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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
2022 national-A-science

14 maths questions

Q1 5 marks Complex Numbers Arithmetic Complex Division/Multiplication Simplification View
If $z = - 1 + \sqrt { 3 } \mathrm { i }$ , then $\frac { z } { z \bar { z } - 1 } =$
A. $- 1 + \sqrt { 3 } \mathrm { i }$
B. $- 1 - \sqrt { 3 } \mathrm { i }$
C. $- \frac { 1 } { 3 } + \frac { \sqrt { 3 } } { 3 } \mathrm { i }$
D. $- \frac { 1 } { 3 } - \frac { \sqrt { 3 } } { 3 } \mathrm { i }$
Q5 5 marks Curve Sketching Identifying the Correct Graph of a Function View
The graph of the function $y = \left( 3 ^ { x } - 3 ^ { - x } \right) \cos x$ on the interval $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is approximately: (see figures A, B, C, D)
Q6 5 marks Radians, Arc Length and Sector Area View
As shown in the figure, $O$ is the center of a circle, $OA$ is the radius, arc $AB$ is part of the circle with center $O$ and radius $OA$, $C$ is the midpoint of chord $AB$, $D$ is on arc $AB$, and $CD \perp AB$. The formula for calculating the chord value $s$ is: $s = AB + \frac { CD ^ { 2 } } { OA }$. When $OA = 2$ and $\angle AOB = 60 ^ { \circ }$, then $s =$
A. $\frac { 11 - 3 \sqrt { 3 } } { 2 }$
B. $\frac { 11 - 4 \sqrt { 3 } } { 2 }$
C. $\frac { 9 - 3 \sqrt { 3 } } { 2 }$
D. $\frac { 9 - 4 \sqrt { 3 } } { 2 }$
Q10 5 marks Conic sections Eccentricity or Asymptote Computation View
For the ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$, let $A$ be the left vertex. Points $P$ and $Q$ are both on $C$ and symmetric about the $y$-axis. If the product of the slopes of $AP$ and $AQ$ is $\frac { 1 } { 4 }$, then the eccentricity of $C$ is:
A. $\frac { \sqrt { 3 } } { 2 }$
B. $\frac { \sqrt { 2 } } { 2 }$
C. $\frac { 1 } { 2 }$
D. $\frac { 1 } { 3 }$
Q13 5 marks Vectors Introduction & 2D Dot Product Computation View
Let vectors $\boldsymbol { a }$ and $\boldsymbol { b }$ have an angle whose cosine is $\frac { 1 } { 3 }$, and $| \boldsymbol { a } | = 1$, $| \boldsymbol { b } | = 3$. Then $( 2 \boldsymbol { a } + \boldsymbol { b } ) \cdot \boldsymbol { b } =$ $\_\_\_\_$
Q14 5 marks Conic sections Tangent and Normal Line Problems View
If the asymptotes of the hyperbola $y ^ { 2 } - \frac { x ^ { 2 } } { m ^ { 2 } } = 1 ( m > 0 )$ are tangent to the circle $x ^ { 2 } + y ^ { 2 } - 4 y + 3 = 0$, then $m =$ $\_\_\_\_$
Q15 5 marks Combinations & Selection Combinatorial Probability View
If 4 vertices are randomly selected from the 8 vertices of a cube, the probability that these 4 points lie on the same plane is $\_\_\_\_$
Q16 5 marks Sine and Cosine Rules Multi-step composite figure problem View
In $\triangle ABC$, point $D$ is on side $BC$, $\angle ADB = 120 ^ { \circ }$, $AD = 2$, and $CD = 2 BD$. Then $BD =$ $\_\_\_\_$.
Q17 12 marks Arithmetic Sequences and Series Multi-Part Structured Problem on AP View
Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given that $\frac { 2 S _ { n } } { n } + n = 2 a _ { n } + 1$.
(1) Prove that $\left\{ a _ { n } \right\}$ is an arithmetic sequence;
(2) If $a _ { 4 }$, $a _ { 7 }$, $a _ { 9 }$ form a geometric sequence, find the minimum value of $S _ { n }$.
Q18 12 marks 3x3 Matrices Geometric Interpretation of 3×3 Systems View
In the pyramid $P - ABCD$, $PD \perp$ base $ABCD$, $CD \parallel AB$, $AD = DC = CB = 1$, $AB = 2$.
(1) Prove that $BD \perp PA$;
(2) Find the sine of the angle between $PD$ and plane $PAB$.
Q19 12 marks Conic sections Chord Properties and Midpoint Problems View
(1) Find the equation of $C$;
(2) Let the lines $MD$ and $ND$ intersect $C$ at another point $A$ and $B$ respectively. Denote the inclination angles of lines $MN$ and $AB$ as $\alpha$ and $\beta$ respectively. When $\alpha - \beta$ attains its maximum value, find the equation of line $AB$.
Q21 12 marks Applied differentiation Existence and number of solutions via calculus View
Given the function $f ( x ) = \frac { e ^ { x } } { x } - \ln x + x - a$.
(1) If $f ( x ) \geq 0$, find the range of values for $a$;
(2) Prove that if $f ( x )$ has two zeros $x _ { 1 }$ and $x _ { 2 }$, then $x _ { 1 } x _ { 2 } < 1$.
Q22 10 marks Parametric curves and Cartesian conversion View
[Elective 4-4: Coordinate Systems and Parametric Equations] (10 points) In the rectangular coordinate system $xOy$, the parametric equation of curve $C _ { 1 }$ is $\left\{ \begin{array} { l } x = \frac { 2 + t } { 6 } \\ y = \sqrt { t } \end{array} \right.$ ($t$ is the parameter), and the parametric equation of curve $C _ { 2 }$ is $\left\{ \begin{array} { l } x = - \frac { 2 + s } { 6 } \\ y = - \sqrt { s } \end{array} \right.$ ($s$ is the parameter).
(1) Write the Cartesian equation of $C _ { 1 }$;
(2) With the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C _ { 3 }$ is $2 \cos \theta - \sin \theta = 0$. Find the rectangular coordinates of the intersection points of $C _ { 3 }$ with $C _ { 1 }$, and the rectangular coordinates of the intersection points of $C _ { 3 }$ with $C _ { 2 }$.
Q23 10 marks Proof Direct Proof of an Inequality View
[Elective 4-5: Inequalities] (10 points) Given that $a$, $b$, $c$ are all positive numbers and $a ^ { 2 } + b ^ { 2 } + 4 c ^ { 2 } = 3$, prove that:
(1) $a + b + 2 c \leq 3$;
(2) If $b = 2 c$, then $\frac { 1 } { a } + \frac { 1 } { c } \geq 3$.