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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
2015 jiangsu

21 maths questions

Q1 Probability Definitions Set Operations View
1. Given sets $A = \{ 1,2,3 \} , B = \{ 2,4,5 \}$, then the number of elements in set $A \cup B$ is $\_\_\_\_$ .
Q2 Measures of Location and Spread View
2. Given a set of data $4,6,5,8,7,6$, then the mean of this set of data is $\_\_\_\_$ .
Q3 Complex Numbers Arithmetic Modulus Computation View
3. Let the complex number z satisfy $z ^ { 2 } = 3 + 4 i$ (where $i$ is the imaginary unit), then the modulus of z is $\_\_\_\_$ .
Q5 Probability Definitions Finite Equally-Likely Probability Computation View
5. A bag contains 4 balls of identical shape and size, including 1 white ball, 1 red ball, and 2 yellow balls. If 2 balls are randomly drawn at once, then the probability that the 2 balls have different colors is $\_\_\_\_$ .
Q6 Vectors Introduction & 2D Expressing a Vector as a Linear Combination View
6. Given vectors $\vec { a } = ( 2,1 ) , \vec { a } = ( 1 , - 2 )$, if $m \vec { a } + n \vec { b } = ( 9 , - 8 ) ( m n \in R )$, then the value of $\mathrm { m } - \mathrm { n }$ is $\_\_\_\_$ .
Q7 Exponential Functions Exponential Equation Solving View
7. The solution set of the inequality $2 ^ { x ^ { 2 } - x } < 4$ is $\_\_\_\_$ .
Q8 Addition & Double Angle Formulae Addition/Subtraction Formula Evaluation View
8. Given $\tan \alpha = - 2 , \tan ( \alpha + \beta ) = \frac { 1 } { 7 }$, then the value of $\tan \beta$ is $\_\_\_\_$ .
Q10 Circles Optimization on a Circle View
10. In the rectangular coordinate system $x O y$, among all circles with center at point $( 1,0 )$ and tangent to the line $m x - y - 2 m - 1 = 0 ( m \in R )$, the standard equation of the circle with the largest radius is $\_\_\_\_$.
Q11 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
11. The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$ and $a _ { n + 1 } - a _ { n } = n + 1 \quad \left( n \in N ^ { * } \right)$, then the sum of the first 10 terms of the sequence $\left\{ \frac { 1 } { a _ { n } } \right\}$ is $\_\_\_\_$.
Q12 Circles Optimization on Conics View
12. In the rectangular coordinate system $x O y$, let $P$ be a moving point on the right branch of the hyperbola $x ^ { 2 } - y ^ { 2 } = 1$. If the distance from point $P$ to the line $x - y + 1 = 0$ is always greater than or equal to c, then the maximum value of the real number c is $\_\_\_\_$.
Q13 Curve Sketching Number of Solutions / Roots via Curve Analysis View
13. Given functions $f ( x ) = | \ln x | , g ( x ) = \left\{ \begin{array} { c } 0,0 < x \leq 1 \\ \left| x ^ { 2 } - 4 \right| - 2 , x > 1 \end{array} \right.$, then the number of real roots of the equation $| f ( x ) + g ( x ) | = 1$ is $\_\_\_\_$.
Q14 Vectors Introduction & 2D Dot Product Computation View
14. Let vectors $a _ { k } = \left( \cos \frac { k \pi } { 6 } , \sin \frac { k \pi } { 6 } + \cos \frac { k \pi } { 6 } \right) ( k = 0,1,2 , \cdots , 12 )$, then the value of $\sum _ { k = 0 } ^ { 12 } \left( a _ { k } \cdot a _ { k + 1 } \right)$ is $\_\_\_\_$.
Q15 Sine and Cosine Rules Find a side length using the cosine rule View
15. In $\triangle ABC$, given $A B = 2 , A C = 3 , A = 60 ^ { \circ }$ .
(1) Find the length of BC;
(2) Find the value of $\sin 2 C$.
Q16 Vectors 3D & Lines Perpendicularity Proof in 3D Geometry View
16. As shown in the figure, in the right triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$, given $A C \perp B C$. Let D be the midpoint of $A B _ { 1 }$, $B _ { 1 } C \cap B C _ { 1 } = E$ . Prove: (1) $D E \parallel$ plane $A A _ { 1 } C C _ { 1 }$
(2) $B C _ { 1 } \perp A B _ { 1 }$ [Figure]
Q17 Straight Lines & Coordinate Geometry Exponential Growth/Decay Modelling with Contextual Interpretation View
17. (This problem is worth 14 points) There are two mutually perpendicular straight-line highways on the periphery of a mountainous area. To further improve the traffic situation in the mountainous area, a plan is made to build a straight-line highway connecting the two highways and the boundary of the mountainous area. Let the two mutually perpendicular highways be $l _ { 1 } , l _ { 2 }$, the boundary curve of the mountainous area be C, and the planned highway be l. As shown in the figure, $\mathrm { M } , \mathrm { N }$ are two endpoints of C. It is measured that the distances from point M to $l _ { 1 } , l _ { 2 }$ are 5 kilometers and 40 kilometers respectively, and the distances from point N to $l _ { 1 } , l _ { 2 }$ are 20 kilometers and 2.5 kilometers respectively. Taking the lines where $l _ { 1 } , l _ { 2 }$ are located as the $\mathrm { x } , \mathrm { y }$ axes respectively, establish a rectangular coordinate system xOy. Assume that the curve C conforms to the function model $y = \frac { a } { x ^ { 2 } + b }$ (where $\mathrm { a } , \mathrm { b }$ are constants). (I) Find the values of $\mathrm { a } , \mathrm { b }$; (II) Let the highway l be tangent to curve C at point P, and the x-coordinate of P is t.
(1) Write out the function expression $f ( t )$ for the length of highway l and its domain;
(2) When t takes what value is the length of highway l shortest? Find the shortest length.
Q18 Conic sections Circle Equation Derivation View
18. (This problem is worth 16 points) As shown in the figure, in the rectangular coordinate system xOy, given that the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has eccentricity $\frac { \sqrt { 2 } } { 2 }$, and the distance from the right focus F to the left directrix l is 3. [Figure]
(1) Find the standard equation of the ellipse;
(2) A line through F intersects the ellipse at points $\mathrm { A } , \mathrm { B }$. The perpendicular bisector of segment AB intersects the line l and AB at points $\mathrm { P } , \mathrm { C }$ respectively. If $\mathrm { PC } = 2 \mathrm { AB }$, find the equation of line AB.
Q19 Stationary points and optimisation Count or characterize roots using extremum values View
19. Given the function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b ( a , b \in R )$.
(1) Discuss the monotonicity of $f ( x )$;
(2) If $b = c - a$ (where the real number c is a constant independent of a), when the function $f ( x )$ has three distinct zeros, the range of a is exactly $( - \infty , - 3 ) \cup \left( 1 , \frac { 3 } { 2 } \right) \cup \left( \frac { 3 } { 2 } , + \infty \right)$, find the value of c.
Q20 Arithmetic Sequences and Series Properties of AP Terms under Transformation View
20. Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ be terms of an arithmetic sequence with positive terms and common difference $\mathrm { d } ( d \neq 0 )$
(1) Prove that $2 ^ { a _ { 1 } } , 2 ^ { a _ { 2 } } , 2 ^ { a _ { 3 } } , 2 ^ { a _ { 4 } }$ form a geometric sequence in order
(2) Do there exist $a _ { 1 } , d$ such that $a _ { 1 } , a _ { 2 } { } ^ { 2 } , a _ { 3 } { } ^ { 3 } , a _ { 4 } { } ^ { 4 }$ form a geometric sequence in order? Explain your reasoning
(3) Do there exist $a _ { 1 } , d$ and positive integers $n , k$ such that $a _ { 1 } { } ^ { n } , a _ { 2 } { } ^ { n + k } , a _ { 3 } { } ^ { n + 3 k } , a _ { 4 } { } ^ { n + 5 k }$ form a geometric sequence in order? Explain your reasoning
Supplementary Problems
Q21 Circles Inscribed/Circumscribed Circle Computations View
21. (Multiple Choice) This problem includes four sub-problems A, B, C, and D. Please select two of them and answer in the corresponding areas. If more are done, the first two sub-problems will be graded. When solving, you should write out text explanations, proofs, or calculation steps.
A. [Elective 4-1: Geometric Proof Selection] (This problem is worth 10 points) In $\triangle A B C$, $A B = A C$, the chord $A E$ of the circumcircle O of $\triangle A B C$ intersects $B C$ at point D. Prove: $\triangle A B D \sim \triangle A E B$ [Figure]
B. [Elective 4-2: Matrices and Transformations] (This problem is worth 10 points) Given $x , y \in R$, the vector $\alpha = \left[ \begin{array} { c } 1 \\ - 1 \end{array} \right]$ is an eigenvector of the matrix $A = \left[ \begin{array} { c c } x & 1 \\ y & 0 \end{array} \right]$ corresponding to the eigenvalue $- 2$. Find the matrix A and its other eigenvalue.
C. [Elective 4-4: Coordinate Systems and Parametric Equations]
The polar equation of circle C is $\rho ^ { 2 } + 2 \sqrt { 2 } \rho \sin \left( \theta - \frac { \pi } { 4 } \right) - 4 = 0$. Find the radius of circle C.
D. [Elective 4-5: Inequalities Selection] Solve the inequality $x + | 2 x + 3 | \geq 3$
Q22 Vectors 3D & Lines Multi-Part 3D Geometry Problem View
22. As shown in the figure, in the quadrangular pyramid $P - A B C D$, given that $P A \perp$ plane $A B C D$, and the quadrilateral $A B C D$ is a right trapezoid, $\angle A B C = \angle B A D = \frac { \pi } { 2 } , P A = A D = 2 , A B = B C = 1$
(1) Find the cosine of the dihedral angle between plane $P A B$ and plane $P C D$;
(2) Point Q is a moving point on segment BP. When the angle between line CQ and DP is minimized, find the length of segment BQ. [Figure]
Q23 Proof by induction Prove a summation identity by induction View
23. Given the set $X = \{ 1,2,3 \} , Y _ { n } = \{ 1,2,3 , m , n \} \left( n \in N ^ { * } \right)$, let $S _ { n } = \left\{ ( a , b ) \mid a \text{ divides } b \text{ or } b \text{ divides } a , a \in X , b \in Y _ { n } \right\}$, and let $f ( n )$ denote the number of elements in set $S _ { n }$.
(1) Write out the value of $f ( 6 )$;
(2) When $n \geq 6$, write out the expression for $f ( n )$ and prove it using mathematical induction.