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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
2020 national-I-arts

20 maths questions

Q1 5 marks Inequalities Set Operations Using Inequality-Defined Sets View
Given set $A = \left\{ x \mid x ^ { 2 } - 3 x - 4 < 0 \right\} , B = \{ - 4,1,3,5 \}$ , then $A \cap B =$
A. $\{ - 4,1 \}$
B. $\{ 1,5 \}$
C. $\{ 3,5 \}$
D. $\{ 1,3 \}$
Q2 5 marks Complex Numbers Arithmetic Modulus Computation View
If $z = 1 + 2 \mathrm { i } + \mathrm { i } ^ { 3 }$ , then $| z | =$
A. 0
B. 1
C. $\sqrt { 2 }$
D. 2
Q3 5 marks Solving quadratics and applications Finding a ratio or relationship between variables from an equation View
The Great Pyramid of Egypt is one of the ancient wonders of the world. Its shape can be viewed as a regular square pyramid. The area of a square with side length equal to the height of the pyramid equals the area of one lateral triangular face of the pyramid. The ratio of the height of a lateral triangular face to the side length of the base square is
A. $\frac { \sqrt { 5 } - 1 } { 4 }$
B. $\frac { \sqrt { 5 } - 1 } { 2 }$
C. $\frac { \sqrt { 5 } + 1 } { 4 }$
D. $\frac { \sqrt { 5 } + 1 } { 2 }$
Q4 5 marks Combinations & Selection Combinatorial Probability View
Let $O$ be the center of square $A B C D$. If we randomly select 3 points from $O , A , B , C , D$, the probability that the 3 points are collinear is
A. $\frac { 1 } { 5 }$
B. $\frac { 2 } { 5 }$
C. $\frac { 1 } { 2 }$
D. $\frac { 4 } { 5 }$
Q5 5 marks Linear regression View
A study group at a school conducted seed germination experiments at 20 different temperature conditions to investigate the relationship between the germination rate $y$ of a certain crop seed and temperature $x$ (in ${}^{\circ}\mathrm{C}$). From the experimental data $\left( x _ { i } , y _ { i } \right) ( i = 1,2 , \cdots , 20 )$, a scatter plot was obtained. Based on this scatter plot, between $10^{\circ}\mathrm{C}$ and $40^{\circ}\mathrm{C}$, which of the following four regression equation types is most suitable as the regression equation type for the germination rate $y$ and temperature $x$?
A. $y = a + b x$
B. $y = a + b x ^ { 2 }$
C. $y = a + b \mathrm { e } ^ { x }$
D. $y = a + b \ln x$
Q6 5 marks Circles Chord Length and Chord Properties View
Given the circle $x ^ { 2 } + y ^ { 2 } - 6 x = 0$ , the minimum length of the chord cut by this circle from a line passing through the point $( 1,2 )$ is
A. 1
B. 2
C. 3
D. 4
Q7 5 marks Trig Graphs & Exact Values View
The function $f ( x ) = \cos \left( \omega x + \frac { \pi } { 6 } \right)$ has a graph on $[ - \pi , \pi ]$ as shown. The minimum positive period of $f ( x )$ is
A. $\frac { 10 \pi } { 9 }$
B. $\frac { 7 \pi } { 6 }$
C. $\frac { 4 \pi } { 3 }$
D. $\frac { 3 \pi } { 2 }$
Q8 5 marks Laws of Logarithms Express One Logarithm in Terms of Another View
If $a \log _ { 3 } 4 = 2$ , then $4 ^ { - a } =$
A. $\frac { 1 } { 16 }$
B. $\frac { 1 } { 9 }$
C. $\frac { 1 } { 8 }$
D. $\frac { 1 } { 6 }$
Q10 5 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
Let $\left\{ a _ { n } \right\}$ be a geometric sequence with $a _ { 1 } + a _ { 2 } + a _ { 3 } = 1 , a _ { 2 } + a _ { 3 } + a _ { 4 } = 2$ , then $a _ { 6 } + a _ { 7 } + a _ { 8 } =$
A. 12
B. 24
C. 30
D. 32
Q11 5 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
Let $F _ { 1 } , F _ { 2 }$ be the two foci of the hyperbola $C : x ^ { 2 } - \frac { y ^ { 2 } } { 3 } = 1$ , $O$ be the origin, and point $P$ on $C$ with $| O P | = 2$ . The area of $\triangle P F _ { 1 } F _ { 2 }$ is
A. $\frac { 7 } { 2 }$
B. 3
C. $\frac { 5 } { 2 }$
D. 2
Q13 5 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
If $x , y$ satisfy the constraint conditions $\left\{ \begin{array} { l } 2 x + y - 2 \leqslant 0 , \\ x - y - 1 \geqslant 0 , \\ y + 1 \geqslant 0 , \end{array} \right.$ then the maximum value of $z = x + 7 y$ is $\_\_\_\_$
Q14 5 marks Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Let vectors $\boldsymbol { a } = ( 1 , - 1 ) , \boldsymbol { b } = ( m + 1,2 m - 4 )$ . If $\boldsymbol { a } \perp \boldsymbol { b }$ , then $m =$ $\_\_\_\_$.
Q15 5 marks Tangents, normals and gradients Find tangent line with a specified slope or from an external point View
A tangent line to the curve $y = \ln x + x + 1$ has slope 2. The equation of this tangent line is $\_\_\_\_$.
Q16 5 marks Arithmetic Sequences and Series Find Specific Term from Given Conditions View
The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { n + 2 } + ( - 1 ) ^ { n } a _ { n } = 3 n - 1$ . The sum of the first 16 terms is 540. Then $a _ { 1 } =$ $\_\_\_\_$.
Q17 12 marks Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
A factory accepted a processing contract. The processed products (unit: pieces) are classified into four grades: A, B, C, and D according to standards. According to the contract: for grade A, B, and C products, the customer pays processing fees of 90 yuan, 50 yuan, and 20 yuan per piece respectively; for grade D products, the factory must compensate 50 yuan per piece for raw material loss. The factory has two branch factories, Factory A and Factory B, that can undertake the processing contract. Factory A has a processing cost of 25 yuan per piece, and Factory B has a processing cost of 20 yuan per piece. To decide which branch factory should undertake the contract, the factory conducted trial processing of 100 pieces of this product at each branch factory and recorded the grades of these products, as shown below:
Frequency Distribution Table of Product Grades for Factory A:
GradeABCD
Frequency40202020

Frequency Distribution Table of Product Grades for Factory B:
GradeABCD
Frequency28173421

(1) Estimate the probability that a product from Factory A and Factory B respectively is grade A;
(2) Find the average profit for 100 products from Factory A and Factory B respectively. Based on average profit, which branch factory should the factory choose to undertake the contract?
Q18 12 marks Sine and Cosine Rules Compute area of a triangle or related figure View
In $\triangle A B C$, the sides opposite to angles $A , B , C$ are $a , b , c$ respectively. Given $B = 150 ^ { \circ }$ ,
(1) If $a = \sqrt { 3 } c , b = 2 \sqrt { 7 }$ , find the area of $\triangle A B C$ ;
(2) If $\sin A + \sqrt { 3 } \sin C = \frac { \sqrt { 2 } } { 2 }$ , find $C$ .
Q19 12 marks Vectors: Lines & Planes Prove Perpendicularity/Orthogonality of Line and Plane View
As shown in the figure, $D$ is the apex of the cone, $O$ is the center of the base of the cone, $\triangle A B C$ is an equilateral triangle inscribed in the base, and $P$ is a point on $D O$ with $\angle A P C = 90 ^ { \circ }$ .
(1) Prove that plane $P A B \perp$ plane $P A C$ ;
(2) Given $D O = \sqrt { 2 }$ and the lateral surface area of the cone is $\sqrt { 3 } \pi$ , find the volume of the triangular pyramid $P - A B C$ .
Q20 12 marks Applied differentiation Existence and number of solutions via calculus View
Given the function $f ( x ) = \mathrm { e } ^ { x } - a ( x + 2 )$ .
(1) When $a = 1$ , discuss the monotonicity of $f ( x )$ ;
(2) If $f ( x )$ has two zeros, find the range of values for $a$ .
Q21 12 marks Conic sections Fixed Point or Collinearity Proof for Line through Conic View
Let $A , B$ be the left and right vertices of the ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + y ^ { 2 } = 1 ( a > 1 )$ respectively, $G$ be the upper vertex of $E$ , and $\overrightarrow { A G } \cdot \overrightarrow { G B } = 8$ . $P$ is a moving point on the line $x = 6$ , the other intersection point of $P A$ with $E$ is $C$ , and the other intersection point of $P B$ with $E$ is $D$ .
(1) Find the equation of $E$ ;
(2) Prove that the line $C D$ passes through a fixed point.
Q22 10 marks Polar coordinates View
[Elective 4-4: Coordinate Systems and Parametric Equations] (10 points)
In the rectangular coordinate system $x O y$, the parametric equation of curve $C _ { 1 }$ is $\left\{ \begin{array} { l } x = \cos ^ { k } t , \\ y = \sin ^ { k } t \end{array} \right.$ ($t$ is the parameter). Establishing a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C _ { 2 }$ is $$4 \rho \cos \theta - 16 \rho \sin \theta + 3 = 0$$
(1) When $k = 1$ , what type of curve is $C _ { 1 }$?
(2) When $k = 4$ , find the rectangular coordinates of the common points of $C _ { 1 }$ and $C _ { 2 }$ .