1. Before answering, you must fill in your name and seat number in the designated places on the test paper and answer sheet, and carefully verify that the name and seat number on the barcode pasted on the answer sheet match your name and seat number. You must also fill in your name and the last two digits of your seat number on the back of the answer sheet in the designated place.

1. Before answering, write your name and admission ticket number on both the examination paper and answer sheet, and paste the barcode of your admission ticket at the designated position on the answer sheet. Use a 2B pencil to fill in the box after test paper type A on the answer sheet.

1. Before answering, candidates must fill in their admission ticket number and name on the answer sheet. Candidates should carefully verify that the barcode on the answer sheet shows the correct ``admission ticket number, name, and examination subject'' that match their own admission ticket number and name.

1. This test paper is divided into Section I (Multiple Choice) and Section II (Non-Multiple Choice). Section I is on pages 1-3, and Section II is on pages 3-5.

1. Before answering, candidates must use a 0.5 mm black pen to write their name, seat number, candidate number, county/district, and subject category in the designated positions on the answer sheet and test paper.

1. Before answering, candidates must use a 0.5 mm black pen to write their name, seat number, candidate number, county/district, and subject category in the designated positions on the answer sheet and test paper.

1. Let set $A = \{ x \mid - 1 < x < 2 \}$ and set $B = \{ x \mid 1 < x < 3 \}$. Then $A \cup B =$
(A) $\{ x \mid - 1 < x < 3 \}$
(B) $\{ x \mid - 1 < x < 1 \}$
(C) $\{ x \mid 1 < x < 2 \} \{ x \mid 2 < x < 3 \}$
(D)

2. For Section I, after selecting the answer for each question, use a 2B pencil to blacken the corresponding answer option on the answer sheet. If you need to make changes, erase it cleanly with an eraser and then select another answer option.

2. If $x , y$ satisfy $\left\{ \begin{array} { l } x - y \leqslant 0 , \\ x + y \leqslant 1 , \\ x \geqslant 0 , \end{array} \right.$ then the maximum value of $z = x + 2 y$ is
A. $0$
B. $1$
C. $\frac { 3 } { 2 }$
D. $2$

2. In the ancient Chinese mathematical classic ``The Nine Chapters on the Mathematical Art,'' there is a problem called ``Grain and Chaff Separation'': A grain warehouse receives 1534 shi of rice. Upon inspection, the rice contains chaff. A sample of one handful of rice is taken, and among 254 grains, 28 are chaff. Estimate the amount of chaff in this batch of rice as
A. $ 134$ shi
B. $ 169$ shi
C. $ 338$ shi
D. $ 1365$ shi

2. For multiple choice questions: After selecting the answer for each question, use a 2B pencil to fill in the corresponding answer letter on the answer sheet. Answers written on the examination paper, scratch paper, and non-answer areas on the answer sheet are invalid.

2. For Section I, after selecting the answer to each question, use a 2B pencil to blacken the corresponding answer choice on the answer sheet; if changes are needed, erase cleanly with an eraser before marking another answer choice. Section II must be answered using a 0.5 mm black pen. Answers written on the test paper are invalid.

2. Before answering, candidates must fill in their name and admission ticket number in the corresponding positions on this test paper.

2. For Section I, after selecting the answer to each question, use a 2B pencil to mark the corresponding answer choice on the answer sheet; if changes are needed, erase cleanly with an eraser and then mark another answer choice. Answers written on the test paper are invalid.

2. For Section I, after selecting the answer for each question, use a 2B pencil to mark the corresponding answer choice on the answer sheet; if changes are needed, erase completely with an eraser before marking another answer choice. Answers written on the test paper are invalid.

2. The three-view drawing of a certain solid is shown in the figure (unit: cm). The volume of this solid is
A. $8 \mathrm {~cm} ^ { 3 }$
B. $12 \mathrm {~cm} ^ { 3 }$
C. $\frac { 32 } { 3 } \mathrm {~cm} ^ { 3 }$
D. $\frac { 40 } { 3 } \mathrm {~cm} ^ { 3 }$

3. For Section II, you must use a 0.5 mm black ink pen to write on the answer sheet. The handwriting must be neat and clear. For drawing questions, you may first use a pencil to draw in the designated position on the answer sheet, and after confirmation, describe it clearly with a 0.5 mm black ink pen. You must answer in the answer area indicated by the question number. Answers written outside the answer area are invalid, and answers written on the test paper or draft paper are invalid.

3. Executing the flowchart shown in the figure, the output result is
A. $( - 2,2 )$
B. $( - 4,0 )$
C. $( - 4 , - 4 )$
D. $( 0 , - 8 )$ [Figure]

3. For fill-in-the-blank and solution questions: Use a pen to write your answers directly in the corresponding answer areas on the answer sheet. Answers written on the examination paper, scratch paper, and non-answer areas on the answer sheet are invalid.

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

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 [Figure]
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 [Figure]
(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 $\mathrm { n } =$ [Figure]
(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 $r =$
[Figure]
Front View
[Figure]
(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

2015 National College Entrance Examination

Liberal Arts Mathematics

Section II

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.

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 \_\_\_\_\_\_.

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. [Figure]
\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

3. All answers must be completed on the answer sheet. Answers written on this test paper are invalid.

3. Section II must be answered with a 0.5 mm black pen. Answers must be written in the designated areas for each question on the answer sheet and not on the test paper. If changes are needed, first cross out the original answer and then write the new answer. Do not use correction fluid, tape, or correction tape. Answers not following the above requirements are invalid.

3. Section II must be answered with a 0.5 mm black pen. Answers must be written in the designated areas for each question on the answer sheet, not on the test paper. If changes are needed, cross out the original answer first, then write the new answer. Do not use correction fluid, tape, or correction tape. Answers not following the above requirements are invalid.

3. A school wants to investigate whether there is a significant difference in vision among students in grades 3, 6, and 9. They plan to sample students from these three grades in proportion to their numbers. The most appropriate sampling method is
(A) Lottery method
(B) Systematic sampling
(C) Stratified sampling
(D) Random number method

3. By reading the program flowchart on the right and running the corresponding program, the output value of $i$ is
(A) 2
(B) 3
(C) 4
(D) 5