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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
2024 national-I_github

19 maths questions

Q1 5 marks Probability Definitions Set Operations View
Given sets $A = \left\{ x \mid - 5 < x ^ { 3 } < 5 \right\} , B = \{ - 3 , - 1,0,2,3 \}$ , then $A \cap B =$
A. $\{ - 1,0 \}$
B. $\{ 2,3 \}$
C. $\{ - 3 , - 1,0 \}$
D. $\{ - 1,0,2 \}$
Q2 5 marks Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
If $\frac { z } { z - 1 } = 1 + \mathrm { i }$ , then $z =$
A. $- 1 - \mathrm { i }$
B. $- 1 + \mathrm { i }$
C. $1 - \mathrm{i}$
D. $1 + \mathrm { i }$
Q3 5 marks Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Given vectors $\boldsymbol { a } = ( 0,1 ) , \boldsymbol { b } = ( 2 , x )$ , if $\boldsymbol { b } \perp ( \boldsymbol { b } - 4 \boldsymbol { a } )$ , then $x =$
A. $- 2$
B. $- 1$
C. $1$
D. $2$
Q4 5 marks Addition & Double Angle Formulae Addition/Subtraction Formula Evaluation View
Given $\cos ( \alpha + \beta ) = m , \tan \alpha \tan \beta = 2$ , then $\cos ( \alpha - \beta ) =$
A. $- 3 m$
B. $- \frac { m } { 3 }$
C. $\frac { m } { 3 }$
D. $3 m$
Q5 5 marks Volumes of Revolution Volume of a 3D Geometric Solid (Pyramid/Tetrahedron) View
A cylinder and a cone have equal base radii and equal lateral surface areas, and both have height $\sqrt { 3 }$ . Then the volume of the cone is
A. $2 \sqrt { 3 } \pi$
B. $3 \sqrt { 3 } \pi$
C. $6 \sqrt { 3 } \pi$
D. $9 \sqrt { 3 } \pi$
Q6 5 marks Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Given function $f ( x ) = \left\{ \begin{array} { l l } - x ^ { 2 } - 2 a x - a , & x < 0 , \\ \mathrm { e } ^ { x } + \ln ( x + 1 ) , & x \geqslant 0 \end{array} \right.$ is monotonically increasing on $\mathbb { R }$ , then the range of $a$ is
A. $( - \infty , 0 ]$
B. $[ - 1,0 ]$
C. $[ - 1,1 ]$
D. $[ 0 , + \infty )$
Q7 5 marks Trig Graphs & Exact Values View
When $x \in [ 0,2 \pi ]$ , the number of intersection points of the curves $y = \sin x$ and $y = 2 \sin \left( 3 x - \frac { \pi } { 6 } \right)$ is
A. $3$
B. $4$
C. $6$
D. $8$
Q8 5 marks Proof by induction Prove a summation inequality by induction View
Given that the domain of function $f ( x )$ is $\mathbb { R }$ , $f ( x ) > f ( x - 1 ) + f ( x - 2 )$ , and when $x < 3$ , $f ( x ) = x$ , then the following conclusion that must be correct is
A. $f ( 10 ) > 100$
B. $f ( 20 ) > 1000$
C. $f ( 10 ) < 1000$
D. $f ( 20 ) < 10000$
Q9 6 marks Normal Distribution Multiple-Choice Conceptual Question on Normal Distribution Properties View
To understand the per-acre income (in units of 10,000 yuan) after promoting exports, a sample was taken from the planting area. The sample mean of per-acre income after promoting exports is $\bar { x } = 2.1$ , and the sample variance is $s ^ { 2 } = 0.01$ . The historical per-acre income $X$ in the planting area follows a normal distribution $N \left( 1.8 , ~ 0.1 ^ { 2 } \right)$ . Assume that the per-acre income $Y$ after promoting exports follows a normal distribution $N \left( \bar { x } , s ^ { 2 } \right)$ . Then (if a random variable $Z$ follows a normal distribution $N \left( \mu , \sigma ^ { 2 } \right)$ , then $P ( Z < \mu + \sigma ) \approx 0.8413$ )
A. $P ( X > 2 ) > 0.2$
B. $P ( X > 2 ) < 0.5$
C. $P ( Y > 2 ) > 0.5$
D. $P ( Y > 2 ) < 0.8$
Q10 6 marks Stationary points and optimisation Find critical points and classify extrema of a given function View
Let function $f ( x ) = ( x - 1 ) ^ { 2 } ( x - 4 )$ , then
A. $x = 3$ is a local minimum point of $f ( x )$
B. When $0 < x < 1$ , $f ( x ) < f \left( x ^ { 2 } \right)$
C. When $1 < x < 2$ , $- 4 < f ( 2 x - 1 ) < 0$
D. When $- 1 < x < 0$ , $f ( 2 - x ) > f ( x )$
Q11 6 marks Conic sections Locus and Trajectory Derivation View
The shape ``$\varnothing$'' can be made into a beautiful ribbon. Consider it as part of the curve $C$ in the figure. It is known that $C$ passes through the origin $O$ , and points on $C$ satisfy: the abscissa is greater than $- 2$ , and the product of the distance to point $F ( 2,0 )$ and the distance to the line $x = a ( a < 0 )$ equals 4 . Then
A. $a = - 2$
B. The point $( 2 \sqrt { 2 } , 0 )$ is on $C$
C. The maximum ordinate of points on $C$ in the first quadrant is 1
D. When point $\left( x _ { 0 } , y _ { 0 } \right)$ is on $C$ , $y _ { 0 } \leqslant \frac { 4 } { x _ { 0 } + 2 }$
Q12 5 marks Conic sections Eccentricity or Asymptote Computation View
Let the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ have left and right foci $F _ { 1 } , F _ { 2 }$ respectively. A line through $F _ { 2 }$ parallel to the $y$-axis intersects $C$ at points $A$ and $B$ . If $\left| F _ { 1 } A \right| = 13 , | A B | = 10$ , then the eccentricity of $C$ is $\_\_\_\_$ .
Q13 5 marks Tangents, normals and gradients Common tangent line to two curves View
If the tangent line to the curve $y = \mathrm { e } ^ { x } + x$ at the point $( 0,1 )$ is also a tangent line to the curve $y = \ln ( x + 1 ) + a$ , then $a = $ $\_\_\_\_$ .
Q14 5 marks Tree Diagrams Multi-Stage Sequential Process View
Person A and Person B each have four cards, with each card labeled with a number. Person A's cards are labeled with the numbers 1, 3, 5, 7, and Person B's cards are labeled with the numbers 2, 4, 6, 8. They play four rounds of competition. In each round, both players randomly select one card from their own cards and compare the numbers. The player with the larger number scores 1 point, and the player with the smaller number scores 0 points. Then each player discards the card used in that round (discarded cards cannot be used in subsequent rounds). The probability that Person A's total score after four rounds is at least 2 is $\_\_\_\_$ .
Q15 13 marks Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
(13 points) Let the sides opposite to angles $A , B , C$ of $\triangle A B C$ be $a , b , c$ respectively. Given $\sin C = \sqrt { 2 } \cos B , a ^ { 2 } + b ^ { 2 } - c ^ { 2 } = \sqrt { 2 } a b$ .
(1) Find $B$ ;
(2) If the area of $\triangle A B C$ is $3 + \sqrt { 3 }$ , find $c$ .
Q16 15 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
(15 points) Given that $A ( 0,3 )$ and $P \left( 3 , \frac { 3 } { 2 } \right)$ are two points on the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ .
(1) Find the eccentricity of $C$ ;
(2) If a line $l$ through $P$ intersects $C$ at another point $B$ , and the area of $\triangle A B P$ is 9 , find the equation of $l$ .
Q17 15 marks Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
(15 points) As shown in the figure, in the quadrangular pyramid $P - A B C D$ , $P A \perp$ base $A B C D , P A = A C = 2$ , $B C = 1 , A B = \sqrt { 3 }$ .
(1) If $A D \perp P B$ , prove that $A D \|$ plane $P B C$ ;
(2) If $A D \perp D C$ , and the sine of the dihedral angle $A - C P - D$ is $\frac { \sqrt { 42 } } { 7 }$ , find $A D$ .
Q18 17 marks Applied differentiation Finding parameter values from differentiability or equation constraints View
(17 points) Given function $f ( x ) = \ln \frac { x } { 2 - x } + a x + b ( x - 1 ) ^ { 3 }$ .
(1) If $b = 0$ and $f ^ { \prime } ( x ) \geqslant 0$ , find the minimum value of $a$ ;
(2) Prove that the curve $y = f ( x )$ is centrally symmetric;
(3) If $f ( x ) > - 2$ if and only if $1 < x < 2$ , find the range of $b$ .
Q19 17 marks Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
(17 points) Let $m$ be a positive integer. The sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is an arithmetic sequence with nonzero common difference. If after removing two terms $a _ { i }$ and $a _ { j } ( i < j )$ , the remaining $4 m$ terms can be evenly divided into $m$ groups, and the 4 numbers in each group form an arithmetic sequence, then the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is called an $( i , j )$ -divisible sequence.
(1) Write out all pairs $( i , j )$ with $1 \leqslant i < j \leqslant 6$ such that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 6 }$ is an $( i , j )$ -divisible sequence;
(2) When $m \geqslant 3$ , prove that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is a $( 2,13 )$ -divisible sequence;
(3) From $1,2 , \cdots , 4 m + 2$ , randomly select two numbers $i$ and $j$ ( $i < j$ ) at once. Let $P _ { m }$ denote the probability that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is an $( i , j )$ -divisible sequence. Prove that $P _ { m } > \frac { 1 } { 8 }$ .