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
2025 national-II

18 maths questions

Q1 5 marks Measures of Location and Spread View
The mean of the sample data 2, 8, 14, 16, 20 is ( )
A. 8
B. 9
C. 12
D. 18
Q2 5 marks Complex Numbers Arithmetic Complex Division/Multiplication Simplification View
Given $z = 1 + \mathrm{i}$, then $\frac{1}{z-1} = $ ( )
A. $-i$
B. $i$
C. $-1$
D. $1$
Q3 5 marks Probability Definitions Set Operations View
Given set $A = \{-4, 0, 1, 2, 8\}$, $B = \{x \mid x^3 = x\}$, then $A \cap B = $ ( )
A. $\{0, 1, 2\}$
B. $\{1, 2, 8\}$
C. $\{2, 8\}$
D. $\{0, 1\}$
Q4 5 marks Inequalities Solve Polynomial/Rational Inequality for Solution Set View
The solution set of the inequality $\frac{x-4}{x-1} < 2$ is ( )
A. $\{x \mid -2 \leq x \leq 1\}$
B. $\{x \mid x < -2\}$
C. $\{x \mid -2 \leq x < 1\}$
D. $\{x \mid x > 1\}$
Q5 5 marks Sine and Cosine Rules Find an angle using the cosine rule View
In $\triangle ABC$, $BC = 2$, $AC = 1 + \sqrt{3}$, $AB = \sqrt{6}$, then $A = $ ( )
A. $45°$
B. $60°$
C. $120°$
D. $135°$
Q6 5 marks Conic sections Focal Distance and Point-on-Conic Metric Computation View
Let the parabola $C: y^2 = 2px$ $(p > 0)$ have focus $F$. Point $A$ is on $C$. A perpendicular is drawn from $A$ to the directrix of $C$, with foot $B$. If the equation of line $BF$ is $y = -2x + 2$, then $|AF| = $ ( )
A. $3$
B. $4$
C. $5$
D. $6$
Q7 5 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $S_3 = 6$, $S_5 = -5$, then $S_6 = $ ( )
A. $-20$
B. $-15$
C. $-10$
D. $-5$
Q8 5 marks Addition & Double Angle Formulae Multi-Step Composite Problem Using Identities View
Given $0 < a < \pi$, $\cos \frac{a}{2} = \frac{\sqrt{5}}{5}$, then $\sin\left(a - \frac{\pi}{4}\right) = $
A. $\frac{\sqrt{2}}{10}$
B. $\frac{\sqrt{2}}{5}$
C. $\frac{3\sqrt{2}}{10}$
D. $\frac{7\sqrt{2}}{10}$
Q9 6 marks Geometric Sequences and Series True/False or Multiple-Statement Verification View
Let $S_n$ denote the sum of the first $n$ terms of a geometric sequence $\{a_n\}$, and let $q$ be the common ratio of $\{a_n\}$, $q > 0$. If $S_3 = 7$, $a_3 = 1$, then
A. $q = \frac{1}{2}$
B. $a_5 = \frac{1}{9}$
C. $S_5 = 8$
D. $a_n + S_n = 8$
Q10 6 marks Curve Sketching Function Properties from Symmetry or Parity View
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = (x^2 - 3)e^x + 2$, then
A. $f(0) = 0$
B. When $x < 0$, $f(x) = -(x^2 - 3)e^{-x} - 2$
C. $f(x) < 0$ if and only if $x > \sqrt{3}$
D. $x = -1$ is a local maximum point of $f(x)$
Q11 6 marks Conic sections Eccentricity or Asymptote Computation View
For the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, let $F_1, F_2$ be the left and right foci respectively, and $A_1, A_2$ be the left and right vertices respectively. The circle with diameter $F_1F_2$ intersects one asymptote of $C$ at points $M, N$, and $\angle NA_1M = \frac{5\pi}{6}$, then
A. $\angle A_1MA_2 = \frac{\pi}{6}$
B. $|MA_1| = 2|MA_2|$
C. The eccentricity of $C$ is $\sqrt{13}$
D. When $a = \sqrt{2}$, the area of quadrilateral $NA_1MA_2$ is $8\sqrt{3}$
Q12 5 marks Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Given plane vectors $\vec{a} = (x, 1)$, $\vec{b} = (x-1, 2x)$. If $\vec{a} \perp (\vec{a} - \vec{b})$, then $|\vec{a}| = $ \_\_\_\_
Q13 5 marks Stationary points and optimisation Determine parameters from given extremum conditions View
If $x = 2$ is an extremum point of the function $f(x) = (x-1)(x-2)(x-a)$, then $f(0) = $ \_\_\_\_
Q15 13 marks Harmonic Form View
Given the function $f(x) = \cos(2x + \varphi)$ $(0 \leq \varphi < \pi)$, $f(0) = \frac{1}{2}$.
(1) Find $\varphi$;
(2) Let $g(x) = f(x) + f\left(x - \frac{\pi}{6}\right)$. Find the range and monotonic intervals of $g(x)$.
Q16 15 marks Circles Area and Geometric Measurement Involving Circles View
Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $\frac{\sqrt{2}}{2}$ and major axis length 4.
(1) Find the equation of $C$;
(2) A line $l$ passing through the point $(0, -2)$ intersects $C$ at points $A, B$. Let $O$ be the origin. If the area of $\triangle OAB$ is $\sqrt{2}$, find $|AB|$.
Q17 15 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
As shown in the figure, in quadrilateral $ABCD$, $AB \parallel CD$, $\angle DAB = 90°$, $F$ is the midpoint of $CD$, point $E$ is on $AB$, $EF \parallel AD$, $AB = 3AD$, $CD = 2AD$. Fold quadrilateral $EFDA$ along $EF$ to quadrilateral $EFD'A'$ such that the dihedral angle between plane $EFD'A'$ and plane $EFCB$ is $60°$.
(1) Prove: $A'B \parallel$ plane $CD'F$;
(2) Find the sine of the dihedral angle between plane $BCD'$ and plane $EFD'A'$.
Q18 17 marks Applied differentiation Existence and number of solutions via calculus View
Given the function $f(x) = \ln(1+x) - x + \frac{1}{2}x^2 - kx^3$, where $0 < k < \frac{1}{3}$.
(1) Prove: $f(x)$ has a unique extremum point and a unique zero point on the interval $(0, +\infty)$;
(2) Let $x_1, x_2$ be the extremum point and zero point of $f(x)$ on the interval $(0, +\infty)$ respectively.
(i) Let $g(t) = f(x_1 + t) - f(x_1 - t)$. Prove: $g(t)$ is monotonically decreasing on the interval $(0, x_1)$;
(ii) Compare the sizes of $2x_1$ and $x_2$, and prove your conclusion.
Q19 17 marks Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
Two people, A and B, practice table tennis. The winner of each ball scores 1 point, the loser scores 0 points. Let the probability that A wins each ball be $p$ $\left(\frac{1}{2} < p < 1\right)$, the probability that B wins be $q$, with $p + q = 1$. The outcome of each ball is independent. For a positive integer $k \geq 2$, let $p_k$ denote the probability that after $k$ balls, A has scored at least 2 more points than B, and let $q_k$ denote the probability that after $k$ balls, B has scored at least 2 more points than A.
(1) Find $p_3, p_4$ (expressed in terms of $p$).
(2) If $\frac{p_4 - p_3}{q_4 - q_3} = 4$, find $p$.
(3) Prove: For any positive integer $m$, $p_{2m+1} - q_{2m+1} < p_{2m} - q_{2m} < p_{2m+2} - q_{2m+2}$.