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Papers (71)
2025
national-I 18 national-II 18 national-I_eol 18
2024
beijing 19 national-II 18 national-I_github 18
2023
national-A-science 20 national-B-science 15 national-II 7
2022
national-A-arts 13 national-A-science 11 national-B-arts 19 national-B-science 17 national-I 18 national-II 7
2021
national-A-arts 19 national-A-science 16 national-I 12 national-II 10
2020
national-I-arts 21 national-I-science 9 national-II-science 14 national-III-arts 22 national-III-science 16 shanghai 14
2019
beijing-science 17 national-I-arts 17 national-I-science 13 national-I-science_gkztc 18 national-II-arts 13 national-II-science 8 national-II-science_gkztc 14 national-III-arts 11 national-III-science 16 national-III-science_gkztc 15
2018
national-I-arts 14 national-I-science 13 national-II-arts 20 national-II-science 19 national-III-science 17
2017
national-I-arts 12 national-I-science 13 national-II-arts 14 national-II-science 9
2016
national-I 7
2015
anhui-arts 14 beijing-arts 18 beijing-science 16 chongqing-arts 16 chongqing-science 13 fujian-science 18 hubei-arts 18 hubei-science 15 hunan-arts 20 hunan-science 11 jiangsu 21 national-II-arts 16 national-II-science 16 shaanxi-science 18 sichuan-arts 16 sichuan-science 21 tianjin-arts 18 tianjin-science 17 zhejiang-arts 14 zhejiang-science 15
2011
shanghai-arts 21
2010
shanghai-arts 20 shanghai-science 12
2004
shanghai-science 17
Unknown
sample_questions 3 sample_questions_testmocks 6
2022 national-I

18 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. $2 m + 3 n$
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 Exponential Functions 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. $a < c < b$
Q9 Vectors 3D & Lines 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 Circles 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 Arithmetic Sequences and Series 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 Addition & Double Angle Formulae 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 $\frac { a ^ { 2 } + b ^ { 2 } } { c ^ { 2 } }$ .
Q19 12 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
19. (12 points) As shown in the figure, a right triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$ has volume 4, and the area of $\triangle A _ { 1 } B C$ is $2 \sqrt { 2 }$.
(1) Find the distance from $A$ to plane $A _ { 1 } B C$;
(2) Let $D$ be the midpoint of $A _ { 1 } C$, with $A A _ { 1 } = A B$ and plane $A _ { 1 } B C \perp$ plane [Figure] $A B B _ { 1 } A _ { 1 }$. Find the sine of the dihedral angle $A - B D - C$.
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 ) } \