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-B-science

17 maths questions

Q1 5 marks Probability Definitions Set Operations View
Let the universal set $U = \{ 1,2,3,4,5 \}$, and set $M$ satisfies $C_U M = \{ 1,3 \}$. Then
A. $2 \in M$
B. $3 \in M$
C. $4 \notin M$
D. $5 \in M$
Q2 5 marks Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
Given $z = 1 - 2i$, and $z + a\bar{z} + b = 0$, where $a, b$ are real numbers, then
A. $a = 1, b = -2$
B. $a = -1, b = 2$
C. $a = 1, b = 2$
D. $a = -1, b = -2$
Q3 5 marks Vectors Introduction & 2D Dot Product Computation View
Given vectors $\boldsymbol{a}, \boldsymbol{b}$ satisfy $|\boldsymbol{a}| = 1, |\boldsymbol{b}| = \sqrt{3}, |\boldsymbol{a} - 2\boldsymbol{b}| = 3$, then $\boldsymbol{a} \cdot \boldsymbol{b} =$
A. $-2$
B. $-1$
C. $1$
D. $2$
Q4 5 marks Sequences and Series Recurrence Relations and Sequence Properties View
After completing its lunar exploration mission, the Chang'e-2 satellite continued deep space exploration and became China's first artificial planet orbiting the sun. To study the ratio of Chang'e-2's orbital period around the sun to Earth's orbital period around the sun, the sequence $\{b_n\}$ is used: $b_1 = 1 + \frac{1}{a_1}, b_2 = 1 + \frac{1}{a_1 + \frac{1}{a_2}}, b_3 = 1 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3}}}, \cdots$, and so on, where $a_k \in \mathbf{N}^* (k = 1,2,\cdots)$. Then
A. $b_1 < b_5$
B. $b_3 < b_8$
C. $b_6 < b_2$
D. $b_4 < b_7$
Q5 5 marks Conic sections Focal Distance and Point-on-Conic Metric Computation View
Let $F$ be the focus of the parabola $C: y^2 = 4x$, point $A$ is on $C$, point $B(3,0)$. If $|AF| = |BF|$, then $|AB| =$
A. $2$
B. $2\sqrt{2}$
C. $3$
D. $3\sqrt{2}$
Q7 5 marks Vectors: Lines & Planes Coplanarity and Relative Position of Planes View
In the cube $ABCD-A_1B_1C_1D_1$, $E, F$ are the midpoints of $AB, BC$ respectively. Then
A. Plane $B_1EF \perp$ plane $BDD_1$
B. Plane $B_1EF \perp$ plane $A_1BD$
C. Plane $B_1EF \parallel$ plane $A_1AC$
D. Plane $B_1EF \parallel$ plane $A_1C_1D$
Q8 5 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
Given that the sum of the first 3 terms of a geometric sequence $\{a_n\}$ is $168$, and $a_2 - a_5 = 42$, then $a_6 =$
A. $14$
B. $12$
C. $6$
D. $3$
Q9 5 marks Stationary points and optimisation Geometric or applied optimisation problem View
Given that sphere $O$ has radius $1$, and a quadrangular pyramid has vertex at $O$ with the four vertices of its base all on the surface of sphere $O$. When the volume of this quadrangular pyramid is maximum, its height is
A. $\frac{1}{3}$
B. $\frac{1}{2}$
C. $\frac{\sqrt{3}}{3}$
D. $\frac{\sqrt{2}}{2}$
Q10 5 marks Conditional Probability Optimization of Probability over Arrangements/Orderings View
A chess player plays one game each against three chess players A, B, and C, with the results of each game being independent. The probabilities that the player wins against A, B, and C are $p_1, p_2, p_3$ respectively, where $p_3 > p_2 > p_1 > 0$. Let $p$ denote the probability that the player wins two consecutive games. Then
A. $p$ is independent of the order of games against A, B, and C
B. $p$ is maximum when the player plays against A in the second game
C. $p$ is maximum when the player plays against B in the second game
D. $p$ is maximum when the player plays against C in the second game
Q11 5 marks Conic sections Eccentricity or Asymptote Computation View
For an ellipse $C$, $F_1, F_2$ are its two foci, $M, N$ are two points on the ellipse. If $\cos \angle F_1NF_2 = \frac{3}{5}$, then the eccentricity of $C$ is
A. $\frac{1}{2}$
B. $\frac{3}{2}$
C. $\frac{\sqrt{13}}{2}$
D. $\frac{\sqrt{17}}{2}$
Q12 5 marks Sequences and Series Evaluation of a Finite or Infinite Sum View
Given that $f(x), g(x)$ have domain $\mathbf{R}$, and $f(x) + g(2-x) = 5, g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, and $g(2) = 4$, then $\sum_{k=1}^{22} f(k) =$
A. $-21$
B. $-22$
C. $-23$
D. $-24$
Q13 5 marks Combinations & Selection Combinatorial Probability View
From 5 classmates including A and B, 3 are randomly selected to participate in community service work. The probability that both A and B are selected is $\_\_\_\_$.
Q14 5 marks Circles Circle Equation Derivation View
The equation of a circle passing through three of the four points $(0,0), (4,0), (-1,1), (4,2)$ is $\_\_\_\_$.
Q15 5 marks Standard trigonometric equations Determine parameters of a trigonometric function from given properties View
Let the function $f(x) = \cos(\omega x + \varphi)$ ($\omega > 0, 0 < \varphi < \pi$) have minimum positive period $T$. If $f(T) = \frac{\sqrt{3}}{2}$ and $x = \frac{\pi}{6}$ is a zero of $f(x)$, then the minimum value of $\omega$ is $\_\_\_\_$.
Q16 5 marks Stationary points and optimisation Determine parameters from given extremum conditions View
Given that $x = x_1$ and $x = x_2$ are the local minimum and local maximum points respectively of the function $f(x) = 2a^x - ex^2$ ($a > 0$ and $a \neq 1$). If $x_1 < x_2$, then the range of $a$ is $\_\_\_\_$.
Q17 12 marks Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
(12 points) Let the sides opposite to angles $A, B, C$ of $\triangle ABC$ be $a, b, c$ respectively. Given $$\sin C \sin(A - B) = \sin B \sin(C - A)$$ (1) Prove: $2a^2 = b^2 + c^2$;
(2) If $a = 5, \cos A = \frac{25}{31}$, find the perimeter of $\triangle ABC$.
Q18 12 marks Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
(12 points) As shown in the figure, in tetrahedron $ABCD$, $AD \perp CD, AD = CD, \angle ADB = \angle BDC$, and $E$ is the midpoint of $AC$.
(1) Prove: Plane $BED \perp$ plane $ACD$;
(2) Given $AB = BD = 2, \angle ACB = 60°$, point $F$ is on $BD$. When the area of $\triangle AFC$ is minimum, find the sine of the angle between $CF$ and plane $ABD$.