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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
2025 national-I

19 maths questions

Q1 5 marks Complex Numbers Arithmetic Identifying Real/Imaginary Parts or Components View
The imaginary part of $(1 + 5\mathrm{i})\mathrm{i}$ is
A. $-1$
B. $0$
C. $1$
D. $6$
Q2 5 marks Probability Definitions Set Operations View
Let the universal set $U = \{x \mid x \text{ is a positive integer less than } 9\}$, and set $A = \{1,3,5\}$. Then the number of elements in $\complement_U A$ is
A. $2$
B. $3$
C. $5$
D. $8$
Q3 5 marks Conic sections Eccentricity or Asymptote Computation View
If the imaginary axis length of hyperbola $C$ is $\sqrt{7}$ times the real axis length, then the eccentricity of $C$ is
A. $\sqrt{2}$
B. $2$
C. $\sqrt{7}$
D. $2\sqrt{2}$
Q4 5 marks Trig Graphs & Exact Values View
If the point $(a, 0)$ $(a > 0)$ is a center of symmetry of the graph of the function $y = 2\tan\left(x - \frac{\pi}{3}\right)$, then the minimum value of $a$ is
A. $\frac{\pi}{4}$
B. $\frac{\pi}{2}$
C. $\frac{\pi}{3}$
D. $\frac{4\pi}{3}$
Q5 5 marks Function Transformations View
Let $f(x)$ be an even function defined on $\mathbb{R}$ with period $2$. When $2 \leq x \leq 3$, $f(x) = 5 - 2x$. Then $f\left(-\frac{3}{4}\right) =$
A. $-\frac{1}{2}$
B. $-\frac{1}{4}$
C. $\frac{1}{4}$
D. $\frac{1}{2}$
Q6 5 marks Vectors Introduction & 2D Vector Word Problem / Physical Application View
In sailing competitions, athletes can use an anemometer to measure wind speed and direction. The measured result is called apparent wind speed in nautical science. The vector corresponding to apparent wind speed is the sum of the vector corresponding to true wind speed and the vector corresponding to ship's wind speed, where the vector corresponding to ship's wind speed has the same magnitude as the vector corresponding to ship's velocity but opposite direction. Figure 1 shows the correspondence between part of the wind force levels, names, and wind speeds. An athlete measured the vector corresponding to apparent wind speed and the vector corresponding to ship's velocity as shown in Figure 2 (the magnitude of wind speed and the magnitude of the vector are the same, unit: $\mathrm{m/s}$). Then the true wind is
LevelWind Speed $\mathrm{m/s}$Name
2$1.1 \sim 3.3$Light Breeze
3$3.4 \sim 5.4$Gentle Breeze
4$5.5 \sim 7.9$Moderate Wind
5$8.0 \sim 10.1$Fresh Wind

A. Light Breeze
B. Gentle Breeze
C. Moderate Wind
D. Fresh Wind
Q7 5 marks Circles Circle-Line Intersection and Point Conditions View
If the circle $x^2 + (y+2)^2 = r^2$ $(r > 0)$ has exactly $2$ points at distance $1$ from the line $y = \sqrt{3}x + 2$, then the range of $r$ is
A. $(0,1)$
B. $(1,3)$
C. $(3, +\infty)$
D. $(0, +\infty)$
Q8 6 marks Exponential Equations & Modelling Properties of Logarithmic Functions and Statement Verification View
If real numbers $x, y, z$ satisfy $2 + \log_2 x = 3 + \log_3 y = 5 + \log_5 z$, then the size relationship of $x, y, z$ that is impossible is
A. $x > y > z$
B. $x > z > y$
C. $y > x > z$
D. $y > z > x$
Q9 6 marks Vectors 3D & Lines MCQ: Perpendicularity or Parallelism of Lines and Planes View
In the right triangular prism $ABC - A_1B_1C_1$, let $D$ be the midpoint of $BC$. Then
A. $AD \perp A_1C$
B. $BC \perp$ plane $AA_1D$
C. $CC_1 \parallel$ plane $AA_1D$
D. $AD \parallel A_1B_1$
Q10 6 marks Conic sections Focal Chord and Parabola Segment Relations View
Let the focus of parabola $C: y^2 = 6x$ be $F$. A line through $F$ intersects $C$ at $A$ and $B$. A line through $F$ perpendicular to $AB$ intersects the directrix $l: x = -\frac{3}{2}$ at $E$. From point $A$, draw a perpendicular to the directrix $l$ with foot $D$. Then
A. $|AD| = |AF|$
B. $|AE| = |AB|$
C. $|AB| \geq 6$
D. $|AE| \cdot |BE| \geq 18$
Q11 6 marks Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
Given that the area of $\triangle ABC$ is $\frac{1}{4}$, if $\cos 2A + \cos 2B + 2\sin C = 2$, $\cos A \cos B \sin C = \frac{1}{4}$, then
A. $\sin C = \sin^2 A + \sin^2 B$
B. $AB = \sqrt{2}$
C. $\sin A + \sin B = \frac{\sqrt{6}}{2}$
D. $AC^2 + BC^2 = 3$
Q12 5 marks Tangents, normals and gradients Determine unknown parameters from tangent conditions View
If the line $y = 2x + 5$ is tangent to the curve $y = \mathrm{e}^x + x + a$, then $a = $ $\_\_\_\_$ .
Q13 5 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
If a positive geometric sequence has the sum of its first 4 terms equal to $4$ and the sum of its first 8 terms equal to $68$, then the common ratio of the geometric sequence is $\_\_\_\_$ .
Q14 5 marks Discrete Random Variables Expectation and Variance via Combinatorial Counting View
A box contains 5 identical balls labeled $1$ to $5$. Drawing with replacement three times, let $X$ denote the number of distinct balls drawn at least once. Then the mathematical expectation $E(X) = $ $\_\_\_\_$ .
Q15 13 marks Chi-squared test of independence View
To study the relationship between a certain disease and ultrasound examination results, 1000 people who had undergone ultrasound examination were randomly surveyed, yielding the following contingency table:
\backslashbox{Category}{Ultrasound Result}NormalAbnormalTotal
Has Disease20180200
No Disease78020800
Total8002001000

(1) Let $P$ denote the probability that a person with abnormal ultrasound results has the disease. Find the estimated value of $P$.
(2) Based on the significance level $\alpha = 0.001$ for independence testing, analyze whether the ultrasound examination result is related to having the disease.
Attachment: $\chi^2 = \frac{n(ad - bc)^2}{(a+b)(c+d)(a+c)(b+d)}$,
$P(\chi^2 \geq k)$0.0050.0100.001
$k$3.8416.63510.828
Q16 15 marks Arithmetic Sequences and Series Prove a Sequence is Arithmetic View
Let the sequence $\{a_n\}$ satisfy $a_1 = 3$, $\frac{a_{n+1}}{n} = \frac{a_n}{n+1} + \frac{1}{n(n+1)}$.
(1) Prove that $\{na_n\}$ is an arithmetic sequence.
(2) Let $f(x) = a_1 x + a_2 x^2 + \cdots + a_m x^m$. Find $f'(-2)$.
Q17 15 marks Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
In the quadrangular pyramid $P - ABCD$ shown in the figure, $PA \perp$ plane $ABCD$, $BC \parallel AD$, $AB \perp AD$.
(1) Prove that plane $PAB \perp$ plane $PAD$.
(2) If $PA = AB = \sqrt{2}$, $AD = \sqrt{3} + 1$, $BC = 2$, and $P, B, C, D$ lie on the same sphere with center $O$.
(i) Prove that $O$ lies on plane $ABCD$.
(ii) Find the cosine of the angle between line $AC$ and line $PO$.
Q18 17 marks Conic sections Optimization on Conics View
Let the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ have eccentricity $\frac{2\sqrt{2}}{3}$, with lower vertex $A$ and right vertex $B$, $|AB| = \sqrt{10}$.
(1) Find the standard equation of the ellipse.
(2) Let $P$ be a moving point not on the $y$-axis, and let $R$ be a point on the ray $AP$ satisfying $|AR| \cdot |AP| = 3$.
(i) If $P(m, n)$, find the coordinates of point $R$ (expressed in terms of $m, n$).
(ii) Let $O$ be the origin, and $Q$ be a moving point on $C$. The slope of line $OR$ is 3 times the slope of line $OP$. Find the maximum value of $|PQ|$.
Q19 17 marks Trig Proofs Extremal Value of Trigonometric Expression View
Let the function $f(x) = 5\cos x - \cos 5x$.
(1) Find the maximum value of $f(x)$ on $\left[0, \frac{\pi}{4}\right]$.
(2) Given $\theta \in (0, \pi)$ and $a$ is a real number, prove that there exists $y \in [a - \theta, a + \theta]$ such that $\cos y \leq \cos \theta$.
(3) If there exists $\varphi$ such that for all $x$, $5\cos x - \cos(5x + \varphi) \leq b$, find the minimum value of $b$.