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
2019 beijing-science

15 maths questions

Q1 5 marks Complex Numbers Arithmetic Modulus Computation View
Given the complex number $z = 2 + \mathrm { i }$, then $z \cdot \bar { z } =$ (A) $\sqrt { 3 }$ (B) $\sqrt { 5 }$ (C) 3 (D) 5
Q3 5 marks Straight Lines & Coordinate Geometry Point-to-Line Distance Computation View
The parametric equation of line $l$ is $\left\{ \begin{array} { l } x = 1 + 3 t , \\ y = 2 + 4 t \end{array} \right.$ ($t$ is the parameter). The distance from point $(1,0)$ to line $l$ is (A) $\frac { 1 } { 5 }$ (B) $\frac { 2 } { 5 }$ (C) $\frac { 4 } { 5 }$ (D) $\frac { 6 } { 5 }$
Q4 5 marks Circles Circle Identification and Classification View
Given that the eccentricity of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ is $\frac { 1 } { 2 }$, then (A) $a ^ { 2 } = 2 b ^ { 2 }$ (B) $3 a ^ { 2 } = 4 b ^ { 2 }$ (C) $a = 2 b$ (D) $3 a = 4 b$
Q5 5 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
If $x , y$ satisfy $| x | \leqslant 1 - y$ and $y \geqslant - 1$, then the maximum value of $3 x + y$ is (A) $- 7$ (B) 1 (C) 5 (D) 7
Q6 5 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
In astronomy, the brightness of a celestial body can be described by magnitude or luminosity. The magnitude and luminosity of two stars satisfy $m _ { 2 } - m _ { 1 } = \frac { 5 } { 2 } \lg \frac { E _ { 1 } } { E _ { 2 } }$, where the luminosity of a star with magnitude $m _ { k }$ is $E _ { k } ( k = 1,2 )$. Given that the magnitude of the Sun is $- 26.7$ and the magnitude of Sirius is $- 1.45$, the ratio of the luminosity of the Sun to that of Sirius is (A) $10 ^ { 10.1 }$ (B) 10.1 (C) $\lg 10.1$ (D) $10 ^ { - 10.1 }$
Q7 5 marks Vectors Introduction & 2D Angle or Cosine Between Vectors View
Let points $A , B , C$ be non-collinear. Then ``the angle between $\overrightarrow { A B }$ and $\overrightarrow { A C }$ is acute'' is ``$| \overrightarrow { A B } + \overrightarrow { A C } | > | \overrightarrow { B C } |$'' a (A) sufficient but not necessary condition (B) necessary but not sufficient condition (C) necessary and sufficient condition (D) neither sufficient nor necessary condition
Q8 5 marks Curve Sketching Lattice Points and Counting via Graph Geometry View
There are many beautifully shaped and meaningful curves in mathematics. The curve $C : x ^ { 2 } + y ^ { 2 } = 1 + | x | y$ is one of them (as shown in the figure). Three conclusions are given: (1) The curve $C$ passes through exactly 6 lattice points (points with both integer coordinates); (2) The distance from any point on curve $C$ to the origin does not exceed $\sqrt { 2 }$; (3) The area of the ``heart-shaped'' region enclosed by curve $C$ is less than 3. The sequence numbers of all correct conclusions are (A) (1) (B) (2) (C) (1)(2) (D) (1)(2)(3)
Q9 5 marks Trig Graphs & Exact Values View
The minimum positive period of the function $f ( x ) = \sin ^ { 2 } 2 x$ is $\_\_\_\_$.
Q10 5 marks Arithmetic Sequences and Series Multi-Part Structured Problem on AP View
Let $\left\{ a _ { n } \right\}$ be an arithmetic sequence with the sum of the first $n$ terms being $S _ { n }$. If $a _ { 2 } = - 3 , S _ { 5 } = - 10$, then $a _ { 5 } =$ $\_\_\_\_$, and the minimum value of $S _ { n }$ is $\_\_\_\_$.
Q13 5 marks Exponential Functions Parameter Determination from Conditions View
Let the function $f ( x ) = \mathrm { e } ^ { x } + a \mathrm { e } ^ { - x }$ ($a$ is a constant). If $f ( x )$ is an odd function, then $a =$ $\_\_\_\_$; if $f ( x )$ is an increasing function on $\mathbb { R }$, then the range of $a$ is $\_\_\_\_$.
Q15 13 marks Sine and Cosine Rules Find a side length using the cosine rule View
In $\triangle A B C$, $a = 3 , \quad b - c = 2 , \quad \cos B = - \frac { 1 } { 2 }$. (I) Find the values of $b$ and $c$; (II) Find the value of $\sin ( B - C )$.
Q16 14 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
As shown in the figure, in the quadrangular pyramid $P - A B C D$, $P A \perp$ plane $A B C D$, $A D \perp C D$, $A D \| B C$, $P A = A D = C D = 2$, $B C =$ 3. $E$ is the midpoint of $P D$, and point $F$ is on $P C$ such that $\frac { P F } { P C } = \frac { 1 } { 3 }$. (I) Prove that: $C D \perp$ plane $P A D$; (II) Find the cosine of the dihedral angle $F - A E - P$; (III) Let point $G$ be on $P B$ such that $\frac { P G } { P B } = \frac { 2 } { 3 }$. Determine whether line $A G$ lies in plane $A E F$, and explain the reason.
Q17 13 marks Modelling and Hypothesis Testing View
Since the reform and opening up, people's payment methods have undergone tremendous changes. In recent years, mobile payment has become one of the main payment methods. To understand the usage of two mobile payment methods, $A$ and $B$, among students at a certain school last month, 100 students were randomly selected from the entire school. It was found that 5 people in the sample used neither method. The distribution of payment amounts for students in the sample who used only method $A$ and only method $B$ is as follows:
Payment Amount (yuan)$( 0,1000 ]$$( 1000,2000 ]$Greater than 2000
Payment Method
Using only $A$18 people9 people3 people

(I) Randomly select 1 student from the entire school. Estimate the probability that this student used both payment methods $A$ and $B$ last month; (II) Randomly select 1 student each from the sample students who used only $A$ and only $B$. Let $X$ denote the number of people among these 2 people whose payment amount last month exceeded 1000 yuan. Find the probability distribution and mathematical expectation of $X$; (III) It is known that the payment methods of sample students did not change this month. Now, 3 students are randomly selected from the sample students who used only method $A$, and it is found that their payment amounts this month all exceeded 2000 yuan. Based on the sampling results, can we conclude that the number of students using only method $A$ in the sample whose payment amount this month exceeded 2000 yuan has changed? Explain the reasoning.
Q18 14 marks Circles Circle Equation Derivation View
The parabola $C : x ^ { 2 } = - 2 p y$ passes through the point $( 2 , - 1 )$. (I) Find the equation of parabola $C$ and the equation of its directrix; (II) Let $O$ be the origin. A line $l$ with non-zero slope passes through the focus of parabola $C$ and intersects parabola $C$ at two points $M , N$. The line $y = - 1$ intersects lines $O M$ and $O N$ at points $A$ and $B$ respectively. Prove that the circle with $A B$ as diameter passes through two fixed points on the $y$-axis.
Q19 14 marks Stationary points and optimisation Find absolute extrema on a closed interval or domain View
Given the function $f ( x ) = \frac { 1 } { 4 } x ^ { 3 } - x ^ { 2 } + x$. (I) Find the equation of the tangent line to the curve $y = f ( x )$ with slope 1; (II) When $x \in [ - 2,4 ]$, prove that: $x - 6 \leqslant f ( x ) \leqslant x$; (III) Let $F ( x ) = | f ( x ) - ( x + a ) | ( a \in \mathbb { R } )$. Let $M ( a )$ denote the maximum value of $F ( x )$ on the interval $[ - 2,4 ]$. When $M ( a )$ is minimized, find the value of $a$.