\draw (135:1) node {$1$};\n\\draw [->] (0,0) -- (120:1);\n\\draw (0,-2) node [below] {甲};\n\\end{tikzpicture}\n\\hspace*{3em}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0) circle (2);\n\\draw (-2,0) -- (2,0) (0,-2) -- (0,2);\n\\draw (45:1) node {$2$};\n\\draw (-45:1) node {$3$};\n\\draw (-135:1) node {$4$};\n\\draw (135:1) node {$1$};\n\\draw [->] (0,0) -- (-35:1);\n\\draw (0,-2) node [below] {乙};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433112,7 +436596,9 @@ "id": "017094", "content": "设$A$、$B$为两个事件, 且$P(A)>0$, 若$P(A \\cap B)=\\dfrac{1}{3}$, $P(A)=\\dfrac{2}{3}$, 则$P(B | A)$等于\\bracket{20}.\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{2}{9}$}{$\\dfrac{1}{9}$}{$\\dfrac{4}{9}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -433132,7 +436618,9 @@ "id": "017095", "content": "某气象台统计, 该地区下雨的概率为$\\dfrac{4}{15}$, 既刮四级以上的风又下雨的概率为$\\dfrac{1}{10}$. 设事件$A$为该地区下雨, 事件$B$为该地区刮四级以上的风, 则$P(B | A)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433152,7 +436640,9 @@ "id": "017096", "content": "一袋中装有$10$个球, 其中$3$个黑球、 $7$个白球, 从中先后随意各取一球 (不放回), 则第二次取到的是黑球的概率为\\bracket{20}.\n\\fourch{$\\dfrac{2}{9}$}{$\\dfrac{3}{9}$}{$\\dfrac{3}{10}$}{$\\dfrac{7}{10}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -433172,7 +436662,9 @@ "id": "017097", "content": "有一批同一型号的产品, 已知其中由一厂生产的占$30 \\%$, 二厂生产的占$50 \\%$, 三厂生产的占$20 \\%$, 又知这三个厂的产品次品率分别为$2 \\%, 1 \\%, 1 \\%$, 则从这批产品中任取一件是次品的概率是\\bracket{20}.\n\\fourch{$0.013$}{$0.04$}{$0.002$}{$0.003$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -433192,7 +436684,9 @@ "id": "017098", "content": "现有$6$个节目准备参加比赛, 其中$4$个舞蹈节目, $2$个语言类节目, 如果不放回地依次抽取$2$个节目, 求:\\\\\n(1) 第$1$次抽到舞蹈节目的概率;\\\\\n(2) 第$1$次和第$2$次都抽到舞蹈节目的概率;\\\\\n(3) 在第$1$次抽到舞蹈节目的条件下, 第$2$次抽到舞蹈节目的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433212,7 +436706,9 @@ "id": "017099", "content": "一个盒子中有$6$只白球、 $4$只黑球, 从中不放回地每次任取$1$只, 连取$2$次. 求:\\\\\n(1) 第一次取得白球的概率;\\\\\n(2) 第一、第二次都取得白球的概率;\\\\\n(3) 第一次取得黑球而第二次取得白球的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433232,7 +436728,9 @@ "id": "017100", "content": "$1$号箱中有$2$个白球和$4$个红球, $2$号箱中有$5$个白球和$3$个红球, 现随机地从$1$号箱中取出一球放人$2$号箱, 然后从$2$号箱随机取出一球, 问:\\\\\n(1) 从$1$号箱中取出的是红球的条件下, 从$2$号箱取出红球的概率是多少?\\\\\n(2) 从$2$号箱取出红球的概率是多少?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433252,7 +436750,9 @@ "id": "017101", "content": "甲箱的产品中有$5$个正品和$3$个次品, 乙箱的产品中有$4$个正品和$3$个次品.\\\\\n(1) 从甲箱中任取$2$个产品, 求这$2$个产品都是次品的概率;\\\\\n(2) 若从甲箱中任取$2$个产品放入乙箱中, 然后再从乙箱中任取一个产品, 求取出的这个产品是正品的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433272,7 +436772,9 @@ "id": "017102", "content": "某种电子玩具按下按钮后, 会出现红球或绿球. 已知按钮第一次按下后, 出现红球与绿球的概率都是$\\dfrac{1}{2}$, 从按钮第二次按下起, 若第一次出现红球, 则下一次出现红球、绿球的概率分别为$\\dfrac{1}{3}$、$\\dfrac{2}{3}$, 若第一次出现绿球, 则下一次出现红球、绿球的概率分别为$\\dfrac{3}{5}$、$\\dfrac{2}{5}$, 记第$n$($n \\in \\mathbf{N}$, $n \\geq 1$)次按下按钮后出现红球的概率为$P_{1 n}$.\\\\\n(1) $P_2$的值为\\blank{50};\\\\\n(2) 若$n \\in \\mathbf{N}$, $n \\geq 2$, 用$P_{n-1}$表示$P_n$的表达式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433292,7 +436794,9 @@ "id": "017103", "content": "已知$P(B | A)=\\dfrac{1}{3}$, $P(A)=\\dfrac{2}{5}$, 则$P(A \\cap B)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433312,7 +436816,9 @@ "id": "017104", "content": "某人忘记了一个电话号码的最后一个数字, 只好去试拨, 他第一次失败、第二次成功的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433332,7 +436838,9 @@ "id": "017105", "content": "袋中有$5$个小球 ($3$白$2$黑), 现从袋中每次取一个球, 不放回地抽取两次, 则在第一次取到白球的条件下, 第二次取到白球的概率是\\blank{50}, 两次都取到白球的概率是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433352,7 +436860,9 @@ "id": "017106", "content": "袋中装有编号为$1,2, \\cdots, N$的$N$个球, 先从袋中任取一球, 如该球不是$1$号球就放回袋中, 是$1$号球就不放回, 然后再摸一次, 则取到$2$号球的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433372,7 +436882,9 @@ "id": "017107", "content": "有朋自远方来, 乘火车、船、汽车、飞机来的概率分别为$0.3$、$0.2$、$0.1$、$0.4$, 迟到的概率分别为$0.25$、$0.3$、$0.1$、$0$, 则他迟到的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433392,7 +436904,9 @@ "id": "017108", "content": "若从数字$1,2,3,4$中任取一个数, 记为$x$, 再从$1,2, \\cdots, x$中任取一个数记为$y$, 则$y=2$的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433412,7 +436926,9 @@ "id": "017109", "content": "将两枚质地均匀的骰子各第一次, 设事件$A$表示``出现两个点数互不相同'', $B$表示``出现一个$5$点'', 则$P(B | A)$等于\\bracket{20}.\n\\fourch{$\\dfrac{1}{3}$}{$\\dfrac{5}{18}$}{$\\dfrac{1}{6}$}{$\\dfrac{1}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -433432,7 +436948,9 @@ "id": "017110", "content": "设有来自三个地区的各$10$名, $15$名和$25$名考生的报名表, 其中女生报名表分别为$3$份、$7$份和$5$份, 随机地取一个地区的报名表, 从中先后取出两份, 则先取到的一份为女生表的概率为\\bracket{20}.\n\\fourch{$\\dfrac{3}{10}$}{$\\dfrac{21}{100}$}{$\\dfrac{7}{30}$}{$\\dfrac{29}{90}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -433452,7 +436970,9 @@ "id": "017111", "content": "设袋中有$12$个球, $9$个新球, $3$个旧球, 第一次比赛取$3$球, 比赛后放回, 第二次比赛再任取$3$球, 则第二次比赛取得$3$个新球的概率为\\bracket{20}.\n\\fourch{$\\dfrac{441}{3025}$}{$\\dfrac{193}{220}$}{$\\dfrac{1}{11}$}{$\\dfrac{7}{60}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -433472,7 +436992,9 @@ "id": "017112", "content": "设$b$和$c$分别是抛掷一枚骰子先后得到的点数.\\\\\n(1) 求方程$x^2+b x+c=0$有实根的概率;\\\\\n(2) 求在先后两次出现的点数中有$5$的条件下, 方程$x^2+b x+c=0$有实根的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433492,7 +437014,9 @@ "id": "017113", "content": "某商店收进甲厂生产的产品$30$箱, 乙厂生产的同种产品$20$箱, 甲厂每箱装$100$个, 废品率为$0.06$, 乙厂每箱装$120$个, 废品率为$0.05$, 求:\\\\\n(1) 任取一箱, 从中任取一个为废品的概率;\\\\\n(2) 若将所有产品开箱混放, 求任取一个为废品的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433512,7 +437036,9 @@ "id": "017114", "content": "假设某市场供应的智能手机中, 市场占有率和优质率的信息如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline 品牌 & 甲 & 乙 & 其他 \\\\\n\\hline 市场占有率 & $50 \\%$ & $30 \\%$ & $20 \\%$ \\\\\n\\hline 优质率 & $95 \\%$ & $90 \\%$ & $70 \\%$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n在该市场中任意买一部智能手机, 求买到的是优质品的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433532,7 +437058,9 @@ "id": "017115", "content": "设某项试验的成功率是失败率的$2$倍, 用随机变量$X$描述一次试验的成功次数, 则$P(X=0)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433552,7 +437080,9 @@ "id": "017116", "content": "抛掷一枚硬币, 规定正面向上得$1$分, 反面向上得$-1$分, 则得分$X$的均值为\\bracket{20}.\n\\fourch{$0$}{$\\dfrac{1}{2}$}{1}{$-1$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -433572,7 +437102,9 @@ "id": "017117", "content": "随机变量$X$的取值为$0,1,2$, 若$P(X=0)=\\dfrac{1}{5}$, $E[X]=1$, 则$D[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433592,7 +437124,9 @@ "id": "017118", "content": "从次品率为$0.1$的一批产品中任取$4$件, 恰有两件次品的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433612,7 +437146,9 @@ "id": "017119", "content": "在$100$张奖券中, 有$4$张能中奖, 从中任取$2$张, 则$2$张都能中奖的概率是\\bracket{20}.\n\\fourch{$\\dfrac{1}{50}$}{$\\dfrac{1}{25}$}{$\\dfrac{1}{825}$}{$\\dfrac{1}{4950}$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -433632,7 +437168,9 @@ "id": "017120", "content": "春节期间, 小王用私家车送$4$个朋友到三个旅游景点去游玩, 每个朋友在每一个景点下车的概率均为$\\dfrac{1}{3}$, 用$X$表示$4$个朋友在第三个景点下车的人数, 求:\\\\\n(1) 随机变量$X$的分布列;\n(2) 随机变量$X$的均值.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433652,7 +437190,9 @@ "id": "017121", "content": "一名学生每天骑自行车上学, 从家到学校的途中有$5$个交通岗, 假设他在各交通岗遇到红灯的事件是相互独立的, 并且概率都是$\\dfrac{1}{3}$.\\\\\n(1) 求这名学生在途中遇到红灯的次数$X$的均值;\\\\\n(2) 求这名学生在首次遇到红灯或到达目的地停车前经过的路口数$Y$的分布列;\\\\\n(3) 求这名学生在途中至少遇到一次红灯的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433672,7 +437212,9 @@ "id": "017122", "content": "从$4$名男生和$2$名女生中任选$3$人参加演讲比赛, 设随机变量$X$表示所选$3$人中女生的人数.\\\\\n(1) 求$X$的分布列;\\\\\n(2) 求``所选$3$人中女生人数$X \\leq 1$''的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -433692,7 +437234,9 @@ "id": "017123", "content": "编号为$1,2,3$的三名学生随意人座编号为$1,2,3$的三个座位, 每名学生坐一个座位, 设与座位编号相同的学生的人数是$X$, 则$E[X]=$\\blank{50} , $D[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433712,7 +437256,9 @@ "id": "017124", "content": "若随机变量$X$服从两点分布, 且$P(X=0)=0.8$, $P(X=1)=0.2$. 令$Y=3X-2$, 则$P(Y=-2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433732,7 +437278,9 @@ "id": "017125", "content": "随机变量$X$的可能取值为$1,2,3,4$, $P(X=k)=a k+b$($k=1,2,3,4$), $E[X]=3$, 则$a=$\\blank{50}, $b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433752,7 +437300,9 @@ "id": "017126", "content": "已知小明投$10$次篮, 每次投篮的命中率均为$0.7$, 记$10$次投篮中命中的次数为$X$, 则$D[X]=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -433772,7 +437322,9 @@ "id": "017127", "content": "甲、乙、丙三人参加某次招聘会, 甲应聘成功的概率为$\\dfrac{4}{9}$, 乙、丙应聘成功的概率均为$\\dfrac{t}{3}$($0=latex, xscale = 0.06, yscale = 80]\n\\draw [->] (0,0) -- (85,0) node [below] {年龄(岁)};\n\\draw [->] (0,0) -- (0,0.04) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {15/0.01,25/0.02,35/0.03,45/0.025,55/0.015}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {15/0.01,55/0.015,25/0.02,45/0.025,35/0.030}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\draw (65,0) node [below] {$65$};\n\\end{tikzpicture}\n\\end{center}\n(1) 分别求出$a$、$b$、$x$、$y$的值;\\\\\n(2) 从第$2$、$3$、$4$组回答正确的人中用分层抽样的方法抽取$6$人, 求第$2$、$3$、$4$组每组各抽取多少人?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434132,7 +437718,9 @@ "id": "017145", "content": "某高校大一新生中, 来自东部地区的学生有$2400$人, 中部地区学生有$1.600$人, 西部地区学生有$1000$人, 从中选取$100$人作样本调研饮食习惯. 为保证调研结果相对准确, 下列判断中正确的是\\bracket{20}.\\\\\n\\textcircled{1} 用分层抽样的方法分别抽取东部地区学生$48$人, 中部地区学生$32$人, 西部地区学生$20$人;\\\\\n\\textcircled{2} 用简单随机抽样的方法从新生中选出$100$人;\\\\\n\\textcircled{3} 西部地区学生小刘被选中的概率为$\\dfrac{1}{50}$;\\\\\n\\textcircled{4} 东部地区学生小张被选中的概率比中部地区的学生小王被选中的概率大.\n\\fourch{\\textcircled{1}\\textcircled{4}}{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}{\\textcircled{2}\\textcircled{3}}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -434152,7 +437740,9 @@ "id": "017146", "content": "``净拣棉花弹细, 相合共雇王孀; 九斤十二是张昌, 李德五斤四两. 纺讫织成布匹, 一百八尺曾量; 两家分布要明彰, 莫使些儿偏向.''这首古算诗题出自《算法统宗》中的《棉布均摊》, 它的意思如下: 张昌拣棉花九斤十二两, 李德拣棉花五斤四两, 共同雇王孀来帮忙细弹、纺线、织布、共织成布匹一百零八尺长, 则\\bracket{20}.(注: 古代一斤是十六两)\n\\twoch{按张昌$37.8$尺, 李德$70.2$尺分配就合理了}{按张昌$70.2$尺, 李德$37.8$尺分配就合理了}{按张昌$42.5$尺, 李德$65.5$尺分配就合理了}{按张昌$65.5$尺, 李德$42.5$尺分配就合理了}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -434172,7 +437762,9 @@ "id": "017147", "content": "某市模考共有$70000$多名学生参加, 教科室为了了解本校$3390$名考生的数学成绩, 从中抽取$300$名考生的数学成绩进行统计分析, 在这个问题中有以下说法, 其中正确的是\\blank{50}.\\\\\n\\textcircled{1} $3390$名考生是总体的一个样本;\\\\\n\\textcircled{2} $3390$名考生的数学成绩是总体;\\\\\n\\textcircled{3} 样本容量是$300$;\\\\\n\\textcircled{4} $70000$多名考生的数学成绩是总体.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434192,7 +437784,9 @@ "id": "017148", "content": "某中学高一年级有$400$人, 高二年级有$320$人, 高三年级有$280$人, 若每人被抽到的可能性都为$0.2$, 用随机数表法在该中学抽取容量为$n$的样本, 则$n$等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434212,7 +437806,9 @@ "id": "017149", "content": "福利彩票的中奖号码是从$1 \\sim 36$个号码中选出$7$个号码来按规则确定中奖情况, 这种从$36$个号码中选$7$个号码的抽样方法是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434232,7 +437828,9 @@ "id": "017150", "content": "某单位有职工$750$人, 其中青年职工$350$人, 中年职工$250$人, 老年职工$150$人, 为了了解该单位职工的健康情况, 用分层抽样的方法从中抽取样本; 若样本中的青年职工为$7$人, 则样本容量为\\blank{50}人.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434252,7 +437850,9 @@ "id": "017151", "content": "我国施行个人所得税专项附加扣除办法, 涉及子女教育、继续教育、大病医疗、住房贷款利息、 住房租金、赡养老人等六项专项附加扣除, 某单位老年、中年、青年员工分别有$80$人、$100$人、$120$人, 现采用分层随机抽样的方法, 从该单位上述员工中抽取$30$人调查专项附加扣除的享受情况, 则应该从青年员工中抽取的人数为 \\blank{50}人.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434272,7 +437872,9 @@ "id": "017152", "content": "某地区有小学$150$所, 中学$75$所, 大学$25$所, 现采用分层抽样的方法从这些学校中抽取$30$所学校对学生进行视力调查, 应从小学中抽取\\blank{50}所学校, 中学中抽取\\blank{50}所学校.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434292,7 +437894,9 @@ "id": "017153", "content": "研究下列问题:\\\\\n\\textcircled{1} 某城市元旦前后的气温;\\\\\n\\textcircled{2} 某种新型电器元件使用寿命的测定;\\\\\n\\textcircled{3} 电视台想知道某一个节目的收视率;\\\\\n\\textcircled{4} 银行在收进储户现金时想知道有没有假钞.\\\\\n一般通过试验获取数据的是\\bracket{20}.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{2}}{\\textcircled{4}}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -434312,7 +437916,9 @@ "id": "017154", "content": "总体由编号$01,02, \\cdots, 19,20$的$20$个个体组成, 利用下面的随机数表选取$5$个个体, 选取方法是随机数表第$1$行的第$5$列和第$6$列数字开始由左到右依次选取两个数字, 则选出来的第$5$个个体的编号为\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline 7816 & 6572 & 0802 & 6314 & 0702 & 4369 & 9728 & 0198 \\\\\n\\hline 3204 & 9234 & 4935 & 8200 & 3623 & 4869 & 6938 & 7481 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\fourch{$08$}{$07$}{$02$}{$01$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -434332,7 +437938,9 @@ "id": "017155", "content": "在$100$个零件中, 有一级品$20$个, 二级品$30$个, 三级品$50$个, 从中抽取$20$个作为样本.\\\\\n方法 1: 采用简单随机抽样的方法, 将零件编号$00,01,02, \\cdots, 99$, 用抽签法抽取$20$个;\\\\\n方法 2: 采用分层随机抽样的方法, 从一级品中随机抽取$4$个, 从二级品中随机抽取$6$个, 从三级品中随机抽取$10$个;\\\\\n对于上述问题, 下列说法中正确的是\\bracket{20}.\\\\\n\\textcircled{1} 不论采用哪种抽样方法, 这$100$个零件中每一个零件被抽到的可能性都是$\\dfrac{1}{5}$;\\\\\n\\textcircled{2} 采用不同的方法, 这$100$个零件中每一个零件被抽到的可能性各不相同;\\\\\n\\textcircled{3} 在上述两种抽样方法中, 方法$2$抽到的样本比方法$1$抽到的样本更能反映总体特征;\\\\\n\\textcircled{4} 在上述抽样方法中, 方法$1$抽到的样本比方法$2$抽到的样本更能反映总体的特征.\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{1}\\textcircled{3}}{\\textcircled{1}\\textcircled{4}}{\\textcircled{2}\\textcircled{3}}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -434352,7 +437960,9 @@ "id": "017156", "content": "某校有$4000$名学生, 从不同班级抽取了$400$名学生进行调查, 下表是这$400$名学生早晨醒来方式的统计表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 醒来方式 & 别人叫醒 & 闹钟 & 自己醒来 & 其他 \\\\\n\\hline 人数 & 172 & 88 & 64 & 76 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n回答下列问题:\\\\\n(1) 该问题中总体是\\blank{50}, 样本是 \\blank{50}; 样本的容量是\\blank{50};\\\\\n(2) 估计全校学生中早晨自己醒来的人数.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434372,7 +437982,9 @@ "id": "017157", "content": "选择合适的抽样方法抽样, 写出抽样过程.\\\\\n(1) 现有一批电子元件$600$个, 从中抽取$6$个进行质量检测;\\\\\n(2) 有甲厂生产的$30$个篮球, 其中一箱$21$个, 另一箱$9$个, 抽取$3$个入样.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434392,7 +438004,9 @@ "id": "017158", "content": "某单位最近组织了一次健身活动, 活动分为登山组和游泳组, 且每个职工只能参加其中一组. 在参加活动的职工中, 青年人占$42.5 \\%$, 中年人占$47.5 \\%$, 老年人占$10 \\%$; 登山组的职工占参加活动总人数的$\\dfrac{1}{4}$, 且该组中, 青年人占$50 \\%$, 中年人占$40 \\%$, 老年人占$10 \\%$. 为了了解各组不同年龄层的职工对本次活动的满意程度, 现用分层随机抽样的方法从参加活动的全体职工中抽取容量为$200$的样本. 试求:\\\\\n(1) 游泳组中, 青年人、中年人、老年人分别所占的比例;\\\\\n(2) 游泳组中, 青年人、中年人、老年人分别应抽取的人数.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434412,7 +438026,9 @@ "id": "017159", "content": "已知五个数据$3,4, x, 6,7$的平均数是$x$, 则该样本标准差为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434432,7 +438048,9 @@ "id": "017160", "content": "如果$x_1, x_2, x_3, x_4$的方差是$\\dfrac{1}{3}$, 则$3 x_1, 3 x_2, 3 x_3, 3 x_4$的方差为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434452,7 +438070,9 @@ "id": "017161", "content": "某中学为了调查该校学生对于新冠肺炎防控的了解情况, 组织了一次新冠肺炎防控知识竞赛, 并从该学校$1500$名参赛学生中随机抽取了$100$名学生, 并统计了这$100$名学生成绩情况 (满分$100$分, 其中$80$分及以上为优秀), 得到了样本频率分布直方图 (如图), 根据频率分布直方图推测, 这$1500$名学生中竞赛成绩为优秀的学生人数大约为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.08, yscale = 140]\n\\draw [->] (20,0) -- (22,0) -- (23,0.001) -- (25,-0.001) -- (26,0) -- (115,0) node [below] {成绩(分)};\n\\draw [->] (20,0) -- (20,0.036) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (20,0) node [below left] {$O$};\n\\foreach \\i/\\j in {30/0.002,40/0.006,50/0.012,60/0.024,70/0.028,80/0.02,90/0.008}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {30/0.002,40/0.006,90/0.008,50/0.012,80/0.02,60/0.024,70/0.028}\n{\\draw [dashed] (\\i,\\j) -- (20,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434472,7 +438092,9 @@ "id": "017162", "content": "某中学有$10$个学生社团, 每个社团的人数分别是$70,60,60,50,60,40,40,30,30,10$, 则这组数据的平均数, 众数, 中位数的和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434492,7 +438114,9 @@ "id": "017163", "content": "甲组数据为: $5,12,16,21,25,37$, 乙组数据为: $1,6,14,18,38,39$, 则甲、乙的平均数、极差及中位数相同的是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434512,7 +438136,9 @@ "id": "017164", "content": "下图是某校高三 (1) 班的一次数学知识竞赛成绩的茎叶图 (图中仅列出$[50,60)$, $[90, 100)$的数据)和频率分布直方图.\n\\begin{center}\n\\begin{tabular}{c|ccccc}\n5 & 4 & 5 & 6 & 8 & 9\\\\\n6 \\\\\n7 \\\\\n8 \\\\\n9 & 2 & 4\n\\end{tabular}\n\\end{center}\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.08, yscale = 80]\n\\draw [->] (40,0) -- (42,0) -- (43,0.002) -- (45,-0.002) -- (46,0) -- (115,0) node [below] {分数};\n\\draw [->] (40,0) -- (40,0.05) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (40,0) node [below left] {$O$};\n\\foreach \\i/\\j in {50/0.02,60/0.024,70/0.036,80/0.012,90/0.008}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {50/0.02,60/0.024,70/0.036,80/0.012/x,90/0.008/y}\n{\\draw [dashed] (\\i,\\j) -- (40,\\j) node [left] {$\\k$};};\n\\draw (100,0) node [below] {$100$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求全班人数以及频率分布直方图中的$x, y$;\\\\\n(2) 估计学生竞赛成绩的平均数和中位数(保留两位小数).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434532,7 +438158,9 @@ "id": "017165", "content": "下面是水稻产量与施化肥量的一组观测数据(单位: 千克/亩):\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline 施化肥量 & 15 & 20 & 25 & 30 & 35 & 40 & 45 \\\\\n\\hline 水稻产量 & 320 & 330 & 360 & 410 & 460 & 470 & 480 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 将上述数据制成散点图;\\\\\n(2) 你能从散点图中发现施化肥量与水稻产量近似成什么关系吗? 水稻产量会一直随施化肥量的增加而增长吗?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434552,7 +438180,9 @@ "id": "017166", "content": "某赛季甲、乙两名篮球运动员每场比赛的得分情况如下:\\\\\n甲的得分: $12,15,24,25,31,31,36,36,37,39,44,49,50$;\\\\\n乙的得分: $8,13,14,16,23,26,28,33,38,39,51$;\\\\\n(1) 画出甲、乙两名运动员得分数据的茎叶图;\\\\\n(2) 根据茎叶图分析甲、乙两名运动员的水平.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434572,7 +438202,9 @@ "id": "017167", "content": "某厂研制了一种生产高精产品的设备, 为检验新设备生产产品的某项指标有无提高, 用一台旧设备和一台新设备各生产了$10$件产品, 得到各件产品该项指标数据如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 旧设备 & 9.8 & 10.3 & 10.0 & 10.2 & 9.9 & 9.8 & 10.0 & 10.1 & 10.2 & 9.7 \\\\\n\\hline 新设备 & 10.1 & 10.4 & 10.1 & 10.0 & 10.1 & 10.3 & 10.6 & 10.5 & 10.4 & 10.5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n旧设备和新设备生产产品的该项指标的样本平均数分别记为$\\overline {x}$和$\\overline {y}$, 样本方差分别记为$s_1^2$和$s_2^2$.\\\\\n(1) 求$\\overline {x}, \\overline {y}, s_1^2, s_2^2$;\\\\\n(2) 判断新设备生产产品的该项指标的均值较旧设备是否有显著提高(如果$\\overline {y}-\\overline {x} \\geq$$2 \\sqrt{\\dfrac{s_1^2+s_2^2}{10}}$, 则认为新设备生产产品的该项指标的均值较旧设备有显著提高, 否则不认为有显著提高).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434592,7 +438224,9 @@ "id": "017168", "content": "设样本数据$x_1, x_2, \\cdots, x_{10}$的平均数和方差分别为$1$和$4$, 若$y_i=x_i+a$($a$为非零常数, $i=1,2, \\cdots, 10$), 则$y_1, y_2, \\cdots, y_{10}$的平均数为\\blank{50}, 方差为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434612,7 +438246,9 @@ "id": "017169", "content": "某台机床加工的$1000$只产品中次品数的频率分布如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 次品数 & 0 & 1 & 2 & 3 & 4 \\\\\n\\hline 频率 & 0.5 & 0.2 & 0.05 & 0.2 & 0.05 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n则次品数的众数为\\blank{50}; 平均数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434632,7 +438268,9 @@ "id": "017170", "content": "若$40$个数据的平方和是$56$, 平均数是$\\dfrac{\\sqrt{2}}{2}$, 则这组数据的方差为\\blank{50}, 标准差为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434652,7 +438290,9 @@ "id": "017171", "content": "某人$5$次上班途中所花的时间 (单位: 分钟) 分别为$x, y, 10,11,9$; 已知这组数据的平均数为$10$, 方差为$2$, 则$x^2+y^2$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434672,7 +438312,9 @@ "id": "017172", "content": "有一笔统计资料, 共有$11$个数据如下 (不完全以大小排列): $2,4,4,5,5,6,7,8,9,11, x$, 已知这组数据的平均数为$6$, 则这组数据的方差为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434692,7 +438334,9 @@ "id": "017173", "content": "某舞蹈学院为了解大一舞蹈专业新生的体重情况, 对报到的$1000$名舞蹈专业生的数据(单位: $\\mathrm{kg}$) 进行统计, 得到如题图所示的体重频率分布直方图, 则体重在$60 \\mathrm{kg}$以上的人数为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.1, yscale = 50]\n\\draw [->] (40,0) -- (41,0) -- (41.5,0.002) -- (42.5,-0.002) -- (43,0) -- (80,0) node [below] {体重/kg};\n\\draw [->] (40,0) -- (40,0.08) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (40,0) node [below left] {$O$};\n\\foreach \\i/\\j in {45/0.034,50/0.056,55/0.060,60/0.040,65/0.010}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (5,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {45/0.034,50/0.056,55/0.060,60/0.040,65/0.010}\n{\\draw [dashed] (\\i,\\j) -- (40,\\j) node [left] {$\\k$};};\n\\draw (70,0) node [below] {$70$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434712,7 +438356,9 @@ "id": "017174", "content": "题图所示的茎叶图记录了甲、乙两组各$5$名工人某日的产量数据 (单位: 件) 若这两组数据的中位数相等, 且平均值也相等, 则$x$和$y$的值分别为\\blank{50}.\n\\begin{center}\n\\begin{tabular}{cc|c|ccc}\n\\multicolumn{2}{r|}{甲组} & & \\multicolumn{3}{l}{乙组} \\\\ \n& 6 & 5 & 9 \\\\\n2 & 5 & 6 & 1 & 7 & $y$ \\\\\n$x$ & 4 & 7 & 8\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434732,7 +438378,9 @@ "id": "017175", "content": "从一批零件中抽取$80$个, 测量其直径 (单位: $\\mathrm{mm}$), 将所得数据分为$9$组: $[5.31,5.33)$, $[5.33 .5 .35)$, $\\cdots$, $[5.45,5.47)$, $[5.47,5.49]$, 并整理得到如下频率分布直方图, 则在被抽取的蕶件中, 直径落在区间$[5.43,5.47)$内的个数为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 40, yscale = 0.4]\n\\draw [->] (5.28,0) -- (5.288,0) -- (5.29,0.2) -- (5.294,-0.2) -- (5.296,0) -- (5.52,0) node [below] {直径/mm};\n\\draw [->] (5.28,0) -- (5.28,11.5) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (5.28,0) node [below left] {$O$};\n\\foreach \\i/\\j in {5.31/1.25,5.33/3.75,5.35/2.50,5.37/7.50,5.39/10.00,5.41/8.75,5.43/6.25,5.45/5.00,5.47/5.00}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (0.02,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {5.31/1.25,5.33/3.75,5.35/2.50,5.37/7.50,5.39/10.00,5.41/8.75,5.43/6.25,5.45/5.00}\n{\\draw [dashed] (\\i,\\j) -- (5.28,\\j) node [left] {$\\k$};};\n\\draw (5.49,0) node [below] {$5.49$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$10$}{$18$}{$20$}{$36$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -434752,7 +438400,9 @@ "id": "017176", "content": "为了解某地农村经济情况, 对该地农户家庭年收入进行抽样调查, 将农户家庭年收入的调查数据整理得到如下频率分布直方图:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.8, yscale = 15]\n\\draw [->] (1.5,0) --(1.6,0) -- (1.7,0.005) -- (1.9,-0.005) -- (2,0) -- (16,0) node [below] {收入/万元};\n\\draw [->] (1.5,0) -- (1.5,0.25) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw (1.5,0) node [below left] {$O$};\n\\foreach \\i/\\j in {2.5/0.02,3.5/0.04,4.5/0.10,5.5/0.14,6.5/0.20,7.5/0.20,8.5/0.10,9.5/0.10,10.5/0.04,11.5/0.02,12.5/0.02,13.5/0.02\n}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);};\n\\foreach \\i/\\j/\\k in {11.5/0.02,10.5/0.04,8.5/0.10,5.5/0.14,6.5/0.20}\n{\\draw [dashed] (\\i,\\j) -- (1.5,\\j) node [left] {$\\k$};};\n\\draw (14.5,0) node [below] {$14.5$};\n\\end{tikzpicture}\n\\end{center}\n根据此频率分布直方图, 下面结论中不正确的是\\bracket{20}.\n\\onech{该地农户家庭年收入低于$4.5$万元的农户比率估计为$6 \\%$}{该地农户家庭年收入不低于$10.5$万元的农户比率估计为$10 \\%$}{估计该地农户家庭年收入的平均值不超过$6.5$万元}{估计该地有一半以上的农户, 其家庭生收入介于$4.5$万元至$8.5$万元之间}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -434772,7 +438422,9 @@ "id": "017177", "content": "为评估一种农作物的种植效果, 选了$n$块地作试验田; 这$n$块地的亩产量 (单位: $\\mathrm{kg}$) 分别为: $x_1, x_2, \\cdots, x_n$, 下面给出的指标中可以用来评估这种农作物亩产量稳定程度的是\\bracket{20}.\n\\twoch{$x_1, x_2, \\cdots, x_n$的平均数}{$x_1, x_2, \\cdots, x_n$的标准差}{$x_1, x_2, \\cdots, x_n$的最大值}{$x_1, x_2, \\cdots, x_n$的中位数}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -434792,7 +438444,9 @@ "id": "017178", "content": "某车间$20$名工人年龄数据如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline 年龄(岁) & 工人数(人) \\\\\n\\hline 19 & 1 \\\\\n\\hline 28 & 3 \\\\\n\\hline 29 & 3 \\\\\n\\hline 30 & 5 \\\\\n\\hline 31 & 4 \\\\\n\\hline 32 & 3 \\\\\n\\hline 40 & 1 \\\\\n\\hline 合计 & 20 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 求这$20$名工人年龄的众数与极差;\\\\\n(2) 以十位数为茎, 个位数为叶, 作出这$20$名工人年龄的茎叶图;\\\\\n(3) 求这$20$名工人年龄的方差.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434812,7 +438466,9 @@ "id": "017179", "content": "在一个容量为$5$的样本中, 数据均为整数, 己测出其平均数为$10$, 但墨水污损了两个数据, 其中一个数据的十位数字$1$水污损, 即$9,10,11,1\\blacksquare, \\blacksquare$; 不妨设前后两个污损的数字分别为$a, b$.\\\\\n(1) 求: $a+b$的值;\\\\\n(2) 当这组数据的方差最大时, 求整数$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434832,7 +438488,9 @@ "id": "017180", "content": "对甲、乙两名自行车赛手在相同条件下进行了$6$次测试, 测得他们的最大速度$(\\mathrm{m} / \\mathrm{s})$的数据如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 甲 & 27 & 38 & 30 & 37 & 35 & 31 \\\\\n\\hline 乙 & 33 & 29 & 38 & 34 & 28 & 36 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 画出茎叶图, 由茎叶图你能获得哪些信息?\\\\\n(2) 分别求出甲、乙两名自行车赛手最大速度$(\\mathrm{m} / \\mathrm{s})$数据的平均数、中位数、标准差, 并判断选谁参加比赛更合适.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434852,7 +438510,9 @@ "id": "017181", "content": "下列说法:\\\\\n\\textcircled{1} ``名师出高徒''可以解释为教师的教学水平与学生的水平成正相关关系;\\\\\n\\textcircled{2} 通过经验回归方程$\\hat{y}=\\hat{b} x+\\hat{a}$可以估计预报变量的取值和变化趋势;\\\\\n\\textcircled{3} 经验回归方程$\\hat{y}=\\hat{b} x+\\hat{a}$中, 若$\\hat{a}<0$, 则变量$x$和$y$负相关;\\\\\n\\textcircled{4} 因为由任何一组观测值都可以求得一个经验回归方程, 所以没有必要进行相关性检验.\\\\\n其中正确的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434872,7 +438532,9 @@ "id": "017182", "content": "关于回归分析:\\\\\n\\textcircled{1} 在回归分析中, 变量间的关系若是非确定性关系, 那么因变量不能由自变量唯一确定;\\\\\n\\textcircled{2} 线性相关系数可以是正的也可以是负的;\\\\\n\\textcircled{3} 在回归分析中, 如果$r^2=1$或$r=\\pm 1$, 说明$x$与$y$之间完全线性相关;\\\\\n\\textcircled{4} 相关系数$r \\in(-1,1)$.\\\\\n以上说法中正确的是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434892,7 +438554,9 @@ "id": "017183", "content": "高二第二学期期中考试, 按照甲、乙两个班学生的数学成绩优秀和及格统计人数后, 得到如下列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 优秀 & 及格 & 总计 \\\\\n\\hline 甲班 & 11 & 34 & 45 \\\\\n\\hline 乙班 & 8 & 37 & 45 \\\\\n\\hline 总计 & 19 & 71 & 90 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n则随机变量$\\chi^2$的值约为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434912,7 +438576,9 @@ "id": "017184", "content": "若变量$y$与$x$的非线性回归方程是$\\hat{y}=2 \\sqrt{x}-1$, 则当$\\hat{y}$的值为$2$时, $x$的估计值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -434932,7 +438598,9 @@ "id": "017185", "content": "维尼纶纤维的耐热水性能的好坏可以用指标``缩醛化度''$y$来衡量, 这个指标越高, 耐热水性能也越好, 而甲醛浓度是影响缩醛化度的重要因素, 在生产中常用甲醛浓度$x(\\mathrm{g} / \\mathrm{L})$去控制这一指标, 为此必须找出它们之间的关系. 现安排一批实验, 获得如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline 甲醛浓度 $(\\mathrm{g} / \\mathrm{L})$ & 18 & 20 & 22 & 24 & 26 & 28 & 30 \\\\\n\\hline 缩醛化度(克分子) & $26.86 \\%$ & $28.35 \\%$ & $28.75 \\%$ & $28.87 \\%$ & $29.75 \\%$ & $30.00 \\%$ & $30.36 \\%$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 画散点图, 并判断成对样本数据是否线性相关;\\\\\n(2) 求相关系数$r$(精确到$0.01$), 并通过样本相关系数判断甲醛浓度与缩醛化度的相关程度和变化趋势的异同.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434952,7 +438620,9 @@ "id": "017186", "content": "某商场为提高服务质量, 随机调查了$50$名男顾客和$50$名女顾客, 每位顾客对该商场的服务给出满意或不满意的评价, 得到下面列联表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline & 满意 & 不满意 \\\\\n\\hline 男顾客 & 40 & 10 \\\\\n\\hline 女顾客 & 30 & 20 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 分别估计男、女顾客对该商场服务满意的概率;\\\\\n(2) 能否有$95 \\%$的把握认为男、女顾客对该商场服务的评价有差异?\n附: $\\chi^2=\\dfrac{n(a d-b c)^2}{(a+b)(c+d)(a+c)(b+d)}$.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline$P(\\chi^2 \\geq k)$ & 0.050 & 0.010 & 0.001 \\\\\n\\hline$k$ & 3.841 & 6.635 & 10.828 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434972,7 +438642,9 @@ "id": "017187", "content": "某学校对教师关于``多媒体教学模式''的使用情况进行了问卷调查. 共调查了$50$人, 其中有老教师$20$人, 青年教师$30$人. 老教师对多媒体教学模式赞同的有$10$人, 不赞同的有$10$人; 青年教师对多媒体教学模式赞同的有$24$人, 不赞同的有$6$人.\\\\\n(1) 根据以上数据建立一个$2 \\times 2$列联表;\\\\\n(2) 依据小概率$\\alpha=0.001$值, 能否推断青年教师和老教师在多媒体教学模式的使用上有差异?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -434992,7 +438664,9 @@ "id": "017188", "content": "为加强环境保护, 治理空气污染, 环境监测部门对某市空气质量进行调研, 随机抽查了$100$天空气中的 PM2.5 和$\\mathrm{SO}_2$浓度(单位: $\\mu \\mathrm{g} / \\mathrm{m}^3$), 得下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline \\backslashbox{PM2.5}{SO${}_2$} & {$[0,50]$} & $(50,150]$ & $(150,475]$ \\\\\n\\hline$[0,35]$ & 32 & 18 & 4 \\\\\n\\hline$(35,75]$ & 6 & 8 & 12 \\\\\n\\hline$(75,115]$ & 3 & 7 & 10 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 估计事件``该市一天空气中 PM2.5 浓度不超过 75 , 且$\\mathrm{SO}_2$浓度不超过 150''的概率;\\\\\n(2) 根据所给数据, 完成下面的$2 \\times 2$列联表;\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline \\backslashbox{PM2.5}{SO${}_2$} & {$[0,150]$} & $(150,475]$ \\\\\n\\hline $[0,75]$ & & \\\\\n\\hline $(75,115]$ & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(3) 根据 (2) 中的列联表, 判断是否有$99 \\%$的把握认为该市一天空气中 PM2.5 浓度与$\\mathrm{SO}_2$浓度有关?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -435012,7 +438686,9 @@ "id": "017189", "content": "已知一组数据点$(x_1, y_1),(x_2, y_2),(x_3, y_3), \\cdots,(x_n, y_n)$, 用最小二乘法得到其线性回归方程为$\\hat{y}=-\\sqrt{2} x+4$, 若数据$x_1, x_2, x_3, \\cdots x_n$的均值为$\\sqrt{2}$, 则可以估计数据$y_1, y_2, y_3, \\cdots y_n$的均值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -435032,7 +438708,9 @@ "id": "017190", "content": "已知变量$x$、$y$相对应的一组数据为$(10,1.5)$, $(11,3.2)$, $(11,8.3)$, $(12.5,14)$, $(13,5)$变量$x'$、$y'$相对应的一组数据为$(10,5)$, $(11.3,4)$, $(11,8.3)$, $(12.5,2)$, $(13,1)$, 用$r_1$表示变量$x$与$y$之间的线性相关系数, 用$r_2$表示变量$x'$与$y'$之间的线性相关系数, 则有\\bracket{20}.\n\\fourch{$r_2=latex,xscale = 0.6, yscale = 8]\n\\draw [->] (0,0) -- (0,0.45) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw [->] (0,0) -- (0.1,0) -- (0.2,-0.02) -- (0.4,0.02) -- (0.6,-0.02) -- (0.8,0) -- (9.5,0) node [below] {百分比};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {6.5/0.05/0.05,5.5/0.1/0.1,1.5/0.15/0.15,4.5/0.2/0.2,3.5/0.3/0.3}\n{\\draw [dashed] (\\i,\\j) -- (0,\\j) node [left] {$\\k$};};\n\\foreach \\i/\\j/\\k in {1.5/0.15/0.15,2.5/0.2/0.2,3.5/0.3/0.3,4.5/0.2/0.2,5.5/0.1/0.1,6.5/0.05/0.05}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);\n};\n\\draw (7.5,0) node [below] {$7.5$};\n\\draw (4.5,-0.1) node {甲离子残留百分比直方图};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,xscale = 0.6, yscale = 8]\n\\draw [->] (0,0) -- (0,0.45) node [left] {$\\dfrac{\\text{频率}}{\\text{组距}}$};\n\\draw [->] (0,0) -- (0.1,0) -- (0.2,-0.02) -- (0.4,0.02) -- (0.6,-0.02) -- (0.8,0) -- (9.5,0) node [below] {百分比};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j/\\k in {2.5/0.05/0.05,3.5/0.1/b,7.5/0.15/0.15,6.5/0.2/0.2,5.5/0.35/a}\n{\\draw [dashed] ({\\i-1},\\j) -- (0,\\j) node [left] {$\\k$};};\n\\foreach \\i/\\j/\\k in {2.5/0.05/0.05,3.5/0.1/0.1,4.5/0.15/0.15,5.5/0.35/0.35,6.5/0.2/0.2,7.5/0.15/0.15}\n{\\draw ({\\i-1},0) node [below] {$\\i$} --++ (0,\\j) --++ (1,0) --++ (0,-\\j);\n};\n\\draw (7.5,0) node [below] {$8.5$};\n\\draw (4.5,-0.1) node {乙离子残留百分比直方图};\n\\end{tikzpicture}\n\\end{center}\n(1) 求乙离子残留百分比直方图中$a$、$b$的值;\\\\\n(2) 分别估计甲、乙离子残留百分比的平均值 (同一组中的数据用该组区间的中点值为代表).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -435612,7 +439346,9 @@ "id": "017219", "content": "一个盒子中有$6$个白球、$4$个黑球, 从中不放回地每次任取$1$个, 连取$2$次. 求:\\\\\n(1) 第一次取得白球的概率;\\\\\n(2) 第一、第二次都取得白球的概率;\\\\\n(3) 第一次取得黑球而第二次取得白球的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -435632,7 +439368,9 @@ "id": "017220", "content": "在心理学研究中, 常采用对比试验的方法评价不同心理暗示对人的影响, 具体方法如下: 将参加试验的志愿者随机分成两组, 一组接受甲种心理暗示, 另一组接受乙种心理暗示, 通过对比这两组志愿者接受心理暗示后的结果来评价两种心理暗示的作用, 现有$6$名男志愿者$A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$和$4$名女志愿者$B_1$, $B_2$, $B_3$, $B_4$, 从中随机抽取$5$人接受甲种心理暗示, 另$5$人接受乙种心理暗示.\\\\\n(1) 求接受甲种心理暗示的志愿者中包含$A_1$但不包含$B_1$的频率.\\\\\n(2) 用$X$表示接受乙种心理暗示的女志愿者人数, 求$X$的分布列与数学期望$E[X]$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -435652,7 +439390,9 @@ "id": "017221", "content": "某学校组织``一带一路''知识竞赛, 有$A$、$B$两类问题, 每个参加比赛的同学先在两类问题中选择一类并从中随机抽取一个问题回答, 若回答错误则该同学比赛结束; 若回答正确则从另一类问题中再随机抽取一个问题回答, 无论回答正确与否, 该同学比赛结束.$A$类问题中的每个问题回答正确得$20$分, 否则得$0$分;$B$类问题中的每个问题回答正确得$80$分, 否则得$0$分, 已知小明能正确回答$A$类问题的概率为$0.8$, 能正确回答$B$类问题的概率为$0.6$, 且能正确回答问题的概率与回答次序无关.\\\\\n(1) 若小明先回答$A$类问题, 记$X$为小明的累计得分, 求$X$的分布列;\\\\\n(2) 为使累计得分的期望最大, 小明应选择先回答哪类问题? 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -435672,7 +439412,9 @@ "id": "017222", "content": "甲、乙、丙三位同学进行羽毛球比赛, 约定赛制如下: 累计负两场者被淘汰; 比赛前抽签决定首先比赛的两人, 另一人轮空; 每场比赛的胜者与轮空者进行下一场比赛, 负者下一场轮空, 直至有一人被淘汰; 当一人被淘汰后, 剩余的两人继续比赛, 直至其中一人被淘汰, 另一人最终获胜, 比赛结束. 经抽签, 甲、乙首先比赛, 丙轮空, 设每场比赛双方获胜的概率都为$\\dfrac{1}{2}$.\\\\\n(1) 求甲连胜四场的概率;\\\\\n(2) 求需要进行第五场比赛的概率;\\\\\n(3) 求丙最终获胜的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "",