<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¸­ć–‡čŻ•ĺŤ·

%\SetExamBoolFalse{answer} %ä¸Ťćľç¤şç­”ćˇ

\SetExamOption{
  binding = 2,     % čŁ…č®˘çşżďĽŚ1 ä»…ç©şç™˝čŻ•ĺŤ·ćś‰ďĽŚ2 čŻ•ĺŤ·ĺ’Śç­”ćˇé˝ćś‰
  scratch = 1,     % čŤ‰ç¨żçş¸ć•°é‡ŹďĽŚä»…ç©şç™˝čŻ•ĺŤ·ćś‰ďĽŚA3 ĺ¤§ĺ°ŹďĽŚĺŹŚéť˘ĺŤ°ĺ·
  seed = 19061116, % éšŹćśşć•°ç§Ťĺ­ďĽŚç”¨äşŽć”ąĺŹ B ĺŤ·ĺ°Źé˘çš„éšŹćśşéˇşĺşŹ
}

\begin{document}

\examtitle{ % ç”źćčŻ•ĺŤ·čˇ¨ĺ¤´
  niandu  = 2017--2018,
  xueqi   = 2,
  kecheng = ĺ¤§ĺ­¦ć•°ĺ­¦,
  zhuanye = ç†ĺ·Ąĺ››ĺ­¦ĺ†, % ĺŹŻä»Ąä¸şç©şç™˝
  jiaoshi = {ĺĽ&nbsp;ä¸‰ďĽŚćťŽĺ››ďĽŚçŽ‹äş”}, % ć•™ĺ¸ĺ§“ĺŤ
  shijian = 2018~ĺą´~06~ćś~28~ć—Ą,
  bixiu   = 1, % 1 ä¸şĺż…äż®ďĽŚ0 ä¸şé€‰äż®
  bijuan  = 1, % 1 ä¸şé—­ĺŤ·ďĽŚ0 ä¸şĺĽ€ĺŤ·
  shijuan = A, % A ć– B ć– C ĺŤ·
  neizhao = 1, % 1 ć‰“ĺ‹ľďĽŚ0 ä¸Ťĺ‹ľ
  waizhao = 0, % 1 ć‰“ĺ‹ľďĽŚ0 ä¸Ťĺ‹ľ
}

\gradetable[total=4] % ç”źćčŻ„ĺ†čˇ¨

\exampart{ĺˇ«ç©şé˘}[ĺ…±~6~ĺ°Źé˘ďĽŚćŻŹĺ°Źé˘~3~ĺ†ďĽŚĺ…±~18~ĺ†]

\answertable[total=6,column=3,strut=3em] % ç”źćç­”é˘ć&nbsp;ŹďĽščˇŚé«3emďĽŚć€»ĺ…±6é˘ďĽŚćŻŹčˇŚ3é˘

\begin{question}
č®ľĺ¸¸ć•°$k&gt;0$ďĽŚĺ‡˝ć•°$f(x)=\ln x-\frac{x}{\e}+k$ĺś¨$(0,+\infty)$ĺ†…é›¶ç‚ąçš„ä¸Şć•°ä¸ş \fillout{$2$}ďĽŽ
\end{question}

\vfill

\begin{question}
č®ľ$\va=(2,1,2)$ďĽŚ$\vb=(4,-1,10)$ďĽŚ$\vc=\vb-\lambda\va$ďĽŚä¸”$\va\bot\vc$ďĽŚĺ™$\lambda=$ \fillout{$3$}ďĽŽ
\end{question}

\vfill

\begin{question}
ĺ·˛çźĄäşŚé¶čˇŚĺ—ĺĽŹ $\left|\begin{array}{cc}
  1 &amp; 2\\
  - 3 &amp; x
\end{array}\right|=0$ďĽŚĺ™ $x=$ \fillout{$-6$}ďĽŽ
\end{question}

\vfill

\begin{question}
ĺ‘é‡Źç»„ $\alpha_1=(1,1,0), \alpha_2=(0,1,1), \alpha_3=(1,0,1)$ďĽŚ
ĺ™ĺ°†ĺ‘é‡Ź $\beta=(4, 5, 3)$ čˇ¨ç¤şä¸ş $\alpha_1, \alpha_2, \alpha_3$
çš„çşżć€§ç»„ĺä¸ş $\beta=$ \fillout{$3\alpha_1+2\alpha_2+\alpha_3$}ďĽŽ
\end{question}

\vfill

\begin{question}
ĺ·˛çźĄéšŹćśşĺŹé‡Ź$\xi$çš„ćśźćś›ĺ’Ść–ąĺ·®ĺ„ä¸ş$E\xi=3, D\xi=2$, ĺ™$E\xi^2=$ \fillout{$11$}ďĽŽ
\end{question}

\vfill

\begin{question}
ĺ·˛çźĄ$\xi$ĺ’Ś$\eta$ç›¸äş’ç‹¬ç«‹ä¸”$\xi\sim N(1,4), \eta\sim N(2,5)$ďĽŚĺ™$\xi-2\eta\sim$ \fillout{$N(-3,24)$}ďĽŽ
\end{question}

\vfill

\newpage

\exampart{ĺŤ•é€‰é˘}[ĺ…±~6~ĺ°Źé˘ďĽŚćŻŹĺ°Źé˘~3~ĺ†ďĽŚĺ…±~18~ĺ†]

\answertable[total=6,column=6] % ç”źćç­”é˘ć&nbsp;ŹďĽšé»č®¤čˇŚé«ďĽŚć€»ĺ…±6é˘ďĽŚćŻŹčˇŚ6é˘

\begin{question}
ĺś¨ä¸‹ĺ—ç­‰ĺĽŹä¸­ďĽŚć­Łçˇ®çš„ç»“ćžśćŻ\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}
ĺ‡č®ľ$F(x)$ćŻčżžç»­ĺ‡˝ć•°$f(x)$çš„ä¸€ä¸ŞĺŽźĺ‡˝ć•°ďĽŚĺ™ĺż…ćś‰\pickout{A}
\begin{abcd}
\item $F(x)$ćŻĺ¶ĺ‡˝ć•° $\Leftrightarrow$ $f(x)$ćŻĺĄ‡ĺ‡˝ć•°
\item $F(x)$ćŻĺĄ‡ĺ‡˝ć•° $\Leftrightarrow$ $f(x)$ćŻĺ¶ĺ‡˝ć•°
\item $F(x)$ćŻĺ‘¨ćśźĺ‡˝ć•° $\Leftrightarrow$ $f(x)$ćŻĺ‘¨ćśźĺ‡˝ć•°
\item $F(x)$ćŻĺŤ•č°ĺ‡˝ć•° $\Leftrightarrow$ $f(x)$ćŻĺŤ•č°ĺ‡˝ć•°
\end{abcd}
\end{question}

\bigskip

\begin{question}
č®ľçź©éµ $A = \left(\begin{array}{ccc}
  1 &amp; 1 &amp; 0\\
  1 &amp; x &amp; 0\\
  0 &amp; 0 &amp; 1
\end{array}\right)$ ĺ…¶ä¸­ä¸¤ä¸Şç‰ąĺľĺ€Ľä¸ş $\lambda_1 = 1$ ĺ’Ś $\lambda_2
= 2$ďĽŚĺ™ $x=$ \pickout{B}
\begin{abcd}
\item $2$
\item $1$
\item $0$
\item $-1$
\end{abcd}
\end{question}

\bigskip

\begin{question}
äşŚć¬ˇĺž‹ $f = 4 x_1^2 - 2 x_1 x_2 + 6 x_2^2$ ĺŻąĺş”çš„çź©éµç­‰äşŽ \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¸‹ĺ—čŻ´ćł•\CJKunderline{ä¸Ťć­Łçˇ®}çš„ćŻ\pickout{B}
\begin{abcd}
\item ĺ¤§ć•°ĺ®šĺľ‹čŻ´ćŽäş†ĺ¤§é‡Źç›¸äş’ç‹¬ç«‹ä¸”ĺŚĺ†ĺ¸çš„éšŹćśşĺŹé‡Źçš„ĺť‡ĺ€Ľçš„ç¨łĺ®šć€§
\item ĺ¤§ć•°ĺ®šĺľ‹čŻ´ćŽĺ¤§é‡Źç›¸äş’ç‹¬ç«‹ä¸”ĺŚĺ†ĺ¸çš„éšŹćśşĺŹé‡Źçš„ĺť‡ĺ€Ľčż‘äĽĽäşŽć­Łć€ĺ†ĺ¸
\item ä¸­ĺżćžé™ĺ®šç†čŻ´ćŽäş†ĺ¤§é‡Źç›¸äş’ç‹¬ç«‹ä¸”ĺŚĺ†ĺ¸çš„éšŹćśşĺŹé‡Źçš„ĺ’Śçš„ç¨łĺ®šć€§
\item ä¸­ĺżćžé™ĺ®šç†čŻ´ćŽĺ¤§é‡Źç›¸äş’ç‹¬ç«‹ä¸”ĺŚĺ†ĺ¸çš„éšŹćśşĺŹé‡Źçš„ĺ’Śčż‘äĽĽäşŽć­Łć€ĺ†ĺ¸
\end{abcd}
\end{question}

\bigskip

\begin{question}
ĺŻąć€»ä˝“$X$ĺ’Ść&nbsp;·ćś¬$(X_1,\cdots,X_n)$çš„čŻ´ćł•ĺ“Şä¸ŞćŻ\CJKunderline{ä¸Ťć­Łçˇ®}çš„\pickout{D}
\begin{abcd}
\item ć€»ä˝“ćŻéšŹćśşĺŹé‡Ź
\item ć&nbsp;·ćś¬ćŻ$n$ĺ…éšŹćśşĺŹé‡Ź
\item $X_1, \cdots, X_n$ç›¸äş’ç‹¬ç«‹
\item $X_1 = X_2 =\cdots = X_n$
\end{abcd}
\end{question}

\bigskip

\newpage

\exampart{č®ˇç®—é˘}[ĺ…±~6~ĺ°Źé˘ďĽŚćŻŹĺ°Źé˘~8~ĺ†ďĽŚĺ…±~48~ĺ†]

\begin{question}
ć±‚ä¸Ťĺ®šç§Żĺ†$\displaystyle\int\e^{2x}\,(\tan x+1)^2\dx$ă€‚
\end{question}

\smallskip

\begin{solution}
\everymath{\displaystyle}%
ĺŽźĺĽŹ \? $=\int\e^{2x}\,\sec^2 x\dx+2\int\e^{2x}\,\tan x\dx$ \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}
ć±‚čż‡ç‚ą$A(1,2,-1), B(2,3,0),C(3,3,2)$ çš„ä¸‰č§’ĺ˝˘$\triangle ABC$ çš„éť˘ç§Żĺ’Śĺ®ä»¬çˇ®ĺ®šçš„ĺąłéť˘ć–ąç¨‹.
\end{question}

\smallskip

\begin{solution}
ç”±é˘č®ľ$\overrightarrow{AB}=(1,1,1),\overrightarrow{AC}=(2,1,3)$, \points{2}
ć•…$\overrightarrow{AB}\times \overrightarrow{AC}=\begin{vmatrix}
\vec{i}&amp;\vec{j} &amp;\vec{k}\\
1&amp;1&amp;1\\
2&amp;1&amp;3\\
\end{vmatrix}=(2,-1,-1)$, \points{4}
ä¸‰č§’ĺ˝˘$\triangle ABC$ çš„éť˘ç§Żä¸ş$S_{\triangle ABC}=\frac{1}{2}\big|\overrightarrow{AB}\times
\overrightarrow{AC}\big|=\frac{1}{2}\sqrt{6}.$ \points{6}
ć‰€ć±‚ĺąłéť˘çš„ć–ąç¨‹ä¸ş$2(x-2)-(y-3)-z=0$, ĺŤł$2x-y-z-1=0$ \points{8}
\end{solution}

\vfill

\newpage

\begin{question}
č®ˇç®—ĺ››é¶čˇŚĺ—ĺĽŹ $A = \left|\begin{array}{cccc}
  0 &amp; 1 &amp; 2 &amp; 3\\
  1 &amp; 2 &amp; 3 &amp; 0\\
  2 &amp; 3 &amp; 0 &amp; 1\\
  3 &amp; 0 &amp; 1 &amp; 2
\end{array}\right|$ çš„ĺ€ĽďĽŽ
\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}
ĺ©ç”¨é…Ťć–ąćł•ďĽŚĺ°†äşŚć¬ˇĺž‹ $f = x_1^2 + 2 x_1 x_2 - 6 x_1 x_3 + 2 x_2^2 - 12
x_2 x_3 + 9 x^2_3$ ĺŚ–ä¸şć&nbsp;‡ĺ‡†ĺ˝˘ $f = d_1 y^2_1 + d_2 y^2_2 + d_3 y^2_3$ ďĽŽ
\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
ĺ™$f = y_1^2 + y_2^2 - 9y_3^2$ä¸şć&nbsp;‡ĺ‡†ĺ˝˘ďĽŽ\points{8}
\end{solution}

\vfill

\newpage

\begin{question}
č®ľćŻŹĺŹ‘ç‚®ĺĽąĺ‘˝ä¸­éŁžćśşçš„ć¦‚çŽ‡ćŻ0.2ä¸”ç›¸äş’ç‹¬ç«‹ďĽŚçŽ°ĺś¨ĺŹ‘ĺ°„100ĺŹ‘ç‚®ĺĽąďĽŽ\par
(1) ç”¨ĺ‡č´ťč°˘ĺ¤«ä¸Ťç­‰ĺĽŹäĽ°č®ˇĺ‘˝ä¸­ć•°ç›®$\xi$ĺś¨10ĺŹ‘ĺ°30ĺŹ‘äą‹é—´çš„ć¦‚çŽ‡ďĽŽ\par
(2) ç”¨ä¸­ĺżćžé™ĺ®šç†äĽ°č®ˇĺ‘˝ä¸­ć•°ç›®$\xi$ĺś¨10ĺŹ‘ĺ°30ĺŹ‘äą‹é—´çš„ć¦‚çŽ‡ďĽŽ
\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»Žć­Łć€ć€»ä˝“$N(\mu,\sigma^2)$ä¸­ćŠ˝ĺ‡şć&nbsp;·ćś¬ĺ®ąé‡Źä¸ş16çš„ć&nbsp;·ćś¬ďĽŚç®—ĺľ—ĺ…¶ĺąłĺť‡ć•°ä¸ş3160ďĽŚć&nbsp;‡ĺ‡†ĺ·®ä¸ş100ďĽŽ
čŻ•ćŁ€éŞŚĺ‡č®ľ$H_0:\mu=3140$ćŻĺ¦ćç«‹($\alpha = 0.01$)ďĽŽ
\end{question}

\smallskip

\begin{solution}
(1) ĺľ…ćŁ€ĺ‡č®ľ $H_0 : \mu = 3140$. \points{1}
(2) é€‰ĺŹ–ç»źč®ˇé‡Ź $T = \frac{\widebar{X}-\mu}{S / \sqrt{n}} \sim t(n-1)$. \points{3}
(3) ćźĄčˇ¨ĺľ—ĺ° $t_{\alpha} = t_{\alpha} (n - 1) = t_{0.01} (15) =2.947$. \points{5}
(4) č®ˇç®—ç»źč®ˇĺ€Ľ $t = \frac{\widebar{x} - \mu_0}{s/\sqrt{n}} =\frac{3160-3140}{100/4} = 0.8$.\points{7}
(5) ç”±äşŽ $| t | &lt; t_{\alpha}$, ć•…ćŽĄĺŹ— $H_0$, ĺŤłĺ‡č®ľćç«‹. \points{8}
\end{solution}

\vfill

\newpage

\exampart{čŻćŽé˘}[ĺ…±~2~ĺ°Źé˘ďĽŚćŻŹĺ°Źé˘~8~ĺ†ďĽŚĺ…±~16~ĺ†]

\SetExamTranslation{solution-Solution=čŻ} % ĺ°†â€śč§Łâ€ťĺ­—ć”ąä¸şâ€śčŻâ€ťĺ­—

\begin{question}
č®ľć•°ĺ—$\{x_n\}$ć»ˇč¶ł$x_1=\sqrt2$ďĽŚ$x_{n+1}=\sqrt{2+x_n}$ďĽŽčŻćŽć•°ĺ—ć”¶ć•›ďĽŚĺą¶ć±‚ĺ‡şćžé™ďĽŽ
\end{question}

\smallskip

\begin{solution}
(1) äş‹ĺ®žä¸ŠďĽŚç”±äşŽ$x_1&lt;2$ďĽŚä¸”$x_k&lt;2$ć—¶
$$x_{k+1}=\sqrt{2+x_k}&lt;\sqrt{2+2}=2,$$
ç”±ć•°ĺ­¦ĺ˝’çşłćł•çźĄĺŻąć‰€ćś‰$n$é˝ćś‰$x_n&lt;2$ďĽŚĺŤłć•°ĺ—ćś‰ä¸Šç•ŚďĽŽ
ĺŹç”±äşŽ
$$\frac{x_{n+1}}{x_n}=\sqrt{\frac{2}{x_n^2}+\frac{1}{x_n}}&gt;\sqrt{\frac{2}{2^2}+\frac{1}{2}}=1,$$
ć‰€ä»Ąć•°ĺ—ĺŤ•č°ĺ˘žĺŠ&nbsp;ďĽŽç”±ćžé™ĺ­ĺś¨ĺ‡†ĺ™IIďĽŚć•°ĺ—ĺż…ĺ®šć”¶ć•›ďĽŽ\points{4}
(2) č®ľć•°ĺ—çš„ćžé™ä¸ş$A$ďĽŚĺŻąé€’ćŽ¨ĺ…¬ĺĽŹä¸¤čľąĺŚć—¶ĺŹ–ćžé™ĺľ—ĺ°
$$A=\sqrt{2+A}.$$
č§Łĺľ—$A=2$ďĽŚĺŤłć•°ĺ—$\{x_n\}$çš„ćžé™ä¸ş$2$ďĽŽ\points{8}
\end{solution}

\vfill

\begin{question}
č®ľäş‹ä»¶$A$ĺ’Ś$B$ç›¸äş’ç‹¬ç«‹ďĽŚčŻćŽ$A$ĺ’Ś$\widebar{B}$ç›¸äş’ç‹¬ç«‹ďĽŽ
\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}
ć‰€ä»Ą$A$ĺ’Ś$\widebar{B}$ç›¸äş’ç‹¬ç«‹ďĽŽ\points{8}
\end{solution}

\vfill

\examdata{ä¸€äş›ĺŹŻč˝ç”¨ĺ°çš„ć•°ćŤ®} %é™„ĺ˝•ć•°ćŤ®

\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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