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+% -*- coding: utf-8 -*-
+% !TEX program = xelatex
+\documentclass[12pt,most]{randexam}
+
+%\answerfalse % hide answers
+
+\SetExamOption{
+ seed = 19061116, % random seed
+}
+
+\begin{document}
+
+\examtitle{name=Math 1906 Final Exam,date=2018-06-28,version=A} % make exam title
+
+\gradetable
+
+\examgroup{Fill in the blanks.}{6 questions; 3 points for each; 18 points in total.}
+
+\answertable[3em]{6}{3} % make answer table: height 3em, six in total, three for each row
+
+\begin{question}
+The first question $k>0$, text $f(x)=\ln x-\frac{x}{\e}+k$ text $(0,+\infty)$
+text text text text text text text text text text text text text text text text
+text text text text text text text text text text text text text \fillout{$2$}.
+\end{question}
+
+\bigskip
+
+\begin{question}
+The second question $\va=(2,1,2)$, $\vb=(4,-1,10)$, $\vc=\vb-\lambda\va$,
+text text text $\va\bot\vc$, text text text text text text text text text
+text text text text text $\lambda=$ \fillout{$3$}.
+\end{question}
+
+\bigskip
+
+\begin{question}
+The third question $\left|\begin{array}{cc}
+ 1 & 2\\
+ -3 & x
+\end{array}\right|=0$, text text text text text text text text
+text text text text text text text text text $x=$ \fillout{$-6$}.
+\end{question}
+
+\bigskip
+
+\begin{question}
+The fourth question $\alpha_1=(1,1,0), \alpha_2=(0,1,1), \alpha_3=(1,0,1)$,
+text $\beta=(4, 5, 3)$ text $\alpha_1, \alpha_2, \alpha_3$ text text text text
+text text text text text text text text text text text text text text text text
+text text text text text text $\beta=$ \fillout{$3\alpha_1+2\alpha_2+\alpha_3$}.
+\end{question}
+
+\bigskip
+
+\begin{question}%[points=3]
+The fifth question $\xi$ text text text $E\xi=3, D\xi=2$, text text text text text
+text text text text text text text text text text $E\xi^2=$ \fillout{$11$}.
+\end{question}
+
+\bigskip
+
+\begin{question}
+The sixth question $\xi$ text text text text $\eta$ text text text text $\xi\sim N(1,4),
+\eta\sim N(2,5)$, text text text text text text text text text text text text text text
+text text text text text text text text text text $\xi-2\eta\sim$ \fillout{$N(-3,24)$}.
+\end{question}
+
+\bigskip
+
+\newpage
+
+\examgroup{Select one answer from four choices.}{6 questions; 3 points for each; 18 points in total.}
+
+\answertable{6}{6} % 生成答题栏:默认行高,总共8题,每行8题
+
+\begin{question}
+The first question text, text text text text text text text text text text
+text text text text text text text text text text text text text \pickout{C}
+\begin{abcd}
+\item first $\int f'(x)\dx=f(x)$
+\item second $\int \d f(x)=f(x)$
+\item third $\frac{\d}{\dx}\big(\int f(x)\dx\big)=f(x)$
+\item fourth $\d\big(\int f(x)\dx\big)=f(x)$
+\end{abcd}
+\end{question}
+
+\bigskip
+
+\begin{question}
+The second question $F(x)$ text $f(x)$ text, text \pickout{A}
+\begin{abcd}
+\item first choice $F(x)$ text $\Leftrightarrow$ $f(x)$ text
+\item second choice $F(x)$ text text $\Leftrightarrow$ $f(x)$ text
+\item third choice $F(x)$ text $\Leftrightarrow$ $f(x)$ text text
+\item fourth choice $F(x)$ text text $\Leftrightarrow$ $f(x)$ text text
+\end{abcd}
+\end{question}
+
+\bigskip
+
+\begin{question}
+The third question $A = \left(\begin{array}{ccc}
+ 1 & 1 & 0\\
+ 1 & x & 0\\
+ 0 & 0 & 1
+\end{array}\right)$ text text $\lambda_1 = 1$ text $\lambda_2
+= 2$, text $x=$ \pickout{B}
+\begin{abcd}
+\item $2$
+\item $1$
+\item $0$
+\item $-1$
+\end{abcd}
+\end{question}
+
+\bigskip
+
+\begin{question}
+The fourth question $f = 4 x_1^2 - 2 x_1 x_2 + 6 x_2^2$ text text text text text text
+text text text text text text text text text text text text text text text \pickout{C}
+\begin{abcd}
+\item $\left(\begin{array}{cc}
+ 4 & - 2\\
+ - 2 & 6
+\end{array}\right)$
+\item $\left(\begin{array}{cc}
+ 2 & - 2\\
+ - 2 & 3
+\end{array}\right)$
+\item $\left(\begin{array}{cc}
+ 4 & - 1\\
+ - 1 & 6
+\end{array}\right)$
+\item $\left(\begin{array}{cc}
+ 2 & - 1\\
+ - 1 & 3
+\end{array}\right)$
+\end{abcd}
+\end{question}
+
+\bigskip
+
+\begin{question}
+The fifth question \underline{wrong} text \pickout{B}
+\begin{abcd}
+\item first choice text text text text text text text text text
+\item second choice text text text text text text text text text
+\item third choice text text text text text text text text text
+\item fourth choice text text text text text text text text text
+\end{abcd}
+\end{question}
+
+\bigskip
+
+\begin{question}
+The sixth question $X$ text $(X_1,\cdots,X_n)$ text \underline{wrong} text text
+text text text text text text text text text text text text text text \pickout{D}
+\begin{abcd}
+\item text text text
+\item text $n$ text
+\item $X_1, \cdots, X_n$ text
+\item $X_1 = X_2 =\cdots = X_n$
+\end{abcd}
+\end{question}
+
+\bigskip
+
+\newpage
+
+\examgroup{Work out math questions.}{6 questions; 8 points for each; 48 points in total.}
+
+\begin{question}
+The first question $\int\e^{2x}\,(\tan x+1)^2\dx$.
+\end{question}
+
+\smallskip
+
+\begin{solution}
+$I$ \? $=\int\e^{2x}\,\sec^2 x\dx+2\int\e^{2x}\,\tan x\dx$ \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}
+The second question $A(1,2,-1), B(2,3,0),C(3,3,2)$ text $\triangle ABC$ text text text text text text.
+\end{question}
+
+\smallskip
+
+\begin{solution}
+Text $\overrightarrow{AB}=(1,1,1),\overrightarrow{AC}=(2,1,3)$, \points{2}
+text $\overrightarrow{AB}\times \overrightarrow{AC}=\begin{vmatrix}
+\vec{i}&\vec{j} &\vec{k}\\
+1&1&1\\
+2&1&3\\
+\end{vmatrix}=(2,-1,-1)$, \points{4}
+text $\triangle ABC$ text $S_{\triangle ABC}=\frac{1}{2}\big|\overrightarrow{AB}\times
+\overrightarrow{AC}\big|=\frac{1}{2}\sqrt{6}.$ \points{6}
+Text text $2(x-2)-(y-3)-z=0$, text $2x-y-z-1=0$. \points{8}
+\end{solution}
+
+\vfill
+
+\newpage
+
+\begin{question}
+The third question $A = \left|\begin{array}{cccc}
+ 0 & 1 & 2 & 3\\
+ 1 & 2 & 3 & 0\\
+ 2 & 3 & 0 & 1\\
+ 3 & 0 & 1 & 2
+\end{array}\right|$ text.
+\end{question}
+
+\smallskip
+
+\begin{solution}
+$A \? = \left|\begin{array}{cccc}
+ 0 & 1 & 2 & 3\\
+ 1 & 2 & 3 & 0\\
+ 2 & 3 & 0 & 1\\
+ 3 & 0 & 1 & 2
+ \end{array}\right| = \left|\begin{array}{cccc}
+ 0 & 1 & 2 & 3\\
+ 1 & 2 & 3 & 0\\
+ 0 & - 1 & - 6 & 1\\
+ 0 & - 6 & - 8 & 2
+ \end{array}\right| = 1 \cdot (- 1)^{2 + 1} \left|\begin{array}{ccc}
+ 1 & 2 & 3\\
+ - 1 & - 6 & 1\\
+ - 6 & - 8 & 2
+ \end{array}\right|$ \points{4}
+\+ $= -\left|\begin{array}{ccc}
+ 1 & 2 & 3\\
+ 0 & - 4 & 4\\
+ 0 & 4 & 20
+ \end{array}\right| = - \left|\begin{array}{cc}
+ - 4 & 4\\
+ 4 & 20
+ \end{array}\right| = -(-4\cdot20-4\cdot4) = 96$ \points{8}
+\end{solution}
+
+\vfill
+
+\begin{question}
+The fourth question, tex text $f = x_1^2 + 2 x_1 x_2 - 6 x_1 x_3 + 2 x_2^2 - 12
+x_2 x_3 + 9 x^2_3$ text text $f = d_1 y^2_1 + d_2 y^2_2 + d_3 y^2_3$.
+\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}
+Text $y_1 = x_1 + x_2 - 3 x_3, y_2 = x_2 - 3 x_3, y_3 = x_3$, \newline
+text $f = y_1^2 + y_2^2 - 9y_3^2$ text.\points{8}
+\end{solution}
+
+\vfill
+
+\newpage
+
+\begin{question}
+The fifth question text text text $0.2$ text text, text text $100$ text text.\par
+(1) text text text text text text $\xi$ text $10$ text $30$ text.\par
+(2) text text text text text $\xi$ text $10$ text $30$ text.
+\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) $P (10 < \xi < 30) = P (|\xi - E\xi| < 10) \ge 1 - \frac{D\xi}{10^2}
+ = 1 - \frac{16}{100} = 0.84$. \points{4}
+(2) $P (10 < \xi < 30) \? \approx \Phi_0\left(\frac{30 - 20}{\sqrt{16}}\right)
+ - \Phi_0\left(\frac{10 - 20}{\sqrt{16}}\right)$ \points{6}
+ \+ $= 2 \Phi_0(2.5) - 1 = 2 \cdot 0.9938 - 1 =0.9876$ \points{8}
+\end{solution}
+
+\vfill
+
+\begin{question}
+The sixth question $N(\mu,\sigma^2)$ text text $16$ text, text text text $3160$, text text $100$.
+Text text $H_0:\mu=3140$ text text ($\alpha = 0.01$).
+\end{question}
+
+\smallskip
+
+\begin{solution}
+(1) Text text $H_0 : \mu = 3140$. \points{2}
+(2) Text text text $T = \frac{\widebar{X}-\mu}{S / \sqrt{n}} \sim t(n-1)$. \points{3}
+(3) Text text $t_{\alpha} = t_{\alpha} (n - 1) = t_{0.01} (15) =2.947$. \points{5}
+(4) Text text text $t = \frac{\widebar{x} - \mu_0}{s/\sqrt{n}} =\frac{3160-3140}{100/4} = 0.8$.\points{7}
+(5) Text $| t | < t_{\alpha}$, text text $H_0$, text text text. \points{8}
+\end{solution}
+
+\vfill
+
+\newpage
+
+\examgroup{Work out math proofs.}{2 questions; 16 points in total.}
+
+\renewcommand{\solutionname}{Proof} % 将“解”字改为“证”字
+
+\begin{question}[points=9]
+The first question $\{x_n\}$ text $x_1=\sqrt2$, $x_{n+1}=\sqrt{2+x_n}$.
+Text text text, text text text.
+\end{question}
+
+\smallskip
+
+\begin{solution}
+(1) Text, text $x_1<2$, text $x_k<2$ text
+$$x_{k+1}=\sqrt{2+x_k}<\sqrt{2+2}=2,$$
+Text text text text text $n$ text $x_n<2$, text text text.
+Text text
+$$\frac{x_{n+1}}{x_n}=\sqrt{\frac{2}{x_n^2}+\frac{1}{x_n}}>\sqrt{\frac{2}{2^2}+\frac{1}{2}}=1,$$
+Text text text text text. Text text text text text, Text text text text.\points{4}
+(2) Text text text text $A$, text text text text text text
+$$A=\sqrt{2+A}.$$
+Text text $A=2$, text text $\{x_n\}$ text text text $2$.\points{8}
+\end{solution}
+
+\vfill
+
+\begin{question}[points=7]
+The second question $A$ text $B$ text, text $A$ text $\widebar{B}$ text.
+\end{question}
+
+\smallskip
+
+\begin{solution}
+\? $P (A \cdot \widebar{B}) = P (A - B) = P (A - A B)$ \points{2}
+\< $= P (A) - P (A B) = P (A) - P (A) P (B)$ \points{4}
+\< $= P (A) (1 - P (B)) = P (A) P (\widebar{B})$ \points{6}
+Text text text $A$ text text text $\widebar{B}$ text text text.\points{8}
+\end{solution}
+
+\vfill
+
+\examdata{Some data may be used in the exam} % appendix data
+
+\begin{tabularx}{\linewidth}{*{4}{>{$}X<{$}}}
+\hline
+\Phi_0(0.5)=0.6915 & \Phi_0(1)=0.8413 & \Phi_0(2)=0.9773 & \Phi_0(2.5)=0.9938 \\
+t_{0.01}(8)=3.355 & t_{0.01}(9)=3.250 & t_{0.01}(15)=2.947 & t_{0.01}(16)=2.921 \\
+\chi_{0.005}^2(8)=22.0 & \chi_{0.005}^2(9)=23.6 & \chi_{0.005}^2(15)=32.8 & \chi_{0.005}^2(16)=34.3 \\
+\chi_{0.995}^2(8)=1.34 & \chi_{0.995}^2(9)=1.73 & \chi_{0.995}^2(15)=4.60 & \chi_{0.995}^2(16)=5.14 \\
+\hline
+\end{tabularx}
+
+\end{document}