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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-III-arts

22 maths questions

Q1 5 marks Inequalities Set Operations Using Inequality-Defined Sets View
Given sets $A = \{ 1,2,3,5,7,11 \} , B = \{ x \mid 3 < x < 15 \}$, the number of elements in $A \cap B$ is
A. 2
B. 3
C. 4
D. 5
Q2 5 marks Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
If $\bar { z } ( 1 + \mathrm { i } ) = 1 - \mathrm { i }$, then $z =$
A. $1 - \mathrm { i }$
B. $1 + \mathrm { i }$
C. $- i$
D. $i$
Q3 5 marks Measures of Location and Spread View
For a sample of data $x _ { 1 } , x _ { 2 } , \cdots , x _ { n }$ with variance 0.01, the variance of the data $10 x _ { 1 } , 10 x _ { 2 } , \cdots , 10 x _ { n }$ is
A. 0.01
B. 0.1
C. 1
D. 10
Q4 5 marks Exponential Equations & Modelling Threshold or Tipping-Point Calculation in Applied Exponential Models View
The Logistic model is one of the commonly used mathematical models and can be applied in epidemiology. Based on published data, scholars established a Logistic model for the cumulative confirmed cases of COVID-19 in a certain region $I ( t )$ (where $t$ is measured in days): $I ( t ) = \frac { K } { 1 + \mathrm { e } ^ { -0.23 ( t - 53 ) } }$, where $K$ is the maximum number of confirmed cases. When $I \left( t ^ { * } \right) = 0.95 K$, it indicates that the epidemic has been initially controlled. Then $t ^ { * }$ is approximately (given $\ln 19 \approx 3$)
A. 60
B. 63
C. 66
D. 69
Q5 5 marks Trig Proofs Trigonometric Identity Simplification View
Given $\sin \theta + \sin \left( \theta + \frac { \pi } { 3 } \right) = 1$, then $\sin \left( \theta + \frac { \pi } { 6 } \right) =$
A. $\frac { 1 } { 2 }$
B. $\frac { \sqrt { 3 } } { 3 }$
C. $\frac { 2 } { 3 }$
D. $\frac { \sqrt { 2 } } { 2 }$
Q6 5 marks Conic sections Locus and Trajectory Derivation View
In the plane, $A , B$ are two fixed points and $C$ is a moving point. If $\overrightarrow { A C } \cdot \overrightarrow { B C } = 1$, then the locus of point $C$ is
A. a circle
B. an ellipse
C. a parabola
D. a line
Q7 5 marks Conic sections Equation Determination from Geometric Conditions View
Let $O$ be the origin of coordinates. The line $x = 2$ intersects the parabola $C : y ^ { 2 } = 2 p x ( p > 0 )$ at points $D , E$. If $O D \perp O E$, then the focus coordinates of $C$ are
A. $\left( \frac { 1 } { 4 } , 0 \right)$
B. $\left( \frac { 1 } { 2 } , 0 \right)$
C. $( 1,0 )$
D. $( 2,0 )$
Q8 5 marks Straight Lines & Coordinate Geometry Point-to-Line Distance Computation View
The maximum distance from the point $( 0 , - 1 )$ to the line $y = k ( x + 1 )$ is
A. 1
B. $\sqrt { 2 }$
C. $\sqrt { 3 }$
D. 2
Q10 5 marks Laws of Logarithms Compare or Order Logarithmic Values View
Let $a = \log _ { 3 } 2 , b = \log _ { 5 } 3 , c = \frac { 2 } { 3 }$. Then
A. $a < c < b$
B. $a < b < c$
C. $b < c < a$
D. $c < a < b$
Q11 5 marks Sine and Cosine Rules Multi-step composite figure problem View
In $\triangle A B C$, $\cos C = \frac { 2 } { 3 } , A C = 4 , B C = 3$. Then $\tan B =$
A. $\sqrt { 5 }$
B. $2 \sqrt { 5 }$
C. $4 \sqrt { 5 }$
D. $8 \sqrt { 5 }$
Q12 5 marks Trig Graphs & Exact Values View
Given the function $f ( x ) = \sin x + \frac { 1 } { \sin x }$, then
A. the minimum value of $f ( x )$ is 2
B. the graph of $f ( x )$ is symmetric about the $y$-axis
C. the graph of $f ( x )$ is symmetric about the line $x = \pi$
D. the graph of $f ( x )$ is symmetric about the line $x = \frac { \pi } { 2 }$
Q13 5 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
If $x , y$ satisfy the constraints $\left\{ \begin{array} { l } x + y \geqslant 0 , \\ 2 x - y \geqslant 0 , \\ x \leqslant 1 , \end{array} \right.$ then the maximum value of $z = 3 x + 2 y$ is $\_\_\_\_$ .
Q14 5 marks Conic sections Eccentricity or Asymptote Computation View
For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$, one asymptote is $y = \sqrt { 2 } x$. Then the eccentricity of $C$ is $\_\_\_\_$ .
Q15 5 marks Differentiating Transcendental Functions Determine parameters from function or curve conditions View
Given the function $f ( x ) = \frac { \mathrm { e } ^ { x } } { x + a }$. If $f ^ { \prime } ( 1 ) = \frac { \mathrm { e } } { 4 }$, then $a =$ $\_\_\_\_$ .
Q16 5 marks Stationary points and optimisation Geometric or applied optimisation problem View
A cone has a base radius of 1 and slant height of 3. The volume of the largest sphere that can be inscribed in this cone is $\_\_\_\_$ .
Q17 12 marks Geometric Sequences and Series Derive General Term from Geometric Property View
Let the geometric sequence $\left\{ a _ { n } \right\}$ satisfy $a _ { 1 } + a _ { 2 } = 4 , a _ { 3 } - a _ { 1 } = 8$ .
(1) Find the general term formula for $\left\{ a _ { n } \right\}$;
(2) Let $S _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ \log _ { 3 } a _ { n } \right\}$. If $S _ { m } + S _ { m + 1 } = S _ { m + 3 }$, find $m$ .
Q18 12 marks Chi-squared test of independence View
A student interest group randomly surveyed the air quality level and the number of people exercising in a certain park each day over 100 days in a city. The data is organized in the following table (unit: days):
Air Quality Level$[ 0,200 ]$$( 200,400 ]$$( 400,600 ]$
1 (Excellent)21625
2 (Good)51012
3 (Slight Pollution)678
4 (Moderate Pollution)720

(1) Estimate the probability that the air quality level on a given day in the city is 1, 2, 3, or 4 respectively;
(2) Find the estimated average number of people exercising in the park on a given day (use the midpoint of each interval as the representative value for data in that interval);
(3) If the air quality level on a given day is 1 or 2, the day is called ``good air quality''; if the air quality level is 3 or 4, the day is called ``poor air quality''. Based on the given data, complete the following $2 \times 2$ contingency table and determine whether there is 95\% confidence to conclude that the number of people exercising in the park on a given day is related to the air quality of the city on that day.
Number of people $\leqslant 400$Number of people $> 400$
Good air quality
Poor air quality

Attachment: $K ^ { 2 } = \frac { n ( a d - b c ) ^ { 2 } } { ( a + b ) ( c + d ) ( a + c ) ( b + d ) }$,
$P \left( K ^ { 2 } \geqslant k \right)$0.0500.0100.001
$k$3.8416.63510.828
.
Q19 12 marks Vectors: Lines & Planes Prove Perpendicularity/Orthogonality of Line and Plane View
As shown in the figure, in the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, points $E , F$ are on edges $D D _ { 1 } , B B _ { 1 }$ respectively, with $2 D E = E D _ { 1 } , B F = 2 F B _ { 1 }$. Prove:
(1) When $A B = B C$, $E F \perp A C$;
(2) Point $C _ { 1 }$ lies in plane $A E F$.
Q20 12 marks Applied differentiation Existence and number of solutions via calculus View
Given the function $f ( x ) = x ^ { 3 } - k x + k ^ { 2 }$ .
(1) Discuss the monotonicity of $f ( x )$;
(2) If $f ( x )$ has three zeros, find the range of values for $k$ .
Q21 12 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
Given the ellipse $C : \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { m ^ { 2 } } = 1 ( 0 < m < 5 )$ with eccentricity $\frac { \sqrt { 15 } } { 4 }$, where $A , B$ are the left and right vertices of $C$ respectively.
(1) Find the equation of $C$;
(2) If point $P$ is on $C$, point $Q$ is on the line $x = 6$, and $| B P | = | B Q | , B P \perp B Q$, find the area of $\triangle A P Q$ .
Q22 10 marks Polar coordinates View
[Elective 4-4: Coordinate Systems and Parametric Equations] In the rectangular coordinate system $x O y$, the parametric equation of curve $C$ is $\left\{ \begin{array} { l } x = 2 - t - t ^ { 2 } , \\ y = 2 - 3 t + t ^ { 2 } \end{array} ( t \right.$ is a parameter and $t \neq 1 )$. $C$ intersects the coordinate axes at points $A , B$.
(1) Find $| A B |$;
(2) With the origin as the pole and the positive $x$-axis as the polar axis, establish a polar coordinate system and find the polar equation of line $A B$ .
Q23 10 marks Proof Direct Proof of an Inequality View
[Elective 4-5: Inequalities] Let $a , b , c \in \mathbf { R } , a + b + c = 0 , a b c = 1$ .
(1) Prove: $a b + b c + c a < 0$;
(2) Let $\max \{ a , b , c \}$ denote the maximum value among $a , b , c$. Prove: $\max \{ a , b , c \} \geqslant \sqrt[3]{\frac{3}{2}}$.