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diff --git a/Master/texmf-dist/doc/latex/jnuexam/examfe-a-answer.tex b/Master/texmf-dist/doc/latex/jnuexam/examfe-a-answer.tex new file mode 100644 index 00000000000..ee35652a916 --- /dev/null +++ b/Master/texmf-dist/doc/latex/jnuexam/examfe-a-answer.tex @@ -0,0 +1,375 @@ +% -*- coding: utf-8 -*- +% !TEX program = xelatex +\documentclass[english]{jnuexam} % 英文试卷 + +%\SetExamBoolFalse{answer} % hide answers + +\SetExamOption{ + binding = 1, % 装订线,1 仅空白试卷有,2 试卷和答案都有 + scratch = 1, % 草稿纸数量,仅空白试卷有,A3 大小,双面印刷 + seed = 19061116, % 随机数种子,用于改变 B 卷小题的随机顺序 +} + +\begin{document} + +% make exam title +\examtitle{ + AcademicYear = 2016--2017, % 教学年度 + Semester = 2, % 第几学期 + CourseTitle = Math 1906, % 课程名称 + Instructor = {Jianrui Lyu}, % 教师姓名 + Date = dd/mm/yyyy, % 考试时间 + Compulsory = 1, % 是否必修:1 为必修,0 为选修 + ClosedBook = 1, % 是否闭卷:1 为闭卷,0 为开卷 + Paper = A, % 试卷版本:A 或 B 或 C 卷 + Mainland = 0, % 内招适用:1 打勾,0 不勾 + NonMainland = 1, % 外招适用:1 打勾,0 不勾 +} + +\gradetable[total=4] + +\exampart{Fill in the blanks.}[6 questions; 3 points for each; 18 points in total.] + +% make answer table: six in total, three for each row, strut height 3em +\answertable[total=6,column=3,strut=3em] + +\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 + +\exampart{Select one answer from four choices.}[6 questions; 3 points for each; 18 points in total.] + +% make answer table: six in total, six for each row, default strut height +\answertable[total=6,column=6] + +\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 + +\exampart{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 + +\exampart{Work out math proofs.}[2 questions; 16 points in total.] + +\SetExamTranslation{solution-Solution=Proof} % rename "Solution" as "Proof" + +\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} |