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Papers (66)
2025
national-I 19 national-II 18 national-I_eol 19
2024
beijing 21 national-II 19 national-I_github 19
2023
national-A-science 18 national-B-science 16 national-II 8
2022
national-A-arts 15 national-A-science 14 national-B-arts 20 national-B-science 17 national-I 19 national-II 7
2021
national-A-arts 19 national-A-science 18 national-I 14 national-II 12
2020
national-I-arts 20 national-I-science 9 national-II-science 17 national-III-arts 22 national-III-science 16 shanghai 17
2019
beijing-science 15 national-I-arts 13 national-I-science 12 national-I-science_gkztc 14 national-II-arts 13 national-II-science 9 national-II-science_gkztc 13 national-III-arts 14 national-III-science 16 national-III-science_gkztc 14
2018
national-I-arts 14 national-I-science 13 national-II-arts 20 national-II-science 18 national-III-science 17
2017
national-I-arts 13 national-I-science 14 national-II-arts 15 national-II-science 8
2016
national-I 7
2015
anhui-arts 14 beijing-arts 15 beijing-science 13 chongqing-arts 20 chongqing-science 15 fujian-science 18 hubei-arts 16 hubei-science 15 hunan-arts 21 hunan-science 11 jiangsu 19 national-II-arts 15 national-II-science 16 shaanxi-science 17 sichuan-arts 15 sichuan-science 21 tianjin-arts 17 tianjin-science 17 zhejiang-arts 15 zhejiang-science 15
Unknown
sample_questions 7
2023 national-B-science

16 maths questions

Q1 Complex Numbers Arithmetic Complex Division/Multiplication Simplification View
Let $z = \frac { 2 + i } { 1 + i ^ { 2 } + i ^ { 5 } }$, then $\bar { z } =$
A. $1 - 2 i$
B. $1 + 2 i$
C. $2 - i$
D. $2 + i$
Q2 Probability Definitions Set Operations View
Let the universal set $U = \mathbb{R}$, set $M = \{ x \mid x < 1 \}$, $N = \{ x \mid - 1 < x < 2 \}$, then $\{ x \mid x \geqslant 2 \} =$
A. $C _ { U } ( M \cup N )$
B. $N \cup C _ { U } M$
C. $C _ { U } ( M \cap N )$
D. $M \cup C _ { U } N$
Q4 Composite & Inverse Functions Symmetry, Periodicity, and Parity from Composition Conditions View
Given that $f ( x ) = \frac { x e ^ { x } } { e ^ { a x } - 1 }$ is an even function, then $a =$
A. $- 2$
B. $- 1$
C. 1
D. 2
Q5 Geometric Probability View
Let $O$ be the origin of the coordinate system. A point is randomly selected in the region $\left\{ ( x , y ) \mid 1 \leqslant x ^ { 2 } + y ^ { 2 } \leqslant 4 \right\}$. The probability that the inclination angle of line $OA$ does not exceed $\frac { \pi } { 4 }$ is
A. $\frac { 1 } { 8 }$
B. $\frac { 1 } { 6 }$
C. $\frac { 1 } { 4 }$
D. $\frac { 1 } { 2 }$
Q6 Standard trigonometric equations Evaluate a trigonometric function at a specific point after determining its form View
Given the function $f ( x ) = \sin ( \omega x + \varphi )$ is monotonically increasing on the interval $\left( \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } \right)$, and the lines $x = \frac { \pi } { 6 }$ and $x = \frac { 2 \pi } { 3 }$ are two axes of symmetry of the graph of $y = f(x)$, then $f \left( - \frac { 5 \pi } { 12 } \right) =$
A. $- \frac { \sqrt { 3 } } { 2 }$
B. $- \frac { 1 } { 2 }$
C. $\frac { 1 } { 2 }$
D. $\frac { \sqrt { 3 } } { 2 }$
Q10 Arithmetic Sequences and Series Properties of AP Terms under Transformation View
Given that the arithmetic sequence $\left\{ a _ { n } \right\}$ has common difference $\frac { 2 \pi } { 3 }$, and the set $S = \left\{ \cos a _ { n } \mid n \in \mathbb{N} ^ { * } \right\}$. If $S = \{ a , b \}$, then $a b =$
A. $- 1$
B. $- \frac { 1 } { 2 }$
C. 0
D. $\frac { 1 } { 2 }$
Q11 Conic sections Chord Properties and Midpoint Problems View
Let $A$ and $B$ be two points on the hyperbola $x ^ { 2 } - \frac { y ^ { 2 } } { 9 } = 1$. Which of the following four points could be the midpoint of segment $AB$?
A. $(1,1)$
B. $( - 1,2 )$
C. $( 1,3 )$
D. $( - 1 , - 4 )$
Q12 Circles Optimization on a Circle View
Circle $\odot O$ has radius 1. Line $PA$ is tangent to $\odot O$ at point $A$. Line $PB$ intersects $\odot O$ at points $B$ and $C$. $D$ is the midpoint of $BC$. If $| P O | = \sqrt { 2 }$, then the maximum value of $\overrightarrow { P A } \cdot \overrightarrow { P D }$ is
A. $\frac { 1 + \sqrt { 2 } } { 2 }$
B. $\frac { 1 + 2 \sqrt { 2 } } { 2 }$
C. $1 + \sqrt { 2 }$
D. $2 + \sqrt { 2 }$
Q13 Conic sections Focal Distance and Point-on-Conic Metric Computation View
Given that point $A ( 1 , \sqrt { 5 } )$ lies on the parabola $C : y ^ { 2 } = 2 p x$, then the distance from $A$ to the directrix of $C$ is \_\_\_\_
Q14 Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
If $x , y$ satisfy the constraints $\left\{ \begin{array} { l } x - 3 y \leqslant - 1 \\ x + 2 y \leqslant 9 \\ 3 x + y \geqslant 7 \end{array} \right.$, then the maximum value of $z = 2 x - y$ is \_\_\_\_
Q15 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
Given that $\left\{ a _ { n } \right\}$ is a geometric sequence with $a _ { 2 } a _ { 4 } a _ { 5 } = a _ { 3 } a _ { 6 }$ and $a _ { 9 } a _ { 10 } = - 8$, then $a _ { 7 } = $ \_\_\_\_
Q16 Exponential Functions Variation and Monotonicity Analysis View
Let $a \in ( 0,1 )$. If the function $f ( x ) = a ^ { x } + ( 1 + a ) ^ { x }$ is monotonically increasing on $( 0 , + \infty )$, then the range of $a$ is \_\_\_\_
Q17 12 marks Measures of Location and Spread View
A factory compares the treatment effects of two processes (Process A and Process B) on the elasticity of rubber products through 10 paired experiments. In each paired experiment, two rubber products of the same material are selected, one is randomly chosen to be treated with Process A and the other with Process B. The elasticity rates of the rubber products treated by Process A and Process B are recorded as $x _ { i } , y _ { i } ( i = 1,2 , \cdots 10 )$ respectively. The experimental results are as follows:
Experiment Number $i$12345678910
Elasticity Rate $x _ { i }$545533551522575544541568596548
Elasticity Rate $y _ { i }$536527543530560533522550576536

Let $z _ { i } = x _ { i } - y _ { i } ( i = 1,2 , \cdots , 10 )$. Let $\bar { z }$ denote the sample mean of $z _ { 1 } , z _ { 2 } , \cdots , z _ { 10 }$ and $s ^ { 2 }$ denote the sample variance.
(1) Find $\bar { z }$ and $s ^ { 2 }$.
(2) Determine whether the elasticity rate of rubber products treated by Process A is significantly higher than that treated by Process B. (If $\bar { z } \geqslant 2 \sqrt { \frac { s ^ { 2 } } { 10 } }$, then it is considered that the elasticity rate of rubber products treated by Process A is significantly higher than that treated by Process B; otherwise, it is not considered to be significantly higher.)
Q18 12 marks Sine and Cosine Rules Multi-step composite figure problem View
In $\triangle A B C$, it is given that $\angle B A C = 120 ^ { \circ } , A B = 2 , A C = 1$.
(1) Find $\sin \angle A B C$.
(2) If $D$ is a point on $BC$ such that $\angle B A D = 90 ^ { \circ }$, find the area of $\triangle A D C$.
Q20 12 marks Conic sections Equation Determination from Geometric Conditions View
Given the ellipse $C : \frac { y ^ { 2 } } { a ^ { 2 } } + \frac { x ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ with eccentricity $\frac { \sqrt { 5 } } { 3 }$, and point $A ( - 2,0 )$ lies on $C$.
(1) Find the equation of $C$.
(2) A line passing through point $( - 2,3 )$ intersects $C$ at points $P$ and $Q$. Lines $AP$ and $AQ$ intersect the $y$-axis at points $M$ and $N$ respectively. Prove that the midpoint of segment $MN$ is a fixed point.
Q21 12 marks Stationary points and optimisation Find critical points and classify extrema of a given function View
Given the function $f ( x ) = \left( \frac { 1 } { x } + a \right) \ln ( 1 + x )$.
(1) When $a = - 1$, find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 1 , f ( 1 ) )$.
(2) Do there exist $a$ and $b$ such that the curve $y = f \left( \frac { 1 } { x } \right)$ is symmetric about the line $x = b$? If they exist, find the values of $a$ and $b$. If they do not exist, explain why.
(3) If $f ( x )$ has an extremum on $( 0 , + \infty )$, find the range of $a$.