<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% -*- coding: utf-8 -*-
% !TEX program = xelatex
\documentclass[chinese]{jnuexam} % ä¸­æ–‡è¯•å·

%\SetExamBoolFalse{answer} %ä¸æ˜¾ç¤ºç­”æ¡ˆ

\SetExamOption{
  binding = 2,     % è£…è®¢çº¿ï¼Œ1 ä»…ç©ºç™½è¯•å·æœ‰ï¼Œ2 è¯•å·å’Œç­”æ¡ˆéƒ½æœ‰
  scratch = 1,     % è‰ç¨¿çº¸æ•°é‡ï¼Œä»…ç©ºç™½è¯•å·æœ‰ï¼ŒA3 å¤§å°ï¼ŒåŒé¢å°åˆ·
  seed = 19061116, % éšæœºæ•°ç§å­ï¼Œç”¨äºŽæ”¹å˜ B å·å°é¢˜çš„éšæœºé¡ºåº
}

\begin{document}

\examtitle{ % ç”Ÿæˆè¯•å·è¡¨å¤´
  niandu  = 2017--2018,
  xueqi   = 2,
  kecheng = å¤§å­¦æ•°å­¦,
  zhuanye = ç†å·¥å››å­¦åˆ†, % å¯ä»¥ä¸ºç©ºç™½
  jiaoshi = {å¼&nbsp;ä¸‰ï¼ŒæŽå››ï¼ŒçŽ‹äº”}, % æ•™å¸ˆå§“å
  shijian = 2018~å¹´~06~æœˆ~28~æ—¥,
  bixiu   = 1, % 1 ä¸ºå¿…ä¿®ï¼Œ0 ä¸ºé€‰ä¿®
  bijuan  = 1, % 1 ä¸ºé—­å·ï¼Œ0 ä¸ºå¼€å·
  shijuan = A, % A æˆ– B æˆ– C å·
  neizhao = 1, % 1 æ‰“å‹¾ï¼Œ0 ä¸å‹¾
  waizhao = 0, % 1 æ‰“å‹¾ï¼Œ0 ä¸å‹¾
}

\gradetable[total=4] % ç”Ÿæˆè¯„åˆ†è¡¨

\exampart{å¡«ç©ºé¢˜}[å…±~6~å°é¢˜ï¼Œæ¯å°é¢˜~3~åˆ†ï¼Œå…±~18~åˆ†]

\answertable[total=6,column=3,strut=3em] % ç”Ÿæˆç­”é¢˜æ&nbsp;ï¼šè¡Œé«˜3emï¼Œæ€»å…±6é¢˜ï¼Œæ¯è¡Œ3é¢˜

\begin{question}
è®¾å¸¸æ•°$k&gt;0$ï¼Œå‡½æ•°$f(x)=\ln x-\frac{x}{\e}+k$åœ¨$(0,+\infty)$å†…é›¶ç‚¹çš„ä¸ªæ•°ä¸º \fillout{$2$}ï¼Ž
\end{question}

\vfill

\begin{question}
è®¾$\va=(2,1,2)$ï¼Œ$\vb=(4,-1,10)$ï¼Œ$\vc=\vb-\lambda\va$ï¼Œä¸”$\va\bot\vc$ï¼Œåˆ™$\lambda=$ \fillout{$3$}ï¼Ž
\end{question}

\vfill

\begin{question}
å·²çŸ¥äºŒé˜¶è¡Œåˆ—å¼ $\left|\begin{array}{cc}
  1 &amp; 2\\
  - 3 &amp; x
\end{array}\right|=0$ï¼Œåˆ™ $x=$ \fillout{$-6$}ï¼Ž
\end{question}

\vfill

\begin{question}
å‘é‡ç»„ $\alpha_1=(1,1,0), \alpha_2=(0,1,1), \alpha_3=(1,0,1)$ï¼Œ
åˆ™å°†å‘é‡ $\beta=(4, 5, 3)$ è¡¨ç¤ºä¸º $\alpha_1, \alpha_2, \alpha_3$
çš„çº¿æ€§ç»„åˆä¸º $\beta=$ \fillout{$3\alpha_1+2\alpha_2+\alpha_3$}ï¼Ž
\end{question}

\vfill

\begin{question}
å·²çŸ¥éšæœºå˜é‡$\xi$çš„æœŸæœ›å’Œæ–¹å·®å„ä¸º$E\xi=3, D\xi=2$, åˆ™$E\xi^2=$ \fillout{$11$}ï¼Ž
\end{question}

\vfill

\begin{question}
å·²çŸ¥$\xi$å’Œ$\eta$ç›¸äº’ç‹¬ç«‹ä¸”$\xi\sim N(1,4), \eta\sim N(2,5)$ï¼Œåˆ™$\xi-2\eta\sim$ \fillout{$N(-3,24)$}ï¼Ž
\end{question}

\vfill

\newpage

\exampart{å•é€‰é¢˜}[å…±~6~å°é¢˜ï¼Œæ¯å°é¢˜~3~åˆ†ï¼Œå…±~18~åˆ†]

\answertable[total=6,column=6] % ç”Ÿæˆç­”é¢˜æ&nbsp;ï¼šé»˜è®¤è¡Œé«˜ï¼Œæ€»å…±6é¢˜ï¼Œæ¯è¡Œ6é¢˜

\begin{question}
åœ¨ä¸‹åˆ—ç­‰å¼ä¸­ï¼Œæ­£ç¡®çš„ç»“æžœæ˜¯\pickout{C}
\begin{abcd}
\item $\int f'(x)\dx=f(x)$
\item $\int \d f(x)=f(x)$
\item $\frac{\d}{\dx}\big(\int f(x)\dx\big)=f(x)$
\item $\d\big(\int f(x)\dx\big)=f(x)$
\end{abcd}
\end{question}

\bigskip

\begin{question}
å‡è®¾$F(x)$æ˜¯è¿žç»­å‡½æ•°$f(x)$çš„ä¸€ä¸ªåŽŸå‡½æ•°ï¼Œåˆ™å¿…æœ‰\pickout{A}
\begin{abcd}
\item $F(x)$æ˜¯å¶å‡½æ•° $\Leftrightarrow$ $f(x)$æ˜¯å¥‡å‡½æ•°
\item $F(x)$æ˜¯å¥‡å‡½æ•° $\Leftrightarrow$ $f(x)$æ˜¯å¶å‡½æ•°
\item $F(x)$æ˜¯å‘¨æœŸå‡½æ•° $\Leftrightarrow$ $f(x)$æ˜¯å‘¨æœŸå‡½æ•°
\item $F(x)$æ˜¯å•è°ƒå‡½æ•° $\Leftrightarrow$ $f(x)$æ˜¯å•è°ƒå‡½æ•°
\end{abcd}
\end{question}

\bigskip

\begin{question}
è®¾çŸ©é˜µ $A = \left(\begin{array}{ccc}
  1 &amp; 1 &amp; 0\\
  1 &amp; x &amp; 0\\
  0 &amp; 0 &amp; 1
\end{array}\right)$ å…¶ä¸­ä¸¤ä¸ªç‰¹å¾å€¼ä¸º $\lambda_1 = 1$ å’Œ $\lambda_2
= 2$ï¼Œåˆ™ $x=$ \pickout{B}
\begin{abcd}
\item $2$
\item $1$
\item $0$
\item $-1$
\end{abcd}
\end{question}

\bigskip

\begin{question}
äºŒæ¬¡åž‹ $f = 4 x_1^2 - 2 x_1 x_2 + 6 x_2^2$ å¯¹åº”çš„çŸ©é˜µç­‰äºŽ \pickout{C}
\begin{abcd}
\item $\left(\begin{array}{cc}
  4 &amp; - 2\\
  - 2 &amp; 6
\end{array}\right)$
\item $\left(\begin{array}{cc}
  2 &amp; - 2\\
  - 2 &amp; 3
\end{array}\right)$
\item $\left(\begin{array}{cc}
  4 &amp; - 1\\
  - 1 &amp; 6
\end{array}\right)$
\item $\left(\begin{array}{cc}
  2 &amp; - 1\\
  - 1 &amp; 3
\end{array}\right)$
\end{abcd}
\end{question}

\bigskip

\begin{question}
ä¸‹åˆ—è¯´æ³•\CJKunderline{ä¸æ­£ç¡®}çš„æ˜¯\pickout{B}
\begin{abcd}
\item å¤§æ•°å®šå¾‹è¯´æ˜Žäº†å¤§é‡ç›¸äº’ç‹¬ç«‹ä¸”åŒåˆ†å¸ƒçš„éšæœºå˜é‡çš„å‡å€¼çš„ç¨³å®šæ€§
\item å¤§æ•°å®šå¾‹è¯´æ˜Žå¤§é‡ç›¸äº’ç‹¬ç«‹ä¸”åŒåˆ†å¸ƒçš„éšæœºå˜é‡çš„å‡å€¼è¿‘ä¼¼äºŽæ­£æ€åˆ†å¸ƒ
\item ä¸­å¿ƒæžé™å®šç†è¯´æ˜Žäº†å¤§é‡ç›¸äº’ç‹¬ç«‹ä¸”åŒåˆ†å¸ƒçš„éšæœºå˜é‡çš„å’Œçš„ç¨³å®šæ€§
\item ä¸­å¿ƒæžé™å®šç†è¯´æ˜Žå¤§é‡ç›¸äº’ç‹¬ç«‹ä¸”åŒåˆ†å¸ƒçš„éšæœºå˜é‡çš„å’Œè¿‘ä¼¼äºŽæ­£æ€åˆ†å¸ƒ
\end{abcd}
\end{question}

\bigskip

\begin{question}
å¯¹æ€»ä½“$X$å’Œæ&nbsp;·æœ¬$(X_1,\cdots,X_n)$çš„è¯´æ³•å“ªä¸ªæ˜¯\CJKunderline{ä¸æ­£ç¡®}çš„\pickout{D}
\begin{abcd}
\item æ€»ä½“æ˜¯éšæœºå˜é‡
\item æ&nbsp;·æœ¬æ˜¯$n$å…ƒéšæœºå˜é‡
\item $X_1, \cdots, X_n$ç›¸äº’ç‹¬ç«‹
\item $X_1 = X_2 =\cdots = X_n$
\end{abcd}
\end{question}

\bigskip

\newpage

\exampart{è®¡ç®—é¢˜}[å…±~6~å°é¢˜ï¼Œæ¯å°é¢˜~8~åˆ†ï¼Œå…±~48~åˆ†]

\begin{question}
æ±‚ä¸å®šç§¯åˆ†$\displaystyle\int\e^{2x}\,(\tan x+1)^2\dx$ã€‚
\end{question}

\smallskip

\begin{solution}
\everymath{\displaystyle}%
åŽŸå¼ \? $=\int\e^{2x}\,\sec^2 x\dx+2\int\e^{2x}\,\tan x\dx$ \points{2}
\+ $=\int\e^{2x}\,\d(\tan x)+ 2\int\e^{2x}\,\tan x\dx$ \points{4}
\+ $=\e^{2x}\,\tan x - 2\int\e^{2x}\,\tan x\dx+ 2\int\e^{2x}\,\tan x\dx$ \points{6}
\+ $=\e^{2x}\,\tan x + C$ \points{8}
\end{solution}

\vfill

\begin{question}
æ±‚è¿‡ç‚¹$A(1,2,-1), B(2,3,0),C(3,3,2)$ çš„ä¸‰è§’å½¢$\triangle ABC$ çš„é¢ç§¯å’Œå®ƒä»¬ç¡®å®šçš„å¹³é¢æ–¹ç¨‹.
\end{question}

\smallskip

\begin{solution}
ç”±é¢˜è®¾$\overrightarrow{AB}=(1,1,1),\overrightarrow{AC}=(2,1,3)$, \points{2}
æ•…$\overrightarrow{AB}\times \overrightarrow{AC}=\begin{vmatrix}
\vec{i}&amp;\vec{j} &amp;\vec{k}\\
1&amp;1&amp;1\\
2&amp;1&amp;3\\
\end{vmatrix}=(2,-1,-1)$, \points{4}
ä¸‰è§’å½¢$\triangle ABC$ çš„é¢ç§¯ä¸º$S_{\triangle ABC}=\frac{1}{2}\big|\overrightarrow{AB}\times
\overrightarrow{AC}\big|=\frac{1}{2}\sqrt{6}.$ \points{6}
æ‰€æ±‚å¹³é¢çš„æ–¹ç¨‹ä¸º$2(x-2)-(y-3)-z=0$, å³$2x-y-z-1=0$ \points{8}
\end{solution}

\vfill

\newpage

\begin{question}
è®¡ç®—å››é˜¶è¡Œåˆ—å¼ $A = \left|\begin{array}{cccc}
  0 &amp; 1 &amp; 2 &amp; 3\\
  1 &amp; 2 &amp; 3 &amp; 0\\
  2 &amp; 3 &amp; 0 &amp; 1\\
  3 &amp; 0 &amp; 1 &amp; 2
\end{array}\right|$ çš„å€¼ï¼Ž
\end{question}

\smallskip

\begin{solution}
$A \? = \left|\begin{array}{cccc}
    0 &amp; 1 &amp; 2 &amp; 3\\
    1 &amp; 2 &amp; 3 &amp; 0\\
    2 &amp; 3 &amp; 0 &amp; 1\\
    3 &amp; 0 &amp; 1 &amp; 2
  \end{array}\right| = \left|\begin{array}{cccc}
    0 &amp; 1 &amp; 2 &amp; 3\\
    1 &amp; 2 &amp; 3 &amp; 0\\
    0 &amp; - 1 &amp; - 6 &amp; 1\\
    0 &amp; - 6 &amp; - 8 &amp; 2
  \end{array}\right| = 1 \cdot (- 1)^{2 + 1} \left|\begin{array}{ccc}
    1 &amp; 2 &amp; 3\\
    - 1 &amp; - 6 &amp; 1\\
    - 6 &amp; - 8 &amp; 2
  \end{array}\right|$ \points{4}
\+ $= -\left|\begin{array}{ccc}
    1 &amp; 2 &amp; 3\\
    0 &amp; - 4 &amp; 4\\
    0 &amp; 4 &amp; 20
  \end{array}\right| = - \left|\begin{array}{cc}
    - 4 &amp; 4\\
    4 &amp; 20
  \end{array}\right| = -(-4\cdot20-4\cdot4) = 96$ \points{8}
\end{solution}

\vfill

\begin{question}
åˆ©ç”¨é…æ–¹æ³•ï¼Œå°†äºŒæ¬¡åž‹ $f = x_1^2 + 2 x_1 x_2 - 6 x_1 x_3 + 2 x_2^2 - 12
x_2 x_3 + 9 x^2_3$ åŒ–ä¸ºæ&nbsp;‡å‡†å½¢ $f = d_1 y^2_1 + d_2 y^2_2 + d_3 y^2_3$ ï¼Ž
\end{question}

\smallskip

\begin{solution}
$f \? = x_1^2 + 2 x_1 x_2 - 6 x_1 x_3 + 2 x_2^2 - 12 x_2 x_3 + 9 x^2_3$ \par
  \+ $= x_1^2 + 2 x_1 (x_2 - 3 x_3) + (x_2 - 3 x_3)^2 + x_2^2 - 6 x_2 x_3 $ \par
  \+ $= (x_1 + x_2 - 3 x_3)^2 + x_2^2 - 6 x_2 x_3$ \points{3}
  \+ $= (x_1 + x_2 - 3 x_3)^2 + x_2^2 - 2 x_2 \cdot 3 x_3 + (3 x_3)^2 - 9x_3^2$ \par
  \+ $= (x_1 + x_2 - 3 x_3)^2 + (x_2 - 3 x_3)^2 - 9 x_3^2$ \points{6}
ä»¤$y_1 = x_1 + x_2 - 3 x_3, y_2 = x_2 - 3 x_3, y_3 = x_3$, \newline
åˆ™$f = y_1^2 + y_2^2 - 9y_3^2$ä¸ºæ&nbsp;‡å‡†å½¢ï¼Ž\points{8}
\end{solution}

\vfill

\newpage

\begin{question}
è®¾æ¯å‘ç‚®å¼¹å‘½ä¸­é£žæœºçš„æ¦‚çŽ‡æ˜¯0.2ä¸”ç›¸äº’ç‹¬ç«‹ï¼ŒçŽ°åœ¨å‘å°„100å‘ç‚®å¼¹ï¼Ž\par
(1) ç”¨åˆ‡è´è°¢å¤«ä¸ç­‰å¼ä¼°è®¡å‘½ä¸­æ•°ç›®$\xi$åœ¨10å‘åˆ°30å‘ä¹‹é—´çš„æ¦‚çŽ‡ï¼Ž\par
(2) ç”¨ä¸­å¿ƒæžé™å®šç†ä¼°è®¡å‘½ä¸­æ•°ç›®$\xi$åœ¨10å‘åˆ°30å‘ä¹‹é—´çš„æ¦‚çŽ‡ï¼Ž
\end{question}

\smallskip

\begin{solution}
$E\xi = n p = 100 \cdot 0.2 = 20, D\xi = n p q = 100 \cdot 0.2 \cdot 0.8 = 16$. \points{2}
(1) $P (10 &lt; \xi &lt; 30) = P (|\xi - E\xi| &lt; 10) \ge 1 - \frac{D\xi}{10^2}
     = 1 - \frac{16}{100} = 0.84$. \points{4}
(2) $P (10 &lt; \xi &lt; 30) \? \approx \Phi_0\left(\frac{30 - 20}{\sqrt{16}}\right)
         - \Phi_0\left(\frac{10 - 20}{\sqrt{16}}\right)$ \points{6}
      \+ $= 2 \Phi_0(2.5) - 1 = 2 \cdot 0.9938 - 1 =0.9876$ \points{8}
\end{solution}

\vfill

\begin{question}
ä»Žæ­£æ€æ€»ä½“$N(\mu,\sigma^2)$ä¸­æŠ½å‡ºæ&nbsp;·æœ¬å®¹é‡ä¸º16çš„æ&nbsp;·æœ¬ï¼Œç®—å¾—å…¶å¹³å‡æ•°ä¸º3160ï¼Œæ&nbsp;‡å‡†å·®ä¸º100ï¼Ž
è¯•æ£€éªŒå‡è®¾$H_0:\mu=3140$æ˜¯å¦æˆç«‹($\alpha = 0.01$)ï¼Ž
\end{question}

\smallskip

\begin{solution}
(1) å¾…æ£€å‡è®¾ $H_0 : \mu = 3140$. \points{1}
(2) é€‰å–ç»Ÿè®¡é‡ $T = \frac{\widebar{X}-\mu}{S / \sqrt{n}} \sim t(n-1)$. \points{3}
(3) æŸ¥è¡¨å¾—åˆ° $t_{\alpha} = t_{\alpha} (n - 1) = t_{0.01} (15) =2.947$. \points{5}
(4) è®¡ç®—ç»Ÿè®¡å€¼ $t = \frac{\widebar{x} - \mu_0}{s/\sqrt{n}} =\frac{3160-3140}{100/4} = 0.8$.\points{7}
(5) ç”±äºŽ $| t | &lt; t_{\alpha}$, æ•…æŽ¥å— $H_0$, å³å‡è®¾æˆç«‹. \points{8}
\end{solution}

\vfill

\newpage

\exampart{è¯æ˜Žé¢˜}[å…±~2~å°é¢˜ï¼Œæ¯å°é¢˜~8~åˆ†ï¼Œå…±~16~åˆ†]

\SetExamTranslation{solution-Solution=è¯} % å°†â€œè§£â€å­—æ”¹ä¸ºâ€œè¯â€å­—

\begin{question}
è®¾æ•°åˆ—$\{x_n\}$æ»¡è¶³$x_1=\sqrt2$ï¼Œ$x_{n+1}=\sqrt{2+x_n}$ï¼Žè¯æ˜Žæ•°åˆ—æ”¶æ•›ï¼Œå¹¶æ±‚å‡ºæžé™ï¼Ž
\end{question}

\smallskip

\begin{solution}
(1) äº‹å®žä¸Šï¼Œç”±äºŽ$x_1&lt;2$ï¼Œä¸”$x_k&lt;2$æ—¶
$$x_{k+1}=\sqrt{2+x_k}&lt;\sqrt{2+2}=2,$$
ç”±æ•°å­¦å½’çº³æ³•çŸ¥å¯¹æ‰€æœ‰$n$éƒ½æœ‰$x_n&lt;2$ï¼Œå³æ•°åˆ—æœ‰ä¸Šç•Œï¼Ž
åˆç”±äºŽ
$$\frac{x_{n+1}}{x_n}=\sqrt{\frac{2}{x_n^2}+\frac{1}{x_n}}&gt;\sqrt{\frac{2}{2^2}+\frac{1}{2}}=1,$$
æ‰€ä»¥æ•°åˆ—å•è°ƒå¢žåŠ&nbsp;ï¼Žç”±æžé™å­˜åœ¨å‡†åˆ™IIï¼Œæ•°åˆ—å¿…å®šæ”¶æ•›ï¼Ž\points{4}
(2) è®¾æ•°åˆ—çš„æžé™ä¸º$A$ï¼Œå¯¹é€’æŽ¨å…¬å¼ä¸¤è¾¹åŒæ—¶å–æžé™å¾—åˆ°
$$A=\sqrt{2+A}.$$
è§£å¾—$A=2$ï¼Œå³æ•°åˆ—$\{x_n\}$çš„æžé™ä¸º$2$ï¼Ž\points{8}
\end{solution}

\vfill

\begin{question}
è®¾äº‹ä»¶$A$å’Œ$B$ç›¸äº’ç‹¬ç«‹ï¼Œè¯æ˜Ž$A$å’Œ$\widebar{B}$ç›¸äº’ç‹¬ç«‹ï¼Ž
\end{question}

\smallskip

\begin{solution}
\? $P (A \cdot \widebar{B}) = P (A - B) = P (A - A B)$ \points{2}
\&lt; $= P (A) - P (A B) = P (A) - P (A) P (B)$ \points{4}
\&lt; $= P (A) (1 - P (B)) = P (A) P (\widebar{B})$ \points{6}
æ‰€ä»¥$A$å’Œ$\widebar{B}$ç›¸äº’ç‹¬ç«‹ï¼Ž\points{8}
\end{solution}

\vfill

\examdata{ä¸€äº›å¯èƒ½ç”¨åˆ°çš„æ•°æ®} %é™„å½•æ•°æ®

\begin{tabularx}{\linewidth}{*{4}{&gt;{$}X&lt;{$}}}
\hline
\Phi_0(0.5)=0.6915 &amp; \Phi_0(1)=0.8413 &amp; \Phi_0(2)=0.9773 &amp; \Phi_0(2.5)=0.9938 \\
t_{0.01}(8)=3.355 &amp; t_{0.01}(9)=3.250 &amp; t_{0.01}(15)=2.947 &amp; t_{0.01}(16)=2.921 \\
\chi_{0.005}^2(8)=22.0 &amp; \chi_{0.005}^2(9)=23.6 &amp; \chi_{0.005}^2(15)=32.8 &amp; \chi_{0.005}^2(16)=34.3 \\
\chi_{0.995}^2(8)=1.34 &amp; \chi_{0.995}^2(9)=1.73 &amp; \chi_{0.995}^2(15)=4.60 &amp; \chi_{0.995}^2(16)=5.14 \\
\hline
\end{tabularx}

\end{document}
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