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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
2021 national-A-science

18 maths questions

Q1 Probability Definitions Set Operations View
1. Let sets $M = \{ x \mid 0 < x < 4 \}, N = \left\{ x \left\lvert \, \frac { 1 } { 3 } \leq x \leq 5 \right. \right\}$, then $M \cap N =$
A. $\left\{ x \left\lvert \, 0 < x \leq \frac { 1 } { 3 } \right. \right\}$
B. $\left\{ x \left\lvert \, \frac { 1 } { 3 } \leq x < 4 \right. \right\}$
C. $\{ x \mid 4 \leq x < 5 \}$
D. $\{ x \mid 0 < x \leq 5 \}$
Q2 Data representation View
2. To understand the rural economic situation in a certain area, a sampling survey was conducted on the annual household income of farmers. The survey data on farmers' annual household income was organized into the following frequency distribution histogram. Based on this frequency distribution histogram, which of the following conclusions is incorrect?
A. The estimated proportion of farmers with annual household income below 4.5 ten thousand yuan is 60\%
B. The estimated proportion of farmers with annual household income not less than 10.5 ten thousand yuan is 10\% [Figure]
C. The estimated average annual household income of farmers in this area does not exceed 6.5 ten thousand yuan
D. It is estimated that more than half of the farmers have annual household income between 4.5 and 8.5 ten thousand yuan
Q3 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
3. Given $(1 - i)^2 z = 3 + 2i$, then $z =$
A. $-1 - \frac{3}{2}i$
B. $-1 + \frac{3}{2}i$
C. $-\frac{3}{2} + i$
D. $-\frac{3}{2} - i$
Q4 10 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
4. Adolescent vision is a matter of widespread social concern. Vision can be measured using a vision chart. Vision data is usually recorded using the five-point recording method and the decimal recording method. The data $l$ in the five-point recording method and the data $v$ in the decimal recording method satisfy $l = 5 + \log_v$. A student's vision data in the five-point recording method is 4.4. Then their vision data in the decimal recording method is approximately ($\sqrt[10]{10} \approx 1.259$)
A. 1.5
B. 1.2
C. 0.8
D. 0.6
Q5 Conic sections Eccentricity or Asymptote Computation View
5. Let $F_1, F_2$ be the two foci of hyperbola $C$. Let $P$ be a point on $C$, and $\angle F_1 P F_2 = 60°$, $|PF_1| = 3|PF_2|$. Then the eccentricity of $C$ is
A. $\frac{\sqrt{7}}{2}$
B. $\frac{\sqrt{13}}{2}$
C. $\sqrt{7}$
D. $\sqrt{13}$
Q7 Geometric Sequences and Series Logical Relationship Between Conditions on Geometric Sequences View
7. For a geometric sequence $\{a_n\}$ with common ratio $q$ and sum of the first $n$ terms $S_n$, let Proposition A: $q > 0$. Proposition B: $\{S_n\}$ is an increasing sequence. Then
A. A is a sufficient but not necessary condition for B
B. A is a necessary but not sufficient condition for B
C. A is a necessary and sufficient condition for B
D. A is neither a sufficient nor a necessary condition for B
Q8 Sine and Cosine Rules Heights and distances / angle of elevation problem View
8. On December 8, 2020, China and Nepal jointly announced that the latest height of Mount Everest is 8848.86 m. The trigonometric height measurement method is one of the methods for measuring the height of Mount Everest. The figure on the right is a schematic diagram of the trigonometric height measurement method. There are three points $A, B, C$, and their projections $A', B', C'$ on the same horizontal plane satisfy $\angle A'C'B' = 45°$, $\angle A'B'C' = 60°$. The angle of elevation from point $C$ to point $B$ is $15°$. The difference between $BB'$ and $CC'$ is 100 m. The angle of elevation from point $B$ to point $A$ is $45°$. Then the height difference between points $A$ and $C$ to the horizontal plane $A'B'C'$ is $AA' - CC'$ approximately equals ($\sqrt{3} \approx 1.732$)
A. 346
B. 373
C. 446
D. 473
Q9 Addition & Double Angle Formulae Trigonometric Equation Solving via Identities View
9. If $a \in \left(0, \frac{\pi}{2}\right)$, $\tan 2a = \frac{\cos a}{2 - \sin a}$, then $\tan a =$
A. $\frac{\sqrt{15}}{15}$
B. $\frac{\sqrt{5}}{5}$
C. $\frac{\sqrt{5}}{3}$
D. $\frac{\sqrt{15}}{3}$
Q10 Permutations & Arrangements Probability via Permutation Counting View
10. Arrange 4 ones and 2 zeros randomly in a row. The probability that the 2 zeros are not adjacent is [Figure]
A. $\frac{1}{3}$
B. $\frac{2}{5}$
C. $\frac{2}{3}$
D. $\frac{4}{5}$
Q12 Function Transformations View
12. Let the domain of function $f(x)$ be $\mathbb{R}$. $f(x+1)$ is an odd function, $f(x+2)$ is an even function. When $x \in [1,2]$, $f(x) = ax^2 + b$. If $f(0) + f(3) = 6$, then $f\left(\frac{4}{2}\right) =$
A. $-\frac{9}{4}$
B. $-\frac{3}{2}$
C. $\frac{7}{4}$
D. $\frac{5}{2}$
II. Fill in the Blank Questions (4 questions in total, 5 points each, 20 points total)
Q13 Tangents, normals and gradients Find tangent line equation at a given point View
13. The equation of the tangent line to the curve $y = \frac{2x - 1}{x + 2}$ at the point $(-1, -3)$ is $\_\_\_\_$.
Q14 Vectors Introduction & 2D Perpendicularity or Parallel Condition View
14. Given vectors $\vec{a} = (3,1)$, $\vec{b} = (1,0)$, $\vec{c} = \vec{a} + k\vec{b}$. If $\vec{a} \perp \vec{c}$, then $k = \_\_\_\_$.
Q15 Circles Area and Geometric Measurement Involving Circles View
15. Let $F_1, F_2$ be the two foci of the ellipse $C: \frac{x^2}{16} + \frac{y^2}{4} = 1$. Let $P, Q$ be two points on $C$ that are symmetric about the origin, and $|PQ| = |F_1F_2|$. Then the area of quadrilateral $PF_1QF_2$ is $\_\_\_\_$. [Figure]
Q16 Trig Graphs & Exact Values View
16. Given that the function $f(x) = 2\cos(\omega x + \varphi)$ has a partial graph as shown in the figure, then the minimum positive integer $x$ satisfying the condition $\left(f(x) - f\left(-\frac{7\pi}{4}\right)\right)\left(f(x) - f\left(\frac{4\pi}{3}\right)\right) > 0$ is $\_\_\_\_$.
III. Solution Questions (70 points total. Show all work, proofs, and calculations. Questions 17-21 are required for all students. Questions 22-23 are optional; students should answer according to requirements.)
Q17 12 marks Chi-squared test of independence View
17. (12 points) Two machine tools, A and B, produce the same type of product. Products are classified by quality into first-grade and second-grade products. To compare the quality of products from the two machine tools, 200 products were produced by each machine tool. The quality statistics are shown in the table below:
First-gradeSecond-gradeTotal
Machine A15050200
Machine B12080200
Total270130400

(1) What are the proportions of first-grade products produced by machine A and machine B respectively?
(2) Can we conclude with 99\% confidence that there is a difference in product quality between machine A and machine B? Attachment: $k^2 = \frac{n(ac - bd)^2}{(a+b)(c+d)(a+c)(b+d)}$,
$P(K^2 \geq k)$0.0500.0100.001
$k$3.8416.63510.828
Q18 12 marks Arithmetic Sequences and Series Prove a Sequence is Arithmetic View
18. (12 points) Given that all terms of the sequence $\{a_n\}$ are positive numbers, and $S_n$ denotes the sum of the first $n$ terms of $\{a_n\}$. Choose two of the following three statements as conditions and prove the remaining one.
(1) The sequence $\{a_n\}$ is an arithmetic sequence;
(2) The sequence $\{\sqrt{S_n}\}$ is an arithmetic sequence;
(3) $a_2 = 3a_1$.
Note: If different combinations are answered correctly, only the first answer will be scored.
Q19 12 marks Areas by integration View
19. (12 points) In a right triangular prism $ABC-A_1B_1C_1$, the lateral face $AA_1B_1B$ is a square, $AB = BC = 2$. $E, F$ are the midpoints of $AC$ and $CC_1$ respectively. $D$ is a point on edge $AB_1$. $BF \perp A_1B_1$.
(1) Prove that $BF \perp CE$;
(2) When $BD$ equals what value, is the sine of the dihedral angle between plane $BCC_1$ and plane $DFE$ minimized? [Figure]
Q20 12 marks Circles Tangent Lines and Tangent Lengths View
20. (12 points) The parabola $C$ has its vertex at the origin $O$ and focus on the $x$-axis. The line $l: x =