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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
2022 national-I

19 maths questions

Q1 Inequalities Set Operations Using Inequality-Defined Sets View
1. If $M = \{ x \mid \sqrt { x } < 4 \}$ and $N = \{ x \mid 3 x \geqslant 1 \}$, then $M \cap N =$
A. $\{ x \mid 0 \leqslant x < 2 \}$
B. $\left\{ x \left\lvert \, \frac { 1 } { 3 } \leqslant x < 2 \right. \right\}$
C. $\{ x \mid 3 \leqslant x < 16 \}$
D. $\left\{ x \left\lvert \, \frac { 1 } { 3 } \leqslant x < 16 \right. \right\}$
Q2 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
2. If $\mathrm { i } ( 1 - z ) = 1$, then $z + \bar { z } =$
A. $- 2$
B. $- 1$
C. $1$
D. $2$
Q3 Vectors Introduction & 2D Expressing a Vector as a Linear Combination View
3. In $\triangle A B C$, point $D$ is on side $A B$ with $B D = 2 D A$. Let $\overrightarrow { C A } = m$ and $\overrightarrow { C D } = n$. Then $\overrightarrow { C B } =$
A. $3 m - 2 n$
B. $- 2 m + 3 n$
C. $3 \boldsymbol { m } + 2 \boldsymbol { n }$
D. $2 m + 3 n$
Q4 Volumes of Revolution Volume of a 3D Geometric Solid (Pyramid/Tetrahedron) View
4. The South-to-North Water Diversion Project has alleviated water shortages in some northern regions, with part of the water stored in a certain reservoir. When the water level of the reservoir is at an elevation of 148.5 m, the corresponding water surface area is $140.0 \mathrm {~km} ^ { 2 }$; when the water level is at an elevation of 157.5 m, the corresponding water surface area is $180.0 \mathrm {~km} ^ { 2 }$. Treating the shape of the reservoir between these two water levels as a frustum, the volume of water added when the water level rises from an elevation of 148.5 m to 157.5 m is approximately ( $\sqrt { 7 } \approx 2.65$ )
A. $1.0 \times 10 ^ { 9 } \mathrm {~m} ^ { 3 }$
B. $1.2 \times 10 ^ { 9 } \mathrm {~m} ^ { 3 }$
C. $1.4 \times 10 ^ { 9 } \mathrm {~m} ^ { 3 }$
D. $1.6 \times 10 ^ { 9 } \mathrm {~m} ^ { 3 }$
Q5 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
5. From 7 integers from 2 to 8, two different numbers are randomly selected. The probability that these two numbers are coprime is
A. $\frac { 1 } { 6 }$
B. $\frac { 1 } { 3 }$
C. $\frac { 1 } { 2 }$
D. $\frac { 2 } { 3 }$
Q6 Trig Graphs & Exact Values View
6. Let the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 4 } \right) + b$ ( $\omega > 0$ ) have minimum positive period $T$. If $\frac { 2 \pi } { 3 } < T < \pi$ and the graph of $y = f ( x )$ is symmetric about the point $\left( \frac { 3 \pi } { 2 } , 2 \right)$, then $f \left( \frac { \pi } { 2 } \right) =$
A. $1$
B. $\frac { 3 } { 2 }$
C. $\frac { 5 } { 2 }$
D. $3$
Q7 Laws of Logarithms Compare or Order Logarithmic Values View
7. Let $a = 0.1 \mathrm { e } ^ { 0.1 }$, $b = \frac { 1 } { 9 }$, $c = - \ln 0.9$. Then
A. $a < b < c$
B. $c < b < a$
C. $c < a < b$
D. $a < c < b$
Q8 Volumes of Revolution Volume of a 3D Geometric Solid (Pyramid/Tetrahedron) View
8. A regular square pyramid has lateral edge length $l$, and all its vertices lie on the same sphere. If the volume of the sphere is $36 \pi$ and $3 \leqslant l \leqslant 3 \sqrt { 3 }$, then the range of the volume of the regular square pyramid is
A. $\left[ 18 , \frac { 81 } { 4 } \right]$
B. $\left[ \frac { 27 } { 4 } , \frac { 81 } { 4 } \right]$
C. $\left[ \frac { 27 } { 4 } , \frac { 64 } { 3 } \right]$
D. $[ 18,27 ]$
II. Multiple Choice Questions: This section contains 4 questions, each worth 5 points, for a total of 20 points. For each question, there may be multiple correct options. Full marks are awarded for selecting all correct options, 2 points for partially correct selections, and 0 points if any incorrect option is selected.
Q9 Vectors Introduction & 2D Angle or Cosine Between Vectors View
9. Given a cube $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, then
A. The angle between lines $B C _ { 1 }$ and $D A _ { 1 }$ is $90 ^ { \circ }$
B. The angle between lines $B C _ { 1 }$ and $C A _ { 1 }$ is $90 ^ { \circ }$
C. The angle between line $B C _ { 1 }$ and plane $B B _ { 1 } D _ { 1 } D$ is $45 ^ { \circ }$
D. The angle between line $B C _ { 1 }$ and plane $A B C D$ is $45 ^ { \circ }$
Q10 Stationary points and optimisation Find critical points and classify extrema of a given function View
10. Given the function $f ( x ) = x ^ { 3 } - x + 1$, then
A. $f ( x )$ has two extreme points
B. $f ( x )$ has three zeros
C. The point $( 0,1 )$ is a center of symmetry of the curve $y = f ( x )$
D. The line $y = 2 x$ is a tangent line to the curve $y = f ( x )$
Q11 Conic sections Tangent and Normal Line Problems View
11. Let $O$ be the origin. Point $A ( 1,1 )$ lies on the parabola $C : x ^ { 2 } = 2 p y$ ( $p > 0$ ). A line through point $B ( 0 , - 1 )$ intersects $C$ at points $P$ and $Q$. Then
A. The directrix of $C$ is $y = - 1$
B. Line $A B$ is tangent to $C$
C. $| O P | \cdot | O Q | > | O A | ^ { 2 }$
D. $| B P | \cdot | B Q | > | B A | ^ { 2 }$
Q12 Function Transformations View
12. Let the function $f ( x )$ and its derivative $f ^ { \prime } ( x )$ both have domain $\mathbf { R }$. Let $g ( x ) = f ^ { \prime } ( x )$. If $f \left( \frac { 3 } { 2 } - 2 x \right)$ and $g ( 2 + x )$ are both even functions, then
A. $f ( 0 ) = 0$
B. $g \left( - \frac { 1 } { 2 } \right) = 0$
C. $f ( - 1 ) = f ( 4 )$
D. $g ( - 1 ) = g ( 2 )$
III. Fill-in-the-Blank Questions: This section contains 4 questions, each worth 5 points, for a total of 20 points.
Q13 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View
13. The coefficient of $x ^ { 2 } y ^ { 6 }$ in the expansion of $\left( 1 - \frac { y } { x } \right) ( x + y ) ^ { 8 }$ is $\_\_\_\_$ (answer with a number).
Q14 Circles Tangent Lines and Tangent Lengths View
14. Write the equation of a line that is tangent to both the circle $x ^ { 2 } + y ^ { 2 } = 1$ and the circle $( x - 3 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 16$: $\_\_\_\_$ .
Q15 Tangents, normals and gradients Existence or count of tangent lines with given properties View
15. If the curve $y = ( x + a ) \mathrm { e } ^ { x }$ has two tangent lines passing through the origin, then the range of $a$ is $\_\_\_\_$ .
Q16 Circles Chord Length and Chord Properties View
16. Given an ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ ( $a > b > 0$ ), with upper vertex $A$, two foci $F _ { 1 }$ and $F _ { 2 }$, and eccentricity $\frac { 1 } { 2 }$. A line through $F _ { 1 }$ perpendicular to $A F _ { 2 }$ intersects $C$ at points $D$ and $E$, with $| D E | = 6$. The perimeter of $\triangle A D E$ is $\_\_\_\_$ .
IV. Solution Questions: This section contains 6 questions, for a total of 70 points. Solutions should include explanations, proofs, or calculation steps.
Q17 10 marks Sequences and series, recurrence and convergence Closed-form expression derivation View
17. (10 points) Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given that $a _ { 1 } = 1$ and $\left\{ \frac { S _ { n } } { a _ { n } } \right\}$ is an arithmetic sequence with common difference $\frac { 1 } { 3 }$.
(1) Find the general term formula for $\left\{ a _ { n } \right\}$;
(2) Prove that $\frac { 1 } { a _ { 1 } } + \frac { 1 } { a _ { 2 } } + \cdots + \frac { 1 } { a _ { n } } < 2$ .
Q18 12 marks Trig Proofs Triangle Trigonometric Relation View
18. (12 points) Let the sides opposite to angles $A$, $B$, $C$ of $\triangle A B C$ be $a$, $b$, $c$ respectively. Given that $\frac { \cos A } { 1 + \sin A } = \frac { \sin 2 B } { 1 + \cos 2 B }$.
(1) If $C = \frac { 2 \pi } { 3 }$, find $B$;
(2) Find the minimum value of $\frac { a ^ { 2 } + b ^ { 2 } } { c ^ { 2 } }$ .
Q20 12 marks Chi-squared test of independence View
20. (12 points)
A medical team conducted a study on the relationship between a certain endemic disease in a region and the hygiene habits of local residents (hygiene habits are classified as either good or not sufficiently good). Among patients with the disease, 100 cases were randomly surveyed (called the case group), and among people without the disease, 100 people were randomly surveyed (called the control group). The following data were obtained:
Not Sufficiently GoodGood
Case Group4060
Control Group1090

(1) Can we conclude with 99\% confidence that there is a difference in hygiene habits between the group with the disease and the group without the disease?
(2) From the population of the region, one person is randomly selected. Let $A$ denote the event ``the selected person has not sufficiently good hygiene habits'' and $B$ denote the event ``the selected person has the disease''. The ratio $\frac { P ( B \mid A ) } { P ( \bar { B } \mid A ) }$ to $\frac { P ( B \mid \bar { A } ) } { P ( \bar { B } \mid \bar { A } ) }$ is a measure of the risk level of the disease associated with not sufficiently good hygiene habits. Let this measure be denoted as $R$.
(i) Prove that $R = \frac { P ( A \mid B ) } { P ( \bar { A } \mid B ) } \