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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
2015 jiangsu

19 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 Conic sections 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 Exponential Equations & Modelling 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 Circles 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
Q22 Moments 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]