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Papers (71)
2025
national-I 18 national-II 18 national-I_eol 18
2024
beijing 19 national-II 18 national-I_github 18
2023
national-A-science 20 national-B-science 15 national-II 7
2022
national-A-arts 13 national-A-science 11 national-B-arts 19 national-B-science 17 national-I 18 national-II 7
2021
national-A-arts 19 national-A-science 16 national-I 12 national-II 10
2020
national-I-arts 21 national-I-science 9 national-II-science 14 national-III-arts 22 national-III-science 16 shanghai 14
2019
beijing-science 17 national-I-arts 17 national-I-science 13 national-I-science_gkztc 18 national-II-arts 13 national-II-science 8 national-II-science_gkztc 14 national-III-arts 11 national-III-science 16 national-III-science_gkztc 15
2018
national-I-arts 14 national-I-science 13 national-II-arts 20 national-II-science 19 national-III-science 17
2017
national-I-arts 12 national-I-science 13 national-II-arts 14 national-II-science 9
2016
national-I 7
2015
anhui-arts 14 beijing-arts 18 beijing-science 16 chongqing-arts 16 chongqing-science 13 fujian-science 18 hubei-arts 18 hubei-science 15 hunan-arts 20 hunan-science 11 jiangsu 21 national-II-arts 16 national-II-science 16 shaanxi-science 18 sichuan-arts 16 sichuan-science 21 tianjin-arts 18 tianjin-science 17 zhejiang-arts 14 zhejiang-science 15
2011
shanghai-arts 21
2010
shanghai-arts 20 shanghai-science 12
2004
shanghai-science 17
Unknown
sample_questions 3 sample_questions_testmocks 6
2023 national-B-science

15 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 Principle of Inclusion/Exclusion 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 Addition & Double Angle Formulae 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 Circles 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 Vectors Introduction & 2D 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 Circles 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 Completing the square and sketching 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 \_\_\_\_
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 Chain Rule 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 $a$.