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
2020 national-III-science

16 maths questions

Q1 5 marks Inequalities Set Operations Using Inequality-Defined Sets View
Given the sets $A = \left\{ ( x , y ) \mid x , y \in \mathbf { N } ^ { * } , y \geqslant x \right\} , B = \{ ( x , y ) \mid x + y = 8 \}$ , the number of elements in $A \cap B$ is
A. 2
B. 3
C. 4
D. 6
Q2 5 marks Complex Numbers Arithmetic Identifying Real/Imaginary Parts or Components View
The imaginary part of the complex number $\frac { 1 } { 1 - 3 \mathrm { i } }$ is
A. $- \frac { 3 } { 10 }$
B. $- \frac { 1 } { 10 }$
C. $\frac { 1 } { 10 }$
D. $\frac { 3 } { 10 }$
Q3 5 marks Measures of Location and Spread View
In a sample of data, the frequencies of $1,2,3,4$ are $p _ { 1 } , p _ { 2 } , p _ { 3 } , p _ { 4 }$ respectively, and $\sum _ { i = 1 } ^ { 4 } p _ { i } = 1$ . Among the following four cases, the one with the largest standard deviation is
A. $p _ { 1 } = p _ { 4 } = 0.1 , p _ { 2 } = p _ { 3 } = 0.4$
B. $p _ { 1 } = p _ { 4 } = 0.4 , p _ { 2 } = p _ { 3 } = 0.1$
C. $p _ { 1 } = p _ { 4 } = 0.2 , p _ { 2 } = p _ { 3 } = 0.3$
D. $p _ { 1 } = p _ { 4 } = 0.3 , p _ { 2 } = p _ { 3 } = 0.2$
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 $I ( t )$ of COVID-19 in a certain region ($t$ 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 ( $\ln 19 \approx 3$ )
A. 60
B. 63
C. 66
D. 69
Q5 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 $D$ and $E$ . 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. $( 2,0 )$
Q6 5 marks Vectors Introduction & 2D Angle or Cosine Between Vectors View
Given vectors $\boldsymbol { a } , \boldsymbol { b }$ satisfying $| \boldsymbol { a } | = 5 , | \boldsymbol { b } | = 6 , \boldsymbol { a } \cdot \boldsymbol { b } = - 6$ , then $\cos \langle \boldsymbol { a } , \boldsymbol { a } + \boldsymbol { b } \rangle =$
A. $- \frac { 31 } { 35 }$
B. $- \frac { 19 } { 35 }$
C. $\frac { 17 } { 35 }$
D. $\frac { 19 } { 35 }$
Q7 5 marks Sine and Cosine Rules Find an angle using the cosine rule View
In $\triangle A B C$ , $\cos C = \frac { 2 } { 3 } , A C = 4 , B C = 3$ , then $\cos B =$
A. $\frac { 1 } { 9 }$
B. $\frac { 1 } { 3 }$
C. $\frac { 1 } { 2 }$
D. $\frac { 2 } { 3 }$
Q9 5 marks Addition & Double Angle Formulae Addition/Subtraction Formula Evaluation View
Given $2 \tan \theta - \tan \left( \theta + \frac { \pi } { 4 } \right) = 7$ , then $\tan \theta =$
A. $- 2$
B. $- 1$
C. $1$
D. $2$
Q10 5 marks Tangents, normals and gradients Common tangent line to two curves View
If line $l$ is tangent to both the curve $y = \sqrt { x }$ and the circle $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 5 }$ , then the equation of $l$ is
A. $y = 2 x + 1$
B. $y = 2 x + \frac { 1 } { 2 }$
C. $y = \frac { 1 } { 2 } x + 1$
D. $y = \frac { 1 } { 2 } x + \frac { 1 } { 2 }$
Q11 5 marks Conic sections Eccentricity or Asymptote Computation View
For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with left and right foci $F _ { 1 } , F _ { 2 }$ respectively, the eccentricity is $\sqrt { 5 }$ . $P$ is a point on $C$ such that $F _ { 1 } P \perp F _ { 2 } P$ . If the area of $\triangle P F _ { 1 } F _ { 2 }$ is 4 , then $a =$
A. $1$
B. $2$
C. $4$
D. $8$
Q12 5 marks Laws of Logarithms Compare or Order Logarithmic Values View
Given $5 ^ { 5 } < 8 ^ { 4 } , 13 ^ { 4 } < 8 ^ { 5 }$ . Let $a = \log _ { 5 } 3 , b = \log _ { 8 } 5 , c = \log _ { 13 } 8$ , then
A. $a < b < c$
B. $b < a < c$
C. $b < c < a$
D. $c < a < b$
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 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
The constant term in the expansion of $\left( x ^ { 2 } + \frac { 2 } { x } \right) ^ { 6 }$ is $\_\_\_\_$ (answer with a number).
Q16 5 marks Trig Graphs & Exact Values View
Regarding the function $f ( x ) = \sin x + \frac { 1 } { \sin x }$ , there are four propositions:
(1) The graph of $f ( x )$ is symmetric about the $y$-axis.
(2) The graph of $f ( x )$ is symmetric about the origin.
(3) The graph of $f ( x )$ is symmetric about the line $x = \frac { \pi } { 2 }$ .
(4) The minimum value of $f ( x )$ is 2 .
The sequence numbers of all true propositions are $\_\_\_\_$ .
Q17 12 marks Proof by induction Prove a closed-form expression for a sequence by induction View
The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 3 , a _ { n + 1 } = 3 a _ { n } - 4 n$ .
(1) Calculate $a _ { 2 } , a _ { 3 }$ , conjecture the general term formula of $\left\{ a _ { n } \right\}$ and prove it;
(2) Find the sum $S _ { n }$ of the first $n$ terms of the sequence $\left\{ 2 ^ { n } a _ { n } \right\}$ .
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 on each of 100 days in a certain city. The organized data is shown in the table below (unit: days):
\backslashbox{Air Quality Level}{Number of Exercisers}[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 value of the 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 based on the contingency table, 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 level of the city on that day?