gaokao

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-arts

15 maths questions

Q1 5 marks Probability Definitions Set Operations View
Let set $A = \{ - 2 , - 1,0,1,2 \} , B = \left\{ x \left\lvert \, 0 \leq x < \frac { 5 } { 2 } \right. \right\}$ , then $A \cap B =$（ ）
A. $\{ 0,1,2 \}$
B. $\{ - 2 , - 1,0 \}$
C. $\{ 0,1 \}$
D. $\{ 1,2 \}$
Q5 5 marks Standard trigonometric equations Graph transformation and phase shift View
The graph of the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 3 } \right) ( \omega > 0 )$ is shifted left by $\frac { \pi } { 2 }$ units. If the minimum value of the resulting curve is $-1$ and the distance between two consecutive minimum points is $\pi$, then the minimum value of $\omega$ is
A. $\frac { 1 } { 6 }$
B. $\frac { 1 } { 4 }$
C. $\frac { 1 } { 3 }$
D. $\frac { 1 } { 2 }$
Q6 5 marks Combinations & Selection Selection with Arithmetic or Divisibility Conditions View
From 6 cards labeled $1,2,3,4,5,6$ respectively, 2 cards are randomly drawn without replacement. The probability that one of the drawn numbers is a multiple of the other is
A. $\frac { 1 } { 5 }$
B. $\frac { 1 } { 3 }$
C. $\frac { 2 } { 5 }$
D. $\frac { 2 } { 3 }$
Q7 5 marks Curve Sketching Identifying the Correct Graph of a Function View
The graph of the function $f ( x ) = \left( 3 ^ { x } - 3 ^ { - x } \right) \cos x$ on the interval $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is approximately [see figures A, B, C, D in the original paper].
Q11 5 marks Conic sections Equation Determination from Geometric Conditions View
The ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has eccentricity $\frac { 1 } { 3 }$. Let $A _ { 1 } , A _ { 2 }$ be the left and right vertices of $C$ respectively, and $B$ be the upper vertex. If $\overrightarrow { B A _ { 1 } } \cdot \overrightarrow { B A _ { 2 } } = - 1$ , then the equation of $C$ is
A. $\frac { x ^ { 2 } } { 18 } + \frac { y ^ { 2 } } { 16 } = 1$
B. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 8 } = 1$
C. $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 2 } = 1$
D. $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 32 } = 1$
Q12 5 marks Exponential Functions Ordering and Comparing Exponential Values View
Given $9 ^ { m } = 10 , a = 10 ^ { m } - 11 , b = 8 ^ { m } - 9$ , then
A. $a > 0 > b$
B. $a > b > 0$
C. $b > a > 0$
D. $b > 0 > a$
Q13 5 marks Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Given vectors $\boldsymbol { a } = ( m , 3 ) , \boldsymbol { b } = ( 1 , m + 1 )$ . If $\boldsymbol { a } \perp \boldsymbol { b }$ , then $m =$ $\_\_\_\_$ .
Q14 5 marks Circles Circle Equation Derivation View
Point $M$ lies on the line $2 x + y - 1 = 0$. Both points $( 3,0 )$ and $( 0,1 )$ lie on circle $\odot M$. Then the equation of $\odot M$ is $\_\_\_\_$ .
Q15 5 marks Conic sections Conic Identification and Conceptual Properties View
For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with eccentricity $e$, write out a value of $e$ that satisfies the condition ``the line $y = kx$ intersects the hyperbola at four distinct points'' and give one such value $\_\_\_\_$ .
Q16 5 marks Sine and Cosine Rules Multi-step composite figure problem View
In $\triangle A B C$, point $D$ is on side $B C$, $\angle A D B = 120 ^ { \circ } , A D = 2 , C D = 2 B D$ . If $S_{\triangle ABD} : S_{\triangle ACD} = 1 : 2$, then $B D =$ $\_\_\_\_$ .
Q18 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 $\frac { 2 S _ { n } } { n } + n = 2 a _ { n } + 1$ .
(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 $S _ { n }$ .
Q19 12 marks Vectors: Lines & Planes Volume of Pyramid/Tetrahedron Using Planes and Lines View
Xiaoming designed a closed packaging box as shown in the figure: the bottom face $ABCD$ is a square with side length 2. Triangles $\triangle E A B , \triangle F B C , \triangle G C D , \triangle H D A$ are all equilateral triangles, and the planes containing them are perpendicular to the bottom face.
(1) Prove that $E F \parallel$ plane $A B C D$ ;
(2) Find the volume of the packaging box (disregarding the thickness of the material).
Q20 12 marks Applied differentiation Tangent line computation and geometric consequences View
Given functions $f ( x ) = x ^ { 3 } - x , g ( x ) = x ^ { 2 } + a$. The tangent line to the curve $y = f ( x )$ at the point $\left( x _ { 1 } , f \left( x _ { 1 } \right) \right)$ is also tangent to the curve $y = g ( x )$ at some point.
(1) If $x _ { 1 } = - 1$ , find $a$ ;
(2) If $x_1 \neq 0$, prove that $a > \frac{1}{4}$ .
Q22 10 marks Parametric curves and Cartesian conversion View
[Elective 4-4: Coordinate Systems and Parametric Equations] In the rectangular coordinate system $x O y$, 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 ordinary 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] Given that $a , b , c$ are all positive numbers and $a ^ { 2 } + b ^ { 2 } + 4 c ^ { 2 } = 3$ , prove:
(1) $a + b + 2 c \leq 3$ ;
(2) If $b = 2 c$ , then $\frac { 1 } { a } + \frac { 1 } { b } + \frac { 4 } { c } \geq 3$ .