From 8ab00417a6279af8bfc6fc370d820b1c619ebeac Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Sat, 24 Jun 2023 20:57:05 +0800 Subject: [PATCH] =?UTF-8?q?=E5=BD=95=E5=85=A52023=E5=8C=97=E4=BA=AC?= =?UTF-8?q?=E9=AB=98=E8=80=83=E7=AD=94=E6=A1=88?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具/文本文件/metadata.txt | 3119 +----------------------------------- 题库0.3/Problems.json | 42 +- 2 files changed, 59 insertions(+), 3102 deletions(-) diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index 56a270e7..a0efef42 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,3128 +1,85 @@ ans - -021441 -错误, 正确, 错误, 错误 - - -021442 -D - - -021443 -C - - -021444 +18237 A -021445 -C - - -021446 +18238 D -021447 -$-390^\circ$ - - -021448 -$304^\circ$, $-56^\circ$ - - -021449 -$-144^\circ$ - - -021450 -二, 四 - - -021451 -(1) $\{\alpha|\alpha=60^\circ+k\cdot 360^\circ, \ k\in \mathbf{Z}\}$, $-300^\circ$, $60^\circ$, $420^\circ$; (2) $\{\alpha|\alpha = -21^\circ+k\cdot 360^\circ, \ k \in \mathbf{Z}\}$, $-21^\circ$, $339^\circ$, $699^\circ$ - - -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} - - -021453 -$-1290^{\circ}$;第二象限 - - -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) $\{\alpha|180^{\circ}+k\cdot 360^{\circ}<\alpha<270^{\circ}+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$. - - -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) $\{\beta|\beta=90^{\circ}-\alpha+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$. - - -021456 -C - - -021457 +18239 B -021458 -$\dfrac{\pi}{12}$; $\dfrac{7\pi}{12}$; $\dfrac{5\pi}{4}$; $300^{\circ}$; $324^{\circ}$; $315^{\circ}$; $(\dfrac{270}{\pi})^{\circ}$ +18240 +C -021459 -(1)$\frac{50\pi+180}{9}$;(2)$\frac{250\pi}{9}$ +18241 +D -021460 -$\sqrt{3}$ +18242 +D -021461 -(1)$\frac{\pi}{3}$;(2)$\frac{2\pi}{3}$ +18243 +B -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. +18244 +C -021463 -$\frac{1}{2}$ +18245 +C -021464 -(1) $\{\alpha|-\frac{\pi}{2}+2k\pi<\alpha<2k\pi,\ k \in \mathbf{Z}\}$;\\ -(2) $\{\alpha|\alpha=\frac{k\pi}{2},\ k \in \mathbf{Z}\}$. +18246 +B -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) $\beta=\alpha+\pi+2k\pi,\ k \in \mathbf{Z}$. - - -021466 -(1) $\{\alpha|-\frac{\pi}{4}+2k\pi \le \alpha \le \frac{\pi}{2}+2k\pi,\ k \in \mathbf{Z}\}$;\\ -(2) $\{\alpha|\frac{\pi}{6}+k\pi \le \alpha \le \frac{5\pi}{6}+k\pi,\ k \in \mathbf{Z}\}$. - - -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. - - -021468 -$A\cap B=\{\alpha | 2k \pi+\dfrac{5\pi}{6}<\alpha<2k \pi+\dfrac{7\pi}{6},\ k \in \mathbf{Z} \}$ - - -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} - - -021470 -$2\sqrt{5}$ - - -021471 -$\frac{2\sqrt{13}}{13}$;$-\frac{2}{3}$ - - -021472 -$ \left( -2,\frac{2}{3} \right)$ - - -021473 -$<$ - - -021474 -5 - - -021475 -2 - - -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, $\cos \alpha=-1$, $\tan \alpha = 0$. - - -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}$, $\tan \alpha=\frac{4}{3}$. - - -021478 -$-\frac{\sqrt{3}}{4}$ - - -021479 -(1) 第四象限； (2) 第一、四象限；(3)第一、三象限；(4)第一、三象限. - - -021480 -$A=\left\{ -2,-0,4 \right\}$ - - -021481 -(1) $\{\alpha|2k\pi \le \alpha \le \frac{\pi}{2}+2k\pi,\ k \in \mathbf{Z}\}$;\\ -(2) $[0,3)$ - - -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} - - -021483 -(1) $\{x|x=\frac{4\pi}{3}+2k \pi$或$ x=\frac{5\pi}{3}+2k \pi,\ k \in \mathbf{Z} \}$;\\ -(2) $\{-\frac{2\pi}{3},-\frac{\pi}{3},\frac{4\pi}{3} ,\frac{5\pi}{3},\frac{10\pi}{3},\frac{11\pi}{3} \}$ - - -021484 -$-\frac{2\sqrt{5}}{5}$;$2$ - - -021485 -\textcircled{2} \textcircled{4} - - -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}$, $\tan \alpha=3$. - - -021487 -$\sin k\pi =0$; $\cos k\pi=\begin{cases}1, & k=2n, \\ -1, & k=2n-1\end{cases}$($n \in \mathbf{Z}$). - - -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) $\{\theta | k \pi+\dfrac{\pi}{3} \le \theta \le k \pi+\dfrac{2\pi}{3},\ k \in \mathbf{Z} \}$. - - -021489 -第二象限 - - -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) $\sin (\cos \alpha) \cdot \cos (\sin \alpha)<0$ - - -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}$. - - -021492 -$-\dfrac{3}{8}$ - - -021493 -$-\dfrac{1}{20}$ - - -021494 -$\dfrac{7\sqrt{2}}{4}$ - - -021495 -$\dfrac{3\sqrt{5}}{5}$ - - -021496 -$11$ - - -021497 -$5$;$-\dfrac{12}{5}$;$\dfrac{4}{9}$ - - -021498 -$\sin ^2 \alpha$ - - -021499 +18247 $1$ -021502 -$-\dfrac{12}{5}$ +18248 +$\dfrac{x^2}{2}-\dfrac{y^2}{2}=1$ -021503 -$-\dfrac{\sqrt{3}}{2}$ +18249 +$\dfrac{9\pi}{4}$, $\dfrac{\pi}{3}$ -021504 -$\dfrac{\sqrt{7}}{2}$;$\dfrac{\sqrt{7}}{4}$ +18250 +$48$, $384$ -021505 -$-\dfrac{\sqrt{11}}{3}$ +18251 +\textcircled{2}\textcircled{3} -021506 -$\dfrac{\pi}{3}$ +18252 +(1) 证明略; (2) $\dfrac{\pi}{3}$ -021507 -$\left[ 0,\pi \right )$ +18253 +(1) $\varphi=-\dfrac{\pi}{3}$; (2) 选条件\textcircled{1}不能使函数$f(x)$存在, 条件\textcircled{2}\textcircled{3}均可解得$\omega = 1$, $\varphi=-\dfrac{\pi}{6}$ -021508 -$-\dfrac{\sqrt{3}}{2}$;$-\dfrac{\sqrt{2}}{2}$;$-\sqrt{3}$;$-\sqrt{3}$ +18254 +(1) $0.4$; (2) $0.168$; (3) 不变的概率最大 -021509 -$69^{\circ}$;$72^{\circ}$;$\dfrac{\pi}{9}$;$\dfrac{7 \pi}{15}$ +18255 +(1) $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$; (2) 证明略 -021510 -$\cot \alpha$ +18256 +(1) $a=-1$, $b=1$; (2) 单调递减区间为$(0,3-\sqrt{3})$和$(3+\sqrt{3},+\infty)$, 单调递增区间为$(-\infty, 0)$和$(3-\sqrt{3}, 3+\sqrt{3})$; (3) $3$个 -021511 -$-1$ +18257 +(1) $r_0=0$, $r_1=1$, $r_2=2$, $r_3=3$; (2) $r_n=n$($n \in \mathbf{N}$); (3) 证明略 -021512 -$-1$ - - -021513 -$ \sin 2-\cos 2$ - - -021514 -$0$ - - -021515 -$0$ - - -021516 -$-\dfrac{\sqrt{1-a^2}}{a}$ - - -021517 -$-\dfrac{2+\sqrt{3}}{3}$ - - -021518 -(1) $\dfrac{\sqrt{3}}{2}$;(2) $\dfrac{1}{4}$. - - -021519 -(1) $-\dfrac{2}{3}$; \\ -(2) $\dfrac{2}{3}$; \\ -(3) $-\dfrac{\sqrt{5}}{3}$;\\ -(4) $\dfrac{\sqrt{5}}{2}$. - - -021520 -(1) $\sin 69^{\circ}$ ; (2) $-\cos 8^{\circ}$ ; -(3) $-\tan \dfrac{\pi}{9}$; (4) $\cot \dfrac{7\pi}{15}$. - - -021521 -$\dfrac{2}{5}$ - - -021522 -$(3,4)$ - - -021523 -$0$ - - -021524 -$\sin \alpha$ - - -021525 -$-\dfrac{1}{5}$ - - -021526 -(1) $\dfrac{\sqrt{6}}{6}-\sqrt{3}$;\\ -(2) $-\dfrac{\sqrt{6}}{3}$;\\ -(3) $1$ - - -021527 -(1) $\dfrac{6 \pi}{5}$; (2) $\dfrac{4 \pi}{5}$; (3) $\dfrac{13 \pi}{10}$; (4) $\dfrac{17 \pi}{10}$. - - -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, $\dfrac{1}{\tan [\dfrac{(2 k+1) \pi}{2}+\alpha]}=\sqrt{3}$. - - -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$x=k \pi - \dfrac{3\pi}{8} ,\ k \in \mathbf{Z}\}$; - - -021530 -(1) $\{ \dfrac{\pi}{12},\dfrac{17\pi}{12} \}$;\\ -(2) $\{ \dfrac{5\pi}{6} \}$;\\ -(3) $\{ \dfrac{\pi}{12},\dfrac{5\pi}{12} \}$;\\ -(4) $\{ \dfrac{5\pi}{6} \}$. - - -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$x=k \pi -\dfrac{ \pi}{6},\ k \in \mathbf{Z}\}$. - - -021532 -$\dfrac{3+4\sqrt{3}}{10}$ - - -021533 -$-1$ - - -021534 -$-\dfrac{33}{50}$ - - -021535 -(1) $\dfrac{\sqrt{6}-\sqrt{2}}{4}$; -(2) $\dfrac{\sqrt{6}+\sqrt{2}}{4}$; -(3) $0$. - - -021536 -(1) $\sqrt{3} \sin \alpha$; -(2) $\cos(\alpha-2\beta)$. - - -021537 -$\dfrac{140}{221}$ - - -021538 -$\dfrac{2\sqrt{6}-1}{6}$ - - -021539 -证明略 - - -021540 -C - - -021541 -A - - -021542 -$\dfrac{3\sqrt{10}+6\sqrt{2}+2\sqrt{14}-\sqrt{70}}{24}$ - - -021543 -$\dfrac{8\sqrt{3}-21}{20}$ - - -021544 -$\dfrac{\pi}{2}$ - - -021545 -$-\dfrac{2+\sqrt{15}}{6}$ - - -021546 -$-\dfrac{\sqrt{2}}{2}$ - - -021547 -$\sin 2\beta$ - - -021548 -$0$ - - -021549 -$2-\sqrt{3}$ - - -021550 -$\dfrac{16}{65}$ - - -021551 -$-\dfrac{7}{25}$ - - -021552 -$-\dfrac{4\sqrt{14}+3\sqrt{2}}{20}$ - - -021553 -$(\dfrac{4\sqrt{3}+3}{2},\dfrac{3\sqrt{3}-4}{2})$ - - -021554 -$-\dfrac{33}{65}$或$\dfrac{63}{65}$ - - -021555 -B - - -021556 -C - - -021557 -$-\dfrac{56}{65}$ - - -021558 -$-3$ - - -021559 -$\dfrac{3}{4}$ - - -021560 -$\dfrac{-6+5\sqrt{3}}{3}$ - - -021561 -$\tan \alpha$ - - -021562 -$\sqrt{3}$ - - -021563 -$-\dfrac{\sqrt{3}}{3}$ - - -021564 -A - - -021565 -$-\dfrac{17}{31}$ - - -021566 -$\dfrac{\pi}{4}$ - - -021567 -(1) $\dfrac{1}{3}$; -(2) $\dfrac{1}{7}$ - - -021568 -$-\dfrac{1}{5}$ - - -021569 -当$CD = 1.4$米时，$\tan \angle ACB$最大 - - -021570 -(1) $2 \sin (\alpha+\dfrac{\pi}{6})$; -(2) $\sqrt{2} \sin (\alpha+\dfrac{7\pi}{4})$. - - -021571 -$6\cos(\alpha+\dfrac{\pi}{3})$ - - -021572 -$2k \pi-\dfrac{\pi}{3}(k\in \mathbf{Z} )$ - - -021573 -B - - -021574 -$\dfrac{1}{3}$ - - -021575 -$\dfrac{\pi}{12}$或$\dfrac{5\pi}{12}$ - - -021576 -$5$ - - -021577 -$\dfrac{13}{3}$ - - -021578 -$-\dfrac{p}{1+q}$ - - -021579 -$\dfrac{3}{5}$ - - -021580 -$\dfrac{24}{7}$ - - -021581 -$-\dfrac{24}{25}$ - - -021582 -$-\dfrac{15}{17}$ - - -021583 -$\sin 2 \varphi=\dfrac{4\sqrt{2}}{9}$; -$\cos 2 \varphi=-\dfrac{7}{9}$; -$\tan 2 \varphi=-\dfrac{4\sqrt{2}}{7}$. - - -021584 -$\dfrac{24}{25}$; $\dfrac{7}{25}$; $\dfrac{24}{7}$ - - -021585 -(1) $-\dfrac{\sqrt{3}}{3}$;\\ -(2) $\dfrac{3}{4}$. - - -021586 -$\dfrac{7}{24}$ - - -021587 -$-\dfrac{2\sqrt{10}}{5}$ - - -021588 -$1$ - - -021590 -$1$或$\dfrac{7}{25}$ - - -021591 -$-\dfrac{\sqrt{2-2a}}{2}$ - - -021592 -第三象限 - - -021593 -当$\dfrac{\theta}{2}$在第二象限时， -$\sin \dfrac{\theta}{2}=\dfrac{\sqrt{3}}{3}$, -$\cos \dfrac{\theta}{2}=-\dfrac{\sqrt{6}}{3}$, -$\tan \dfrac{\theta}{2}=-\dfrac{\sqrt{2}}{2}$;\\ -当$\dfrac{\theta}{2}$在第四象限时， -$\sin \dfrac{\theta}{2}=-\dfrac{\sqrt{3}}{3}$, -$\cos \dfrac{\theta}{2}=\dfrac{\sqrt{6}}{3}$, -$\tan \dfrac{\theta}{2}=-\dfrac{\sqrt{2}}{2}$. - - -021594 -$\dfrac{3}{5}$;$\dfrac{4}{5}$ - - -021596 -$\dfrac{2}{3}$ - - -021597 -$\cos \alpha-\sin \alpha$ - - -021598 -$\sin \dfrac{ \alpha}{2}$ - - -021599 -(1) $\tan \dfrac{\theta}{2}$; (2) $\sin \alpha$. - - -021600 -$\dfrac{\sqrt{6}}{2}$ - - -021601 -$30^{\circ}$或$90^{\circ}$ - - -021602 -$\sqrt{6}$ - - -021603 -$55$ - - -021604 -$\dfrac{\pi}{3}$或$\dfrac{2\pi}{3}$ - - -021605 -$1: \sqrt{3}: 2$ - - -021606 -$2$ - - -021607 -$\dfrac{5}{8}$ - - -021608 -等腰 - - -021609 -$\dfrac{3\sqrt{2}}{2}$ - - -021610 -$\sqrt{3}$ - - -021611 -$\dfrac{7\pi}{12}$ - - -021612 -$\dfrac{2\pi}{3}$ - - -021613 -(1) $\left( 0,9 \right)$; \\ -(2) $\{9\} \cup \left[18,+ \infty \right)$;\\ -(3) $\left( 9,18 \right)$. - - -021614 -$\dfrac{3\sqrt{7}}{8}$ - - -021615 -$\sqrt{17}$或$\sqrt{65}$ - - -021616 -$\dfrac{\pi}{4}$ - - -021617 -\textcircled{1}；\textcircled{2} - - -021618 -$a>3$ - - -021619 -$a=\sqrt{21}$和$\sin B=\dfrac{5\sqrt{7}}{14}$ - - -021620 -$\dfrac{2\pi}{3}$ - - -021621 -$c=\sqrt{6}+\sqrt{2}$;$C=75^\circ$. - - -021622 -$\dfrac{\sqrt{19}}{2}$ - - -021623 -周长的最小值为$12$,此时三角形为正三角形；\\ -面积最大值为$4\sqrt{3}$,此时三角形为正三角形. - - -021624 -$\dfrac{\sqrt{5}}{5}$ - - -021625 -$\dfrac{2\sqrt{5}}{5}$或$-\dfrac{2\sqrt{5}}{25}$ - - -021626 -$\dfrac{\sqrt{5}}{5}$或$\dfrac{11\sqrt{5}}{25}$ - - -021627 -$\left ( 2,2\sqrt{2} \right )$ - - -021628 -(1) 以$C$为直角的直角三角形；\\ -(2) 以$A$为顶角的等腰三角形；\\ -(3) 以$A$为直角的直角三角形. - - -021629 -$a=\sqrt{13}$;$R=\dfrac{\sqrt{39}}{3}$. - - -021630 -$6\sqrt{19}$ - - -021631 -(1) $x=\arcsin \dfrac{2}{5}$或$\pi-\arcsin \dfrac{2}{5}$;\\ -(2) $x=\pi-\arccos \dfrac{2}{3}$或$\pi+\arccos \dfrac{2}{3}$;\\ -(3) $x=k\pi- \arctan \dfrac{1}{2},k \in \mathbf{Z}$. - - -021632 -$300\sqrt{3}$ - - -021633 -证明略 - - -021634 -$\theta=\dfrac{\pi}{12}$;塔高为$1.5$千米. - - -021635 -$64.81$米 - - -021636 -(1) $3.9$千米;(2) $4.0$千米. - - -021637 -$2.4$千米 - - -021638 -$\dfrac{\pi}{2}$ - - -021639 -B - - -021640 -(1) \begin{tikzpicture}[>=latex, scale = 0.7] -\draw [->] (-4,0) -- (4,0) node [below] {$x$}; -\draw [->] (0,-1.5) -- (0,2) node [left] {$y$}; -\draw (0,0) node [below right] {$O$}; -\draw (-pi,0.1) -- (-pi,0) node [below left] {$-\pi$}; -\draw (-0.5*pi,0.1) -- (-0.5*pi,0) node [below] {$-\frac{\pi}{2}$}; -\draw (0.5*pi,0.1) -- (0.5*pi,0) node [below] {$\frac{\pi}{2}$}; -\draw (pi,0.1) -- (pi,0) node [below] {$\pi$}; -\draw (0.1,1) -- (0,1) node [left] {$1$}; -\draw (0.1,-1) -- (0,-1) node [left] {$-1$}; -\draw [domain = -pi:pi,samples = 100] plot (\x,{sin(\x/pi*180)+1}); -\end{tikzpicture}\\ -(2) \begin{tikzpicture}[>=latex, scale = 0.7] -\draw [->] (0,0) -- (7,0) node [below] {$x$}; -\draw [->] (0,-1.5) -- (0,2) node [left] {$y$}; -\draw (0,0) node [below right] {$O$}; -\draw (pi/2,0.1) -- (pi/2,0) node [below] {$\frac{\pi}{2}$}; -\draw (pi,0.1) -- (pi,0) node [below] {$\pi$}; -\draw (1.5*pi,0.1) -- (1.5*pi,0) node [below] {$\frac{3\pi}{2}$}; -\draw (2*pi,0.1) -- (2*pi,0) node [below] {$2\pi$}; -\draw (0.1,1) -- (0,1) node [left] {$1$}; -\draw (0.1,-1) -- (0,-1) node [left] {$-1$}; -\draw [domain = 0:2*pi,samples = 100] plot (\x,{-cos(\x/pi*180)}); -\end{tikzpicture} - - -021641 -(1) 定义域为$\left \{x|x \neq-\dfrac{\pi}{2}+2k\pi,k \in \mathbf{Z} \right \}$;\\ -(2) 定义域为$\left \{x|\dfrac{\pi}{2}+2k\pi \leq x \leq \dfrac{3\pi}{2}+2k\pi,k \in \mathbf{Z} \right \}$. - - -021642 -$\left \{x|\dfrac{\pi}{6} \leq x \leq \dfrac{5\pi}{6},k \in \mathbf{Z} \right \}$ - - -021643 -$2\pi$ - - -021644 -C - - -021645 -C - - -021646 -(1) 当$a \in (-\infty，-\dfrac{\sqrt{2}}{2})\cup (1,+\infty)$ 时，方程实数解个数为$0$个；\\ -当$a \in [-\dfrac{\sqrt{2}}{2},0)\cup \{1\}$ 时，方程实数解个数为$1$个；\\ -当$a \in [0,1)$时，方程实数解个数为$2$个.\\ -(2) 当$a \in (-\infty，-1)\cup (1,+\infty)$ 时，方程实数解个数为$0$个；\\ -当$a \in (0,1]$时，方程实数解个数为$1$个;\\ -当$a \in \{0,-1\}$时，方程实数解个数为$2$个;\\ -当$a \in (-1,0)$时，方程实数解个数为$3$个. - - -021647 -(1) $8\pi$; -(2) $\pi$;(3) $\pi$;(4) $2\pi$. - - -021648 -$3$ - - -021649 -A - - -021650 -C - - -021651 -(1) 假;(1) 假;(3) 真. - - -021652 -D - - -021653 -(1) $\pi$; (2) $\pi$; (3) $\dfrac{\pi}{2}$; (4) $\dfrac{\pi}{|a|}$. - - -021654 -$4\sin(\dfrac{\pi x}{2})-2$ - - -021655 -B - - -021656 -A - - -021657 -(1) $f(3)=-1$; $f(5)=1$; $f(7)=-1$;\\ -(2) $T=4$. - - -021658 -$\left [2,4\right] $ - - -021659 -$\left [-2,2\right] $ - - -021660 -$ [-\dfrac{3}{2},3] $ - - -021661 -$ (-\dfrac{\sqrt{3}}{2},1] $ - - -021662 -$3$; $\left \{x|x=-\dfrac{\pi}{2}+2k\pi,k \in \bf{Z} \right\}$ - - -021663 -$-3$; $\left \{x|x=-\dfrac{\pi}{12}+k\pi,k \in \bf{Z} \right\}$ - - -021664 -当$x=\dfrac{\pi}{2}-\arcsin \dfrac{3\sqrt{13}}{13}+2k\pi,k \in \bf{Z}$时，函数的最大值为$\sqrt{13}$;\\ -当$x=-\dfrac{\pi}{2}-\arcsin \dfrac{3\sqrt{13}}{13}+2k\pi,k \in \bf{Z}$时，函数的最小值为$-\sqrt{13}$. - - -021665 -D - - -021666 -C - - -021667 -当$\alpha=\dfrac{\pi}{2}-\theta$时，竹竿的影子最长，最长为$\dfrac{\sin(\alpha+\theta)}{\sin \theta}*l$. - - -021668 -$[-1,1]$ - - -021669 -$\{x|x\neq 2k\pi,k \in \bf{Z}\}$;$(-\infty,0]$ - - -021670 -$k=3$或$-3$;$b=-1$ - - -021671 -当$x=0$时，函数$y$取到最大值，最大值为$0$;\\ -当$x=\dfrac{\pi}{4}$时，函数$y$取到最小值，最小值为$-1$. - - -021672 -$f(a)=\begin{cases} -a^2+2a+2, & a\leq -1,\\ -1, & -1=latex, scale = 0.7] -\draw [->] (-4,0) -- (4,0) node [below] {$x$}; -\draw [->] (0,-1.5) -- (0,2) node [left] {$y$}; -\draw (0,0) node [below left] {$O$}; -\draw ({-pi/12},0.1) -- ({-pi/12},0) node [below left] {$-\frac{\pi}{12}$}; -\draw ({pi/6},0.1) -- ({pi/6},0) node [below] {$\frac{\pi}{6}$}; -\draw ({5*pi/12},0.1) -- ({5*pi/12},0) node [below] {$\frac{5\pi}{12}$}; -\draw ({2*pi/3},0.1) -- ({2*pi/3},0) node [above] {$\frac{2\pi}{3}$}; -\draw ({11*pi/12},0.1) -- ({11*pi/12},0) node [below right] {$\frac{11\pi}{12}$}; -\draw (0.1,1) -- (0,1) node [left] {$1$}; -\draw (0.1,-1) -- (0,-1) node [left] {$-1$}; -\draw [domain = {-pi/12}:{11*pi/12},samples = 100] plot (\x,{sin(2*\x/pi*180+30)}); -\end{tikzpicture}\\ -(2) \begin{tikzpicture}[>=latex, scale = 0.7] -\draw [->] (-1,0) -- (15,0) node [below] {$x$}; -\draw [->] (0,-3) -- (0,3) node [left] {$y$}; -\draw (0,0) node [below left] {$O$}; -\draw (2*pi,0.1) -- (2*pi,0) node [below] {$2\pi$}; -\draw (pi,0.1) -- (pi,0) node [below] {$\pi$}; -\draw (3*pi,0.1) -- (3*pi,0) node [below] {$\frac{3\pi}{2}$}; -\draw (4*pi,0.1) -- (4*pi,0) node [below] {$4\pi$}; -\draw (0.1,2) -- (0,2) node [left] {$2$}; -\draw (0.1,-2) -- (0,-2) node [left] {$-2$}; -\draw [domain =0:4*pi,samples = 100] plot (\x,{2*sin(0.5*\x/pi*180)}); -\end{tikzpicture}\\ -(3) \begin{tikzpicture}[>=latex, scale = 0.7] -\draw [->] (-1,0) -- (4,0) node [below] {$x$}; -\draw [->] (0,-1) -- (0,1) node [left] {$y$}; -\draw (0,0) node [below right] {$O$}; -\draw (0.25*pi,0.1) -- (0.25*pi,0) node [below] {$\frac{\pi}{4}$}; -\draw (pi,0.1) -- (pi,0) node [below] {$\pi$}; -\draw (0.5*pi,0.1) -- (0.5*pi,0) node [below] {$\frac{\pi}{2}$}; -\draw (0.75*pi,0.1) -- (0.75*pi,0) node [above] {$\frac{3\pi}{4}$}; -\draw (0.1,0.5) -- (0,0.5) node [left] {$\frac{1}{2}$}; -\draw (0.1,-0.5) -- (0,-0.5) node [left] {$-\frac{1}{2}$}; -\draw [domain =0:pi,samples = 100] plot (\x,{0.5*sin(2*\x/pi*180)}); -\end{tikzpicture}\\ -(4) \begin{tikzpicture}[>=latex, scale = 0.7] -\draw [->] (-1.5,0) -- (3.5,0) node [below] {$x$}; -\draw [->] (0,-5.5) -- (0,5.5) node [left] {$y$}; -\draw (0,0) node [below left] {$O$}; -\draw ({-pi/3},0.1) -- ({-pi/3},0) node [below left] {$-\frac{\pi}{3}$}; -\draw ({-pi/12},0.1) -- ({-pi/12},0) node [above left] {$-\frac{\pi}{12}$}; -\draw ({pi/6},0.1) -- ({pi/6},0) node [below right] {$\frac{\pi}{6}$}; -\draw ({5*pi/12},0.1) -- ({5*pi/12},0) node [below] {$\frac{5\pi}{12}$}; -\draw ({2*pi/3},0.1) -- ({2*pi/3},0) node [above right] {$\frac{2\pi}{3}$}; -\draw ({11*pi/12},0.1) -- ({11*pi/12},0) node [below right] {$\frac{11\pi}{12}$}; -\draw (0.1,5) -- (0,5) node [left] {$5$}; -\draw (0.1,-5) -- (0,-5) node [below left] {$-5$}; -\draw [domain = {-4*pi/12}:{2*pi/3},samples = 100] plot (\x,{5*sin(2*\x/pi*180-60)}); -\end{tikzpicture} - - -021695 -$4\pi$;$4$. - - -021696 -$f(x)=4\sin(x+\dfrac{\pi}{6})$ - - -021697 -(1) $f(x)=\dfrac{\sqrt{3}}{2}\sin(3x+\pi)+\dfrac{\sqrt{3}}{2};$\\ -(2) $[-\dfrac{\pi}{2}+\dfrac{2k\pi}{3},-\dfrac{\pi}{6}+\dfrac{2k\pi}{3}],k \in \bf{Z}$;\\ -(3) 函数最大值为$\sqrt{3}$,此时$x$值为${x|x=-\dfrac{\pi}{6}+\dfrac{2k\pi}{3},k \in \bf{Z}}$ - - -021698 -$x=\pi+2k\pi,k \in \bf{Z}$ - - -021699 -纵;伸长; $3$. - - -021700 -缩短; $\dfrac{1}{2}$; 缩短; $\dfrac{1}{3}$. - - -021701 -$f(x)=\sin(\dfrac{1}{2}x-\dfrac{\pi}{3})$ - - -021702 -$f(x)=\sin(\dfrac{1}{2}x-\dfrac{\pi}{6})$ - - -021703 -$f(x)=2\sin(\dfrac{1}{3}x+\dfrac{\pi}{6})$ - - -021704 -$x=\dfrac{\pi}{3}+2k\pi,k \in \bf{Z}$; $(-\dfrac{2\pi}{3}+2k\pi,0),k \in \bf{Z}$. - - -021705 -C - - -021706 -左; $\dfrac{\pi}{8}$. - - -021707 -$f(x)=\sin(2x+\dfrac{\pi}{2})$, -$g(x)=\sin x$. - - -021708 -(1) $\sqrt{2}$; -(2) $g(x)=2\cos(\dfrac{1}{2}x-\dfrac{\pi}{3}) $, 单调递减区间为$[\dfrac{2\pi}{3}+4k\pi,\dfrac{8\pi}{3}+4k\pi],k \in \bf{Z}$. - - -021709 -(1) $2\pi$; (2) $1$; (3) $\dfrac{\pi}{2}$. - - -021710 -(1) $[0,\dfrac{\pi}{2})$, $(\dfrac{3\pi}{2},2\pi]$; \\ -(2) $[0,\dfrac{\pi}{2})$, $(\dfrac{\pi}{2},\pi]$. - - -021711 -(1) 奇函数; (2) 偶函数. - - -021712 -$[-5,+\infty)$ - - -021713 -(1) $<$; (2) $>$; (3) $>$; (4)$<$. - - -021714 -\textcircled{3} - - -021715 -最小值为$-\dfrac{\sqrt{3}}{3}$，此时$x=-\dfrac{\pi}{3}$. - - -021716 -(1) $ \{x|x \neq \dfrac{k\pi}{2},k \in \bf{Z}\} $;\\ -(2) 单调增区间为$(-\dfrac{\pi}{2}+\dfrac{k\pi}{2},\dfrac{k\pi}{2}), k \in \bf{Z}$. - - -021717 -$\{x|x\neq \dfrac{\pi}{4}-\dfrac{1}{2}+\dfrac{k\pi}{2},k \in \bf{Z} \}$ - - -021718 -$(-\dfrac{\pi}{4}+\dfrac{k\pi}{3},\dfrac{\pi}{12}+\dfrac{k\pi}{3}), k \in \bf{Z}$ - - -021719 -B - - -021720 -定义域为$ \{x|x \neq \dfrac{7\pi}{5}+2k\pi,k \in \bf{Z}\} $;\\ -严格增区间为$(-\dfrac{3\pi}{5}+2k\pi,\dfrac{7\pi}{5}+2k\pi), k \in \bf{Z}$. - - -021721 -函数零点为$x=\dfrac{2k\pi}{5}+2k\pi,k \in \bf{Z}$. - - -021722 -(1) 假命题; (2) 假命题; (3) 假命题; (4) 真命题. - - -021723 -$[-4,2+4\sqrt{3}]$ - - -021724 -最大张角的正切值为$\dfrac{\sqrt{2}}{4}$, 此时学生距离时钟$\sqrt{0.18}$米. - - -021726 -A - - -021727 -C - - -021728 -B - - -021729 -单位圆 - - -021730 -B - - -021731 -$\overrightarrow{CD}$ - - -021732 -$\overrightarrow{AC}$ - - -021733 -(1) 假命题; (2) 真命题; (3) 假命题; (4) 假命题. - - -021734 -(1) $\overrightarrow{DB}$; $\overrightarrow{FE}$.\\ -(2) $\overrightarrow{ED}$; $\overrightarrow{CF}$; $\overrightarrow{FA}$.\\ -(3) $\overrightarrow{EF}$; $\overrightarrow{AD}$; $\overrightarrow{DA}$; $\overrightarrow{DB}$; $\overrightarrow{BD}$; $\overrightarrow{AB}$; $\overrightarrow{BA}$. - - -021735 -$40$ - - -021736 -$40$ - - -021737 -$2$ - - -021739 -$-3\overrightarrow {a}+6 \overrightarrow {b}$ - - -021740 -$7 \overrightarrow {a}-2 \overrightarrow {b}- \overrightarrow {c}$ - - -021741 -(1) 假命题; (2) 真命题; (3) 假命题; (4) 真命题. - - -021742 -(1) $\overrightarrow {AB}=\dfrac{1}{2}\overrightarrow {a}-\dfrac{1}{2}\overrightarrow {b}$;\\ -(2) $\overrightarrow {BC}=\dfrac{1}{2}\overrightarrow {a}+\dfrac{1}{2}\overrightarrow {b}$. - - -021743 -$\lambda=\dfrac{1}{3}$ - - -021744 -$x=2$; $y=1$. - - -021745 -(2) $m=1$或$-1$. - - -021746 -$\overrightarrow{DC}=\dfrac{1}{2}\overrightarrow{a}$;\\ $\overrightarrow{DC}=-\dfrac{1}{2}\overrightarrow{a}+\overrightarrow{b}$;\\ -$\overrightarrow{MN}=-\dfrac{1}{4}\overrightarrow{a}-\overrightarrow{b}$. - - -021747 -$\overrightarrow{0}$ - - -021748 -$\dfrac{2}{3}\overrightarrow{a}+\dfrac{1}{3}\overrightarrow{b}$ - - -021749 -A - - -021750 -B - - -021751 -C - - -021752 -$\sqrt{3}$ - - -021753 -$-\dfrac{3\sqrt{3}}{2}$ - - -021754 -等边三角形 - - -021755 -$\dfrac{\pi}{4}$ - - -021756 -$\dfrac{2\pi}{3}$ - - -021757 -$-10\sqrt{2}$ - - -021758 -$\dfrac{4}{3}$ - - -021759 -$-\dfrac{2}{3}\overrightarrow {a}$ - - -021760 -B - - -021761 -B - - -021762 -A - - -021763 -$7$ - - -021764 -$2$ - - -021765 -C - - -021766 -外心; 重心; 垂心. - - -021767 -$\dfrac{\pi}{3}$ - - -021768 -$-25$ - - -021769 -$\lambda=\dfrac{7}{12}$ - - -021770 -$AB=8$ - - -021771 -$t=\dfrac{1}{3}$ - - -021772 -(1) $(\overrightarrow {a}-\overrightarrow {b}) \cdot \overrightarrow {c}=\overrightarrow {a} \cdot \overrightarrow {c}- \overrightarrow {b} \cdot \overrightarrow {c}=1*1*(-\dfrac{1}{2})-1*1*(-\dfrac{1}{2})=0;\\$ -(2) $k<0$或$k>2$. - - -021773 -$[2,5]$ - - -021774 -$\arccos \dfrac{4}{5}$ - - -021775 -$\overrightarrow{OP}=\dfrac{3}{11}\overrightarrow {a}+\dfrac{2}{11}\overrightarrow {b}$ - - -021776 -(1) $(-1,0)$; (2) $(2,\dfrac{1}{2})$; (3) $(2,0)$或 $(-2,0)$; (4) $(\dfrac{3\sqrt{2}}{2},-\dfrac{3\sqrt{2}}{2})$. - - -021777 -(1) 10; (2) $(-\dfrac{4}{5},\dfrac{3}{5})$. - - -021778 -$x=4$, $y=1$. - - -021779 -$(\dfrac{3}{5},-\dfrac{4}{5})$ - - -021780 -$(4,-8)$ - - -021781 -$(1,2)$ - - -021782 -C - - -021783 -A - - -021784 -B - - -021786 -$\lambda=\mu$ 且$\lambda$和$\mu$非零. - - -021787 -(1) 当$t=\dfrac{3}{2}$时,点$P$在$x$轴上; 当$t=\dfrac{1}{3}$时,点$P$在$y$轴上;当$-\dfrac{2}{3}