3. After the examination ends, the invigilator will collect both the test paper and answer sheet.

\section*{Section I}
I. Multiple Choice: This section has 12 questions. Each question is worth 5 points. For each question, only one of the four options is correct.\\
(1) Given the set $A = \{ x \mid x = 3 n + 2 , n \in N \} , B = \{ 6,8,12,14 \}$, the number of elements in the set $A \cap B$ is\\
(A) 5\\
(B) 4\\
(C) 3\\
(D) 2\\
(2) Given points $\mathrm { A } ( 0,1 ) , \mathrm { B } ( 3,2 )$, and vector $\overrightarrow { A C } = ( - 4 , - 3 )$, then $\overrightarrow { B C } =$\\
(A) $( - 7 , - 4 )$\\
(B) $( 7,4 )$\\
(C) $( - 1,4 )$\\
(D) $( 1,4 )$\\
(3) Given that the complex number $z$ satisfies $(z - 1)i = i + 1$, then $z =$\\
(A) $- 2 - 1$\\
(B) $- 2 + 1$\\
(C) $2 - 1$\\
(D) $2 + \mathrm { i }$\\
(4) If 3 integers can serve as the side lengths of a right triangle, these 3 numbers are called a Pythagorean triple. If we select 3 different numbers from 1, 2, 3, 4, 5, the probability that these 3 numbers form a Pythagorean triple is\\
(A) $\frac { 10 } { 3 }$\\
(B) $\frac { 1 } { 5 }$\\
(C) $\frac { 1 } { 10 }$\\
(D) $\frac { 1 } { 20 }$\\
(5) Given that the ellipse $E$ is centered at the origin with eccentricity $\frac { 1 } { 2 }$, the right focus of $E$ coincides with the focus of the parabola $C : y ^ { 2 } = 8 x$. Let $A , B$ be the two foci of $E$ on the directrix of $C$, then $| A B | =$\\
(A) 3\\
(B) 6\\
(C) 9\\
(D) 12\\
(6) The ``Nine Chapters on the Mathematical Art'' is an ancient Chinese mathematical classic with extremely rich content. It contains the following problem: ``Now rice is piled in the corner of a room (as shown in the figure, the rice pile is one-quarter of a cone), the arc at the base of the rice pile is 8 chi, and the height of the rice pile is 5 chi. Question: What is the volume of the rice pile and how many measures of rice are stored?'' It is known that the volume of 1 hu of rice is approximately 1.62 cubic chi, and the circumference ratio is approximately 3. Estimate the amount of rice stored in hu as approximately\\
\includegraphics[max width=\textwidth, alt={}, center]{f131fd8b-5956-4ac8-9498-0208875ce43b-2_511_629_296_383}\\
A. 14 hu\\
B. 22 hu\\
C. 36 hu\\
D. 66 hu\\
(7) Given that $\left\{ a _ { n } \right\}$ is an arithmetic sequence with common difference 1, and $S _ { n }$ is the sum of the first $n$ terms of $\left\{ a _ { n } \right\}$. If $S _ { 8 } = 4 S _ { 4 }$, then $a _ { 10 } =$\\
(A) $\frac { 17 } { 2 }$\\
(B) $\frac { 19 } { 2 }$\\
(C) 10\\
(D) 12\\
(8) The function $f ( x ) = \cos ( \omega x + \varphi )$ has a partial graph shown in the figure. The monotonically decreasing interval of $f ( x )$ is\\
\includegraphics[max width=\textwidth, alt={}, center]{f131fd8b-5956-4ac8-9498-0208875ce43b-2_412_524_1302_351}\\
(A) $\left( \mathrm { k } \pi - - \frac { 1 } { 4 } , \mathrm { k } \pi + - \frac { 3 } { 4 } \right) , \mathrm { k } \in \mathrm { Z }$\\
(A) $\left( 2 \mathrm { k } \pi - - \frac { 1 } { 4 } , 2 \mathrm { k } \pi + - \frac { 3 } { 4 } \right) , \mathrm { k } \in \mathrm { Z }$\\
(A) $\left( \mathrm { k } - - \frac { 1 } { 4 } , \mathrm { k } + - \frac { 3 } { 4 } \right) , \mathrm { k } \in \mathrm { Z }$\\
(A) $\left( 2 \mathrm { k } - \frac { 1 } { 4 } , 2 \mathrm { k } + - \frac { 3 } { 4 } \right) , \mathrm { k } \in \mathrm { Z }$\\
(9) Executing the flowchart on the right, if the input is $\mathrm { t } = 0.01$, then the output is $\mathrm { n } =$\\
\includegraphics[max width=\textwidth, alt={}, center]{f131fd8b-5956-4ac8-9498-0208875ce43b-3_1047_471_285_331}\\
(A) 5\\
(B) 6\\
(C) 7\\
(D) 8\\
(10) Given the function $\mathrm { f } ( \mathrm { x } ) = \left\{ \begin{array} { r } 2 ^ { \mathrm { x } - 1 } - 2 , x \leq 1 \\ - \log _ { 2 } ( \mathrm { x } + 1 ) , x > 1 \end{array} \right.$, and $f(a) = - 3$, then $f(6 - a) =$\\
(A) $- \frac { 7 } { 4 }$\\
(B) $- \frac { 5 } { 4 }$\\
(C) $- \frac { 3 } { 4 }$\\
(D) $- \frac { 1 } { 4 }$\\
(11) A cylinder with a portion cut off by a plane, combined with a hemisphere (radius $r$), forms a geometric solid. The front view and top view of this solid in the three-view diagram are shown in the figure. If the surface area of this geometric solid is $16 + 20 \pi$, then $r =$

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f131fd8b-5956-4ac8-9498-0208875ce43b-3_245_253_1969_344}
\captionsetup{labelformat=empty}
\caption{Front View}
\end{center}
\end{figure}

\includegraphics[max width=\textwidth, alt={}, center]{f131fd8b-5956-4ac8-9498-0208875ce43b-3_398_319_1859_591}\\
(A) 1\\
(B) 2\\
(C) 4\\
(D) 8\\
(12) Let the graph of the function $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$ be symmetric about the line $\mathrm { y } = - \mathrm { x }$, and $\mathrm { f } ( - 2 ) + \mathrm { f } ( - 4 ) = 1$, then $a =$\\
(A) - 1\\
(B) 1\\
(C) 2\\
(D) 4

\section*{2015 National College Entrance Examination}
\section*{Liberal Arts Mathematics}
\section*{Section II}
\section*{Instructions:}
Section II has 3 pages and must be answered using black ink pen on the answer sheet. Answers written on the test paper are invalid.

This section includes both required questions and optional questions. Questions 13 to 21 are required questions that all candidates must answer. Questions 22 to 24 are optional questions; candidates should answer according to the requirements.

\section*{II. Fill-in-the-Blank Questions: This section has 4 questions, each worth 5 points}
(13) In the sequence $\left\{ a _ { n } \right\}$, $a _ { 1 } = 2 , a _ { n + 1 } = 2 a _ { n }$, and $S _ { n }$ is the sum of the first $n$ terms of $\left\{ a _ { n } \right\}$. If $S _ { n } = 126$, then $n = $ \_\_\_\_\_\_.\\
(14) Given that the function $f ( x ) = a x ^ { 3 } + x + 1$ has a tangent line at the point $( 1 , f ( 1 ) )$ that passes through the point $( 2,7 )$, then $\mathrm { a } = $ \_\_\_\_\_\_.\\
(15) $x , y$ satisfy the constraints $\left\{ \begin{array} { l } x + y - 2 \leq 0 \\ x - 2 y + 1 \leq 0 \\ 2 x - y + 2 \geq 0 \end{array} \right.$, then the maximum value of $z = 3 x + y$ is \_\_\_\_\_\_.\\
(16) Let $F$ be the right focus of the hyperbola $\mathrm { C } : \mathrm { x } ^ { 2 } - \frac { y ^ { 2 } } { 8 } = 1$, $P$ is a point on the left branch of $C$, and $\mathrm { A } ( 0,6 \sqrt { 6 } )$. When the perimeter of $\triangle \mathrm { APF }$ is minimized, the area of this triangle is \_\_\_\_\_\_.

\section*{III. Solution Questions: Show all working, proofs, and calculation steps}
(17) (This question is worth 12 points)\\
Let $a , b , c$ be the sides opposite to angles $A , B , C$ respectively in $\triangle A B C$, and $\sin ^ { 2 } B = 2 \sin A \sin C$.\\
(I) If $a = b$, find $\cos B$;\\
(II) If $B = 90 ^ { \circ }$ and $a = \sqrt { 2 }$, find the area of $\triangle A B C$.\\
(18) (This question is worth 12 points)\\
As shown in the figure, quadrilateral ABCD is a rhombus, $G$ is the intersection of $AC$ and $BD$, and $BE \perp$ plane ABCD.\\
(I) Prove that plane $A E C \perp$ plane BED;\\
(II) If $\angle \mathrm { ABC } = 120 ^ { \circ }$, $\mathrm { AE } \perp \mathrm { EC }$, and the volume of the triangular pyramid $E$-$ACD$ is $\frac { \sqrt { 6 } } { 3 }$, find the lateral surface area of this triangular pyramid.\\
(19) (This question is worth 12 points)\\
A company wants to determine the annual advertising expenditure for a certain product in the coming year. It needs to understand the effect of annual advertising expenditure $x$ (in units of thousand yuan) on annual sales volume $y$ (in units of tons) and annual profit $z$ (in units of thousand yuan). The company has conducted preliminary analysis of data on annual advertising expenditure $\mathrm { x } _ { i }$ and annual sales volume $\mathrm { y } _ { i }$ $(i = 1,2, \cdots , 8)$ over the past 8 years, obtaining the following scatter plot and some statistical values.\\
\includegraphics[max width=\textwidth, alt={}, center]{f131fd8b-5956-4ac8-9498-0208875ce43b-5_677_1175_459_349}

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
$\bar { x }$ & $\bar { y }$ & $\bar { w }$ & $\sum _ { i = 1 } ^ { 8 } \left( \mathrm { x } _ { i } - \bar { x } \right