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2025
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2024
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2023
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2022
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2020
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2019
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2015
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Unknown
sample_questions 7
2019 national-I-arts

13 maths questions

Q3 Complex Numbers Arithmetic Powers of i or Complex Number Integer Powers View
3. The main content of this test paper covers all content of the college entrance examination.
Section I
I. Multiple Choice Questions: This section contains 12 questions, each worth 5 points, totaling 60 points. For each question, only one of the four options is correct.
1. The conjugate of the complex number $z = \mathrm { i } ^ { 9 } ( - 1 - 2 \mathrm { i } )$ is
A. $2 + \mathrm { i }$
B. $2 - \mathrm { i }$
C. $- 2 + \mathrm { i }$
D. $- 2 - \mathrm { i }$
2. Let sets $A = \{ a , a + 1 \} , ~ B = \{ 1,2,3 \}$. If $A \cup B$ has 4 elements, then the set of possible values of $a$ is
A. $\{ 0 \}$
B. $\{ 0,3 \}$
C. $\{ 0,1,3 \}$
D. $\{ 1,2,3 \}$
3. For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$, the length of the real axis and the focal distance are 2 and 4 respectively. The asymptote equations of hyperbola $C$ are
A. $y = \pm \frac { \sqrt { 3 } } { 3 } x$
B. $y = \pm \frac { 1 } { 3 } x$ \quad C. $y = \pm \sqrt { 3 } x$
D. $y = \pm 3 x$
Q4 Probability Definitions Set Operations View
4. According to historical records, the ``Hundred Family Names'' was written in the early Northern Song Dynasty. Table 1 records the top 24 surnames from the beginning of the ``Hundred Family Names'':
\begin{table}[h]
Table 1
赵钱孙李周吴郑王冯陈褚卫
蒋沈韩杨朱秦尤许何吕施张

\end{table}
Table 2 records the top 25 most populous surnames in China in 2018:
\begin{table}[h]
Table 2
1: Li2: Wang3: Zhang4: Liu5: Chen
6: Yang7: Zhao8: Huang9: Zhou10: Wu
11: Xu12: Sun13: Hu14: Zhu15: Gao
16: Lin17: He18: Guo19: Ma20: Luo

\end{table}
21: Liang22: Song23: Zheng24: Xie25: Han

If one surname is randomly selected from the top 24 surnames in the ``Hundred Family Names'', the probability that this surname is among the top 24 most populous surnames in China in 2018 is
A. $\frac { 5 } { 12 }$ \quad B. $\frac { 11 } { 24 }$ \quad C. $\frac { 13 } { 24 }$ \quad D. $\frac { 1 } { 2 }$
Q5 Exponential Functions Exponential Equation Solving View
5. The sum of the zeros of the function $f ( x ) = \left\{ \begin{array} { l } 6 ^ { x } - 2 , x > 0 , \\ x + \log _ { 6 } 12 , x \leq 0 \end{array} \right.$ is
A. $-1$ \quad B. $1$ \quad C. $-2$ \quad D. $2$
Q6 Trig Graphs & Exact Values View
6. The monotonically increasing interval of the function $f ( x ) = \cos \left( 3 x + \frac { \pi } { 2 } \right)$ is
A. $\left[ \frac { \pi } { 6 } + \frac { 2 k \pi } { 3 } , \frac { \pi } { 2 } + \frac { 2 k \pi } { 3 } \right] ( k \in \mathbb{Z} )$
[Figure]
Front View
[Figure]
Top View
B. $\left[ \frac { \pi } { 6 } + \frac { k \pi } { 3 } , \frac { \pi } { 2 } + \frac { k \pi } { 3 } \right] ( k \in \mathbb{Z} )$
C. $\left[ \frac { \pi } { 6 } + \frac { k \pi } { 3 } , \frac { \pi } { 6 } + \frac { k \pi } { 3 } \right] ( k \in \mathbb{Z} )$
D. $\left[ - \frac { \pi } { 6 } + \frac { 2 k \pi } { 3 } , \frac { \pi } { 6 } + \frac { 2 k \pi } { 3 } \right] ( k \in \mathbb{Z} )$
Q8 Volumes of Revolution Volume of a Region Defined by Inequalities in 3D View
8. Two unit vectors $e _ { 1 } , e _ { 2 }$ have an angle of $60 ^ { \circ }$ between them. Vector $m = t e _ { 1 } + 2 e _ { 2 } ( t < 0 )$. Then
A. The maximum value of $\frac { | m | } { t }$ is $\frac { \sqrt { 3 } } { 2 }$
B. The minimum value of $\frac { | m | } { t }$ is $- 2$
C. The minimum value of $\frac { | m | } { t }$ is $\frac { \sqrt { 3 } } { 2 }$
D. The maximum value of $\frac { | m | } { t }$ is $- 2$
Q10 Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View
10. The system of inequalities $\left\{ \begin{array} { l } x - 1 \geq 0 , \\ k x - y \leq 0 , \\ x + \sqrt { 3 } y - 3 \sqrt { 3 } \leq 0 \end{array} \right.$ represents a planar region that is an equilateral triangle. The minimum value of $z = x + 3 y$ is
A. $2 + 3 \sqrt { 3 }$ \quad B. $1 + 3 \sqrt { 3 }$ \quad C. $2 + \sqrt { 3 }$ \quad D. $1 + \sqrt { 3 }$
Q11 Inequalities Quadratic Inequality Holding for All x (or a Restricted Domain) View
11. If the range of the function $f ( x ) = a \cdot \left( \frac { 1 } { 3 } \right) ^ { x } \left( \frac { 1 } { 2 } \leq x \leq 1 \right)$ is a subset of the range of the function $g ( x ) = \frac { x ^ { 2 } - 1 } { x ^ { 2 } + x + 1 } ( x \in \mathbb{R} )$, then the range of positive number $a$ is
A. $(0,2]$
B. $(0,1]$
C. $( 0,2 \sqrt { 3 } ]$
D. $( 0 , \sqrt { 3 } ]$
Q12 Circles Inscribed/Circumscribed Circle Computations View
12. In $\triangle A B C$, the sides opposite to angles $A , B , C$ are $a , b , c$ respectively. Given $10 \sin A - 5 \sin C = 2 \sqrt { 6 }$ and $\cos B = \frac { 1 } { 5 }$, then $\frac { c } { a } =$
$$\text { A. } \frac { 6 } { 7 } \quad \text{B.} \frac { 7 } { 6 } \quad \text{C.} \frac { 5 } { 6 } \quad \text{D.} \frac { 6 } { 5 }$$
Section II
II. Fill-in-the-Blank Questions: This section contains 4 questions, each worth 5 points, totaling 20 points. Write your answers on the answer sheet.
Q13 Measures of Location and Spread View
13. A school will select one person from three candidates (A, B, C) to participate in the city-wide middle school boys' 1500-meter race. The mean and variance of their 10 recent training times (in seconds) are shown in the following table:
ABC
Mean280280290
Variance201616

Based on the data in the table, the school should select \_\_\_\_ to participate in the race.
Q14 Addition & Double Angle Formulae Addition/Subtraction Formula Evaluation View
14. Given $\tan \left( \alpha + \frac { \pi } { 4 } \right) = 6$, then $\tan \alpha = $ \_\_\_\_.
Q15 Circles Sphere and 3D Circle Problems View
15. A quadrangular pyramid $P - A B C D$ has all vertices on the surface of sphere $O$. $PA$ is perpendicular to the plane containing rectangle $A B C D$. $AB = 3$, $AD = \sqrt { 3 }$. The surface area of sphere $O$ is $13 \pi$. The length of segment $PA$ is \_\_\_\_.
Q16 Exponential Functions Intersection and Distance between Curves View
16. A line $l$ with slope $k ( k < 0 )$ passes through point $F ( 0,1 )$ and intersects the curve $y = \frac { 1 } { 4 } x ^ { 2 } ( x \geq 0 )$ and the line $y = - 1$ at points $A$ and $B$ respectively. If $| F B | = 6 | F A |$, then $k = $ \_\_\_\_.
III. Solution Questions: This section contains 6 questions, totaling 70 points. Show your working, proofs, or calculation steps. Questions 17-21 are required questions that all candidates must answer. Questions 22 and 23 are optional questions; candidates should answer according to the requirements. (I) Required Questions: Total 60 points.
Q17 12 marks Arithmetic Sequences and Series Sequence Defined by Recurrence with AP Connection View
17. (12 points)
A sequence $\left\{ a _ { n } \right\}$ satisfies $\frac { 1 } { a _ { n + 1 } } - \frac { 2 } { a _ { n } } = 0$, and $a _ { 1 } = \frac { 1 } { 2 }$.
(1) Find the general term formula of the sequence $\left\{ a _ { n } \right\}$;
(2) Find the sum $S _ { n }$ of the first $n$ terms of the sequence $\left\{ \frac { 1 } { a _ { n } } + 2 n \right\}$.