gaokao

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 beijing-science

13 maths questions

Q1 Complex Numbers Arithmetic Complex Division/Multiplication Simplification View
1. $\mathrm { i } ( 2 - \mathrm { i } ) =$
A. $1 + 2 \mathrm { i }$
B. $1 - 2 \mathrm { i }$
C. $- 1 + 2 \mathrm { i }$
D. $- 1 - 2 \mathrm { i }$
Q6 Arithmetic Sequences and Series Properties of AP Terms under Transformation View
6. Let $\left\{ a _ { n } \right\}$ be an arithmetic sequence. The correct conclusion is
A. If $a _ { 1 } + a _ { 2 } > 0$, then $a _ { 2 } + a _ { 3 } > 0$
B. If $a _ { 1 } + a _ { 3 } < 0$, then $a _ { 1 } + a _ { 2 } < 0$
C. If $0 < a _ { 1 } < a _ { 2 }$, then $a _ { 2 } > \sqrt { a _ { 1 } a _ { 3 } }$
D. If $a _ { 1 } < 0$, then $\left( a _ { 2 } - a _ { 1 } \right) \left( a _ { 2 } - a _ { 3 } \right) > 0$
Q7 Curve Sketching Number of Solutions / Roots via Curve Analysis View
7. As shown in the figure, the graph of function $f ( x )$ is the broken line $A C B$. The solution set of the inequality $f ( x ) \geqslant \log _ { 2 } ( x + 1 )$ is [Figure]
A. $\{ x \mid - 1 < x \leqslant 0 \}$
B. $\{ x \mid - 1 \leqslant x \leqslant 1 \}$
C. $\{ x \mid - 1 < x \leqslant 1 \}$
D. $\{ x \mid - 1 < x \leqslant 2 \}$
Q9 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
9. In the expansion of $( 2 + x ) ^ { 5 }$, the coefficient of $x ^ { 3 }$ is $\_\_\_\_$. (Answer with numerals)
Q10 Conic sections Eccentricity or Asymptote Computation View
10. Given that the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - y ^ { 2 } = 1 ( a > 0 )$ has an asymptote $\sqrt { 3 } x + y = 0$, then $a =$ $\_\_\_\_$.
Q12 Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
12. In $\triangle A B C$, $a = 4 , b = 5 , c = 6$, then $\frac { \sin 2 A } { \sin C } =$ $\_\_\_\_$.
Q13 Vectors Introduction & 2D Expressing a Vector as a Linear Combination View
13. In $\triangle A B C$, points $M , N$ satisfy $\overrightarrow { A M } = 2 \overrightarrow { M C } , \overrightarrow { B N } = \overrightarrow { N C }$. If $\overrightarrow { M N } = x \overrightarrow { A B } + y \overrightarrow { A C }$, then $x =$ $\_\_\_\_$ $\_\_\_\_$ ; $y =$ $\_\_\_\_$.
Q14 Exponential Functions Exponential Equation Solving View
14. Let the function $f ( x ) = \left\{ \begin{array} { c c } 2 ^ { x } - a , & x < 1 , \\ 4 ( x - a ) ( x - 2 a ) , & x \geqslant 1 \text { .} \end{array} \right.$
(1) If $a = 1$, then the minimum value of $f ( x )$ is $\_\_\_\_$;
(2) If $f ( x )$ has exactly 2 zeros, then the range of the real number $a$ is $\_\_\_\_$.
III. Answer Questions (6 questions in total, 80 points. Solutions should include written explanations, calculation steps, or proof processes)
Q15 Trig Graphs & Exact Values View
15. (This question is worth 13 points) Given the function $f ( x ) = \sqrt { 2 } \sin \frac { x } { 2 } \cos \frac { x } { 2 } - \sqrt { 2 } \sin ^ { 2 } \frac { x } { 2 }$. (I) Find the minimum positive period of $f ( x )$; (II) Find the minimum value of $f ( x )$ on the interval $[ - \pi , 0 ]$.
Q16 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
16. (This question is worth 13 points) Groups $A$ and $B$ each have 7 patients. Their recovery time (in days) after taking a certain drug is recorded as follows: Group A: $10,11,12,13,14,15,16$ Group B: $12,13,15,16,17,14 , a$ Assume that the recovery times of all patients are mutually independent. Randomly select 1 person from each of groups A and B. The person selected from group A is denoted as patient 甲, and the person selected from group B is denoted as patient 乙. (I) Find the probability that the recovery time of patient 甲 is at least 14 days; (II) If $a = 25$, find the probability that the recovery time of patient 甲 is longer than that of patient 乙; (III) For what value of $a$ are the variances of recovery times for groups A and B equal? (Proof of the conclusion is not required)
Q18 Tangents, normals and gradients Find tangent line equation at a given point View
18. (This question is worth 13 points) Given the function $f ( x ) = \ln \frac { 1 + x } { 1 - x }$. (I) Find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 0 , f ( 0 ) )$; (II) Prove: When $x \in ( 0,1 )$, $f ( x ) > 2 \left( x + \frac { x ^ { 3 } } { 3 } \right)$; (III) Let the real number $k$ be such that $f ( x ) > k \left( x + \frac { x ^ { 3 } } { 3 } \right)$ holds for all $x \in ( 0,1 )$. Find the maximum value of $k$.
Q19 Circles Circle-Related Locus Problems View
19. (This question is worth 14 points) Given the ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ with eccentricity $\frac { \sqrt { 2 } } { 2 }$, point $P ( 0,1 )$ and point $A ( m , n ) ( m \neq 0 )$ are both on the ellipse $C$. The line $P A$ intersects the $x$-axis at point $M$. (I) Find the equation of ellipse $C$ and find the coordinates of point $M$ (expressed in terms of $m , n$); (II) Let $O$ be the origin. Point $B$ is symmetric to point $A$ with respect to the $x$-axis. The line $P B$ intersects the $x$-axis at point $N$. Question: Does there exist a point $Q$ on the $y$-axis such that $\angle O Q M = \angle O N Q$? If it exists, find the coordinates of point $Q$; if it does not exist, explain the reason.
Q20 Sequences and series, recurrence and convergence Applied/contextual sequence problem View
20. (This question is worth 13 points) Given the sequence $\left\{ a _ { n } \right\}$ satisfying: $a _ { 1 } \in \mathbf { N } ^ { * } , a _ { 1 } \leqslant 36$, and $a _ { n + 1 } = \left\{ \begin{array} { l } 2 a _ { n } , a _ { n } \leqslant 18 , \\ 2 a _ { n } - 36 , a _ { n } > 18 \end{array} ( n = 1,2 , \ldots ) \right.$. Let the set $M = \left\{ a _ { n } \mid n \in \mathbf { N } ^ { * } \right\}$. (I) If $a _ { 1