From 6abbf7400ba22674e3c816f8bed325427d9edf85 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Thu, 20 Apr 2023 22:02:16 +0800 Subject: [PATCH] =?UTF-8?q?=E6=B5=A6=E4=B8=9C=E9=AB=98=E4=BA=8C=E7=BB=9F?= =?UTF-8?q?=E8=80=83=E8=AF=95=E5=8D=B7=E6=8C=82=E9=92=A9=E7=AC=AC=E4=B8=83?= =?UTF-8?q?=E5=8D=95=E5=85=83?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 工具/文本文件/metadata.txt | 858 ++----------------------------------- 题库0.3/Problems.json | 84 +++- 2 files changed, 106 insertions(+), 836 deletions(-) diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt index 71d5dd79..f0be1658 100644 --- a/工具/文本文件/metadata.txt +++ b/工具/文本文件/metadata.txt @@ -1,858 +1,86 @@ -ans +tags -021441 -错误, 正确, 错误, 错误 +15311 +第七单元 -021442 -D +15312 +第七单元 -021443 -C +15313 +第七单元 -021444 -A +15314 +第七单元 -021445 -C +15315 +第七单元 -021446 -D +15316 +第七单元 -021447 -$-390^\circ$ +15317 +第七单元 -021448 -$304^\circ$, $-56^\circ$ +15318 +第七单元 -021449 -$-144^\circ$ +15319 +第七单元 -021450 -二, 四 +15320 +第七单元 -021451 -(1) $\{\alpha|\alpha=60^\circ+k\cdot 360^\circ, \ k\in \mathbf{Z}\}$, $-300^\circ$, $60^\circ$, $420^\circ$; (2) $\{\alpha|\alpha = -21^\circ+k\cdot 360^\circ, \ k \in \mathbf{Z}\}$, $-21^\circ$, $339^\circ$, $699^\circ$ +15321 +第七单元 -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} +15322 +第七单元 -021453 -$-1290^{\circ}$;第二象限 +15323 +第七单元 -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) $\{\alpha|180^{\circ}+k\cdot 360^{\circ}<\alpha<270^{\circ}+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$. +15324 +第七单元 -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) $\{\beta|\beta=90^{\circ}-\alpha+k\cdot 360^{\circ}, \ k \in \mathbf{Z}\}$. +15325 +第七单元 -021456 -C +15326 +第七单元 -021457 -B +15327 +第七单元 -021458 -$\dfrac{\pi}{12}$; $\dfrac{7\pi}{12}$; $\dfrac{5\pi}{4}$; $300^{\circ}$; $324^{\circ}$; $315^{\circ}$; $(\dfrac{270}{\pi})^{\circ}$ +15328 +第七单元 -021459 -(1)$\frac{50\pi+180}{9}$;(2)$\frac{250\pi}{9}$ +15329 +第七单元 -021460 -$\sqrt{3}$ +15330 +第七单元 -021461 -(1)$\frac{\pi}{3}$;(2)$\frac{2\pi}{3}$ - - -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. - - -021463 -$\frac{1}{2}$ - - -021464 -(1) $\{\alpha|-\frac{\pi}{2}+2k\pi<\alpha<2k\pi,\ k \in \mathbf{Z}\}$;\\ -(2) $\{\alpha|\alpha=\frac{k\pi}{2},\ k \in \mathbf{Z}\}$. - - -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=\left\{ - \begin{array}{lc} - $1$, & k=2n \\ - $ -1$ , &k=2n-1\\ - \end{array} -\right.$ ($n \in \mathbf{Z}$). - - -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 -$1$ - - -021502 -$-\dfrac{12}{5}$ - - -021503 -$-\dfrac{\sqrt{3}}{2}$ - - -021504 -$\dfrac{\sqrt{7}}{2}$;$\dfrac{\sqrt{7}}{4}$ - - -021505 -$-\dfrac{\sqrt{11}}{3}$ - - -021506 -$\dfrac{\pi}{3}$ - - -021507 -$\left[ 0,\pi \right )$ - - -021508 -$-\dfrac{\sqrt{3}}{2}$;$-\dfrac{\sqrt{2}}{2}$;$-\sqrt{3}$;$-\sqrt{3}$ - - -021509 -$69^{\circ}$;$72^{\circ}$;$\dfrac{\pi}{9}$;$\dfrac{7 \pi}{15}$ - - -021510 -$\cot \alpha$ - - -021511 -$-1$ - - -021512 -$-1$ - - -021513 -$ \sin 2-\cos 2$ - - -021514 -$0$ - - -021515 -$0$ - - -021516 -$-\dfrac{\sqrt{1-a^2}}{a}$ - - -040018 -(1) $\dfrac{\pi}{4}$; (2) $\dfrac{\pi}{6}$; (3) $\dfrac{\pi}{10}$; (4) $\dfrac{\pi}{3}$; (5) $\dfrac{5\pi}{12}$; (6) $\dfrac{\pi}{15}$ - - -040019 -(1) $60^{\circ}$; (2) $36^{\circ}$; (3) $45^{\circ}$; (4) $75^{\circ}$; (5) $40^{\circ}$; (6) $54^{\circ}$ - - -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) $\dfrac{k\pi}{2}+\dfrac{\pi}{6}$ - - -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) $k \times 90^{\circ}+50^{\circ}$ - - -040022 -(1) $330^{\circ}$; (2) $240^{\circ}$; (3) $210^{\circ}$; (4) $300^{\circ}$ - - -040023 -(1) $\dfrac{4\pi}{3}$; (2) $\dfrac{11\pi}{6}$; (3) $10-2\pi$; (4) $-10+4\pi$ - - -040024 -$18$ - - -040025 -$3$,$-2$ - - -040026 -(1) $1037$; (2) $-4k+53$; (3) $500$ - - -040027 -$-2n+10$ - - -040028 -15 - - -040029 -$7$ - - -040030 -$(4,\dfrac{14}{3}]$ - - -040031 -$2n-1$ - - -040032 -$(3,\dfrac{35}{9})$或$(\dfrac{35}{9},3)$ - - -040033 -$200$ - - -040034 -略 - - -040035 -$a_n=\begin{cases}1, & n=1,\\ 2n, & n=2k, \\ 2n-2, & n=2k+1\end{cases}$($k\in \mathbf{N}$, $k\ge 1$) - - -040036 -$6n-3$ - - -040057 -$\dfrac{19}{28}\sqrt{7}$ - - -040058 -$\dfrac{79}{156}$ - - -040059 -$2$ - - -040060 -$-\dfrac{\sqrt{1-m^2}}{m}$ - - -040061 -$-\dfrac{1}{5}, \dfrac{1}{5}$ - - -040062 -$-\dfrac{1}{3}, 3$ - - -040063 -$\dfrac{1}{2}, -2$ - - -040064 -$\dfrac{\sqrt{6}}{3}$ - - -040065 -$\dfrac{1}{3}, -\dfrac{9}{4}$ - - -040066 -$\dfrac{1}{3}, \dfrac{7}{9}$ - - -040067 -$\pm\dfrac{\sqrt{2}}{3}$ - - -040068 -$\dfrac{1}{4}, \dfrac{2}{5}$ - - -040069 -$\dfrac{1-\sqrt{17}}{4}$ - - -040070 -(1) 三; (2) 三 - - -040071 -(1) $[-\dfrac{1}{2},\dfrac{1}{2})\cup\{1\}$; (2) $[-\dfrac{\pi}{3},\dfrac{\pi}{3})$; (3) $\{-\dfrac{1}{2}\}$ - - -040072 -(1) $-\tan \alpha-\cot \alpha$; (2) $-\dfrac{\sqrt{2}}{\sin \alpha}$; (3) $-1$; (4) $0$ - - -040073 -略 - - -040074 -$-\dfrac{10}{9}$ - - -040075 -$a_n=\dfrac{1}{3n-2}$ - - -040076 -$a_n=\dfrac{1}{n}$ - - -040077 -$(n-\dfrac{4}{5})5^n$ - - -040078 -$2^{n+1}-3$ - - -040079 -$1078$ - - -040080 -$S_n=\begin{cases}\dfrac{n^2}{2}+n-\dfrac 23+\dfrac 23\cdot 2^n, & n\text{为偶数},\\ \dfrac{n^2}{2}-\dfrac 76+\dfrac 23\cdot 2^{n+1}, & n\text{为奇数} \end{cases}$ - - -040081 -(1) 略; (2) $n^2$ - - -040082 -(1) 不存在; (2) 存在, 如$c_n=2^{n-1}$ - - -040083 -$\dfrac{\sqrt{3}}{2}$ - - -040084 -$0$ - - -040085 -$\{0,-2\pi\}$ - - -040086 -$-\dfrac{\pi}6,\dfrac 56\pi$ - - -040087 -$\cot \alpha$ - - -040088 -$7+4\sqrt{3}$ - - -040089 -$\dfrac{\sqrt{2}-\sqrt{6}}{4}$ - - -040090 -$\dfrac{\sqrt{3}+\sqrt{35}}{12}$ - - -040091 -$\dfrac 12$ - - -040092 -$5$ - - -040093 -$-\dfrac 12$ - - -040094 -$\dfrac{\pi}{12}$ - - -040095 -$\{x|x=\pm\frac 23 \pi+2k\pi,k \in \mathbf{Z}\}$ - - -040096 -$\dfrac 43 \pi$ - - -040097 -\textcircled{4} - - -040098 -C - - -040099 -$\dfrac{-2\sqrt{2}-\sqrt{3}}6$ - - -040100 -$-\dfrac 7{25}$ - - -040101 -$-\dfrac {\pi}3$ - - -040102 -$(-\dfrac {12}{13}, \dfrac{5}{13})$ - - -040103 -$(\dfrac {5-12\sqrt{3}}{2}, \dfrac{12-5\sqrt{3}}{2})$ - - -040104 -略 - - -040105 -$\dfrac {171} {221}, -\dfrac {21} {221}$ - - -040106 -$\{-\pi\}$ - - -040107 -$\dfrac{8\sqrt{2}-3}{15}$ - - -040108 -$\sin \theta$ - - -040109 -$-\dfrac{56}{65}$ - - -040110 -$\dfrac {\pi}4$ - - -040111 -略 - - -040112 -略 - - -040131 -$-\dfrac{25}{12}$ - - -040132 -$\dfrac 52$ - - -040133 -$-\dfrac{\pi}4$ - - -040134 -$-\dfrac 12$ - - -040135 -$\dfrac 6{19}$ - - -040136 -$-\dfrac {\sqrt{3}}3$ - - -040137 -$\dfrac 3{22}$ - - -040138 -$4$ - - -040139 -$-\dfrac{63}{65}$ - - -040181 -$\dfrac 7{25}$ - - -040182 -$-\dfrac{\pi}3+2k\pi,k \in \mathbf{Z}$ - - -040183 -$\dfrac{4\sqrt{3}-3}{10}$ - - -040184 -$\dfrac 17$ - - -040185 -$4\sqrt{2} \sin(\alpha+\dfrac {7}{4}\pi))$ - - -040186 -$3$ - - -040187 -$\dfrac 32$ - - -040188 -$\sqrt{3}$ - - -040189 -$2$ - - -040190 -$\dfrac {13}{18}$ - - -040191 -$\dfrac{7}{4}\pi$ - - -040192 -$\dfrac{64}{25}$ - - -040193 -C - - -040194 -A - - -040195 -B - - -040196 -C - - -040197 -$-\dfrac{\pi}6$ - - -040198 -$\dfrac 23 \pi$ - - -040199 -$\dfrac 32$ - - -040200 -$\sqrt{1-k}$ - - -040201 -$-\dfrac{484}{729}$ - - -040226 -$\dfrac 49 \sqrt{2}$ - - -040227 -$\sin \theta \cos \theta$ - - -040228 -$-\dfrac1{16}$ - - -040229 -$\dfrac 32$ - - -040230 -$\dfrac{13}{18}$ - - -040231 -$-2-\sqrt{7}$ - - -040232 -$\sin{\dfrac{\alpha}2}$ - - -040233 -$0$ - - -040234 -$\dfrac{120}{169}$ - - -040235 -$3$或$5$ - - -040236 -$\pi-\arcsin{\dfrac{24}{25}}$ - - -040237 -$\arcsin{\dfrac{3\sqrt{10}}{10}}$或$\arcsin{\dfrac{\sqrt{10}}{10}}$ - - -040238 -$60^{\circ}$或$120^{\circ}$ - - -040239 -$\dfrac 23 \pi$ - - -040240 -$8$ - - -040241 -\textcircled{4} - - -040242 -$\dfrac 35$或$\dfrac{24}{25}$或$\dfrac{3\sqrt{10}}{10}$或$\dfrac{\sqrt{10}}{10}$ - - -040243 -(1)$\angle A=75^{\circ}, \angle B=45^{\circ}, a=\sqrt{2}+\sqrt{6}$\\ -(2) $\angle B=60^{\circ}, \angle C=75^{\circ}, c=\sqrt{6}+3\sqrt{2}$或 -$\angle B=120^{\circ}, \angle C=15^{\circ}, c=3\sqrt{2} - \sqrt{6}$ - - -040244 -$\dfrac 12$ - - -040245 -$\dfrac 12 \pm \dfrac{\sqrt{6}}5$ +15331 +第七单元 diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 02673058..77e5c80d 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -378058,7 +378058,9 @@ "id": "015311", "content": "直线$x-y+3=0$的倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378077,7 +378079,9 @@ "id": "015312", "content": "双曲线$\\dfrac{x^2}{2}-\\dfrac{y^2}{3}=1$的焦距为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378096,7 +378100,9 @@ "id": "015313", "content": "过点$(1,1)$且与直线$x+2 y-1=0$平行的直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378115,7 +378121,9 @@ "id": "015314", "content": "己知椭圆$\\dfrac{x^2}{4}+y^2=1$的焦点分别为$F_1$、$F_2$, 过$F_1$的直线交椭圆于$A$、$B$两点, 则$\\triangle ABF_2$的周长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378134,7 +378142,9 @@ "id": "015315", "content": "若抛物线$y^2=8 x$上一点$A$的横坐标为 4 , 则点$A$与抛物线焦点的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378153,7 +378163,9 @@ "id": "015316", "content": "如果方程$(m+1) x^2+(2-m) y^2=1$表示焦点在$y$轴上的双曲线, 则实数$m$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378172,7 +378184,9 @@ "id": "015317", "content": "已知圆$C_1: x^2+y^2=4$和圆$C_2: x^2+y^2-6 x+8 y+25-m^2=0$($m>0$)外切, 则实数$m$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378191,7 +378205,9 @@ "id": "015318", "content": "若直线$a x-y+3=0$与直线$x-2 y+4=0$的夹角为$\\arccos \\dfrac{\\sqrt{5}}{5}$, 则实数$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378210,7 +378226,9 @@ "id": "015319", "content": "己知动点$M(a, b)$在直线$3 x+4 y+10=0$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": "", @@ -378229,7 +378247,9 @@ "id": "015320", "content": "古希腊著名数学家阿波罗尼斯发现: ``平面内到两个定点$A$、$B$的距离之比为定值$\\lambda$($\\lambda \\neq 1$)的点的轨迹是圆''. 后来人们将这个圆以他的名字命名, 称为阿波罗尼斯圆. 在平面直角坐标系$xOy$中, $A(2,0)$, $B(8,0)$, $\\dfrac{|PA|}{|PB|}=\\dfrac{1}{2}$, 则点$P$的轨迹方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378248,7 +378268,9 @@ "id": "015321", "content": "如图所示, 为完成一项探月工程, 某月球探测器飞行到月球附近时, 首先在以月球球心$F$为圆心的圆形轨道 I 上绕月球飞行, 然后在$P$点处变轨进入以$F$为一个焦点的椭圆轨道 II 绕月球飞行, 最后在$Q$点处变轨进入以$F$为圆心的圆形轨道 III 绕月球飞行, 设圆形轨道 I 的半径为$R$, 圆形轨道 III 的半径为$r$, 则椭圆轨道 II 的离心率为\\blank{50}.(用$R$、$r$表示)\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\filldraw (0,0) node [below] {$F$} coordinate (F) circle (0.06);\n\\draw (F) circle (1) circle (3);\n\\draw (1,0) ellipse (2 and {sqrt(3)});\n\\draw [dashed] (-4,0) -- (4,0);\n\\draw (-1,0) node [below left] {$Q$} coordinate (Q) (3,0) node [below right] {$Q$} coordinate (Q);\n\\draw (15:1) node [above right] {III};\n\\draw (1,2) node [left] {II};\n\\draw (0,3) node [above] {I};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378267,7 +378289,9 @@ "id": "015322", "content": "已知点$M$、$N$分别是椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}{3}=1$上两动点, 且直线$OM$、$ON$的斜率的乘积为$-\\dfrac{3}{4}$, 若椭圆一点$P$满足$\\overrightarrow{OP}=\\lambda \\overrightarrow{OM}+\\mu \\overrightarrow{ON}$, 则$\\lambda^2+\\mu^2$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -378286,7 +378310,9 @@ "id": "015323", "content": "若直线$l$经过点$A(2,-3)$、$B(3,1)$, 则以下不是直线$l$的方程的为\\bracket{20}.\n\\fourch{$y+3=4(x-2)$}{$y-1=4(x-3)$}{$4 x-y-11=0$}{$\\dfrac{y+3}{1}=\\dfrac{x-2}{4}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -378305,7 +378331,9 @@ "id": "015324", "content": "在下列双曲线中, 与$x^2-\\dfrac{y^2}{4}=1$共渐近线的为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{16}-\\dfrac{y^2}{4}=1$}{$\\dfrac{x^2}{4}-\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}{2}-y^2=1$}{$x^2-\\dfrac{y^2}{2}=1$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -378324,7 +378352,9 @@ "id": "015325", "content": "已知椭圆$C: \\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$, 直线$l: (m+2) x-(m+4) y+2-m=0$($m \\in \\mathbf{R}$), 则直线$l$与椭圆$C$的位置关系为\\bracket{20}.\n\\fourch{相交}{相切}{相离}{不确定}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -378343,7 +378373,9 @@ "id": "015326", "content": "小明同学在完成教材椭圆和双曲线的相关内容学习后, 提出了新的疑问: 平面上到两个定点距离之积为常数的点的轨迹是什么呢? 又具备哪些性质呢? 老师特别赞赏他的探究精神, 并告诉他这正是历史上法国天文学家卡西尼在研究土星及其卫星的运行规律时发现的, 这类曲线被称为``卡西尼卵形线''. 在老师的鼓励下, 小明决定先从特殊情况开始研究, 假设$F_1(-1,0)$、$F_2(1,0)$是平面直角坐标系$x O y$内的两个定点, 满足$|PF_1| \\cdot|PF_2|=2$的动点$P$的轨迹为曲线$C$, 从而得到以下$4$个结论:\\\\\n\\textcircled{1} 曲线$C$既是轴对称图形, 又是中心对称图形;\\\\\n\\textcircled{2} 动点$P$的横坐标的取值范围是$[-\\sqrt{3}, \\sqrt{3}]$;\\\\\n\\textcircled{3} $|OP|$的取值范围是$[1, \\sqrt{3}]$;\\\\\n\\textcircled{4} $\\triangle PF_1F_2$的面积的最大值为$1$.\\\\\n其中正确结论的个数为\\bracket{20}.\n\\fourch{1}{2}{3}{4}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -378362,7 +378394,9 @@ "id": "015327", "content": "己知直线$l_1: (m-1) x+2 y-m=0$与直线$l_2: x+m y+m-2=0$.\\\\\n(1) 若$l_1$与$l_2$垂直, 求实数$m$的值;\\\\\n(2) 若$l_1$与$l_2$平行, 求实数$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378381,7 +378415,9 @@ "id": "015328", "content": "已知圆$C: x^2+y^2=25$, 点$P(3,4)$.\\\\\n(1) 求过点$P$的圆$C$的切线$l$的方程;\\\\\n(2) 若直线$m$过点$P$且被圆$C$截得的弦长为$8$, 求直线$m$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378400,7 +378436,9 @@ "id": "015329", "content": "已知抛物线$y^2=2 p x$($p>0$), 其焦点$F$到准线的距离为$2$.\\\\\n(1) 求抛物线的标准方程;\\\\\n(2) 若$O$为坐标原点, 斜率为$2$且过焦点$F$的直线$l$交此抛物线于$A$、$B$两点, 求$\\triangle AOB$的面积.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378419,7 +378457,9 @@ "id": "015330", "content": "已知双曲线$C: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的实轴长为$4 \\sqrt{2}$, 离心率为$\\dfrac{\\sqrt{6}}{2}$. 动点$P$是双曲线$C$上任意一点.\\\\\n(1) 求双曲线$C$的标准方程;\\\\\n(2) 已知点$A(3,0)$, 求线段$AP$的中点$Q$的轨迹方程;\\\\\n(3) 已知点$A(3,0)$, 求$|AP|$的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -378438,7 +378478,9 @@ "id": "015331", "content": "已知椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的右顶点为$A(2,0)$, 短轴长为$2 \\sqrt{3}$, $F_1$、$F_2$是椭圆的两个焦点.\\\\\n(1) 求椭圆$C$的方程;\\\\\n(2) 已知$P$是椭圆$C$上的点, 且$\\angle F_1PF_2=\\dfrac{\\pi}{3}$, 求$\\triangle F_1PF_2$的面积;\\\\\n(3) 若过点$G(3,0)$且斜率不为$0$的直线$l$交椭圆$C$于$M$、$N$两点, $O$为坐标原点. 问: $x$轴上是否存在定点$T$, 使得$\\angle MTO=\\angle NTA$恒成立. 若存在, 请求出点$T$的坐标; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "",