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
2018 national-II-science

18 maths questions

Q1 5 marks Complex Numbers Arithmetic Complex Division/Multiplication Simplification View
$\frac { 1 + 2 i } { 1 - 2 i } =$
A. $- \frac { 4 } { 5 } - \frac { 3 } { 5 }$ i
B. $- \frac { 4 } { 5 } + \frac { 3 } { 5 } \mathrm { i }$
C. $- \frac { 3 } { 5 } - \frac { 4 } { 5 } \mathrm { i }$
D. $- \frac { 3 } { 5 } + \frac { 4 } { 5 }$ i
Q2 5 marks Probability Definitions Combinatorial Counting (Non-Probability) View
Given set $A = \left\{ ( x , y ) \left| x ^ { 2 } + y ^ { 2 } \leqslant 3 , x \in \mathbf { Z } , y \in \mathbf { Z } \right. \right\}$, the number of elements in $A$ is
A. 9
B. 8
C. 5
D. 4
Q3 5 marks Curve Sketching Identifying the Correct Graph of a Function View
The graph of function $f ( x ) = \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } } { x ^ { 2 } }$ is approximately
A. [Graph A]
B. [Graph B]
C. [Graph C]
D. [Graph D]
Q4 5 marks Vectors Introduction & 2D Dot Product Computation View
Given vectors $\boldsymbol { a } , \boldsymbol { b }$ satisfying $| \boldsymbol { a } | = 1 , \boldsymbol { a } \cdot \boldsymbol { b } = - 1$, then $\boldsymbol { a } \cdot ( 2 \boldsymbol { a } - \boldsymbol { b } ) =$
A. 4
B. 3
C. 2
D. 0
Q5 5 marks Conic sections Eccentricity or Asymptote Computation View
The eccentricity of the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ is $\sqrt { 3 }$, then its asymptote equation is
A. $y = \pm \sqrt { 2 } x$
B. $y = \pm \sqrt { 3 } x$
C. $y = \pm \frac { \sqrt { 2 } } { 2 } x$
D. $y = \pm \frac { \sqrt { 3 } } { 2 } x$
Q6 5 marks Sine and Cosine Rules Find a side length using the cosine rule View
In $\triangle A B C$, $\cos \frac { C } { 2 } = \frac { \sqrt { 5 } } { 5 } , B C = 1 , A C = 5$, then $A B =$
A. $4 \sqrt { 2 }$
B. $\sqrt { 30 }$
C. $\sqrt { 29 }$
D. $2 \sqrt { 5 }$
Q8 5 marks Probability Definitions Finite Equally-Likely Probability Computation View
Chinese mathematician Chen Jingrun achieved world-leading results in research on Goldbach's conjecture. Goldbach's conjecture states that ``every even number greater than 2 can be expressed as the sum of two prime numbers'', such as $30 = 7 + 23$. Among prime numbers not exceeding 30, if two different numbers are randomly selected, the probability that their sum equals 30 is
A. $\frac { 1 } { 12 }$
B. $\frac { 1 } { 14 }$
C. $\frac { 1 } { 15 }$
D. $\frac { 1 } { 18 }$
Q9 5 marks Vectors 3D & Lines MCQ: Angle Between Skew Lines View
In rectangular prism $A B C D - A _ { 1 } B C _ { 1 } D _ { 1 }$, $A B = B C = 1 , A A _ { 1 } = \sqrt { 3 }$, the cosine of the angle between skew lines $A D _ { 1 }$ and $D B _ { 1 }$ is
A. $\frac { 1 } { 5 }$
B. $\frac { \sqrt { 5 } } { 6 }$
C. $\frac { \sqrt { 5 } } { 5 }$
D. $\frac { \sqrt { 2 } } { 2 }$
Q10 5 marks Addition & Double Angle Formulae Function Analysis via Identity Transformation View
If $f ( x ) = \cos x - \sin x$ is an even function on $[ - a , a ]$, then the maximum value of $a$ is
A. $\frac { \pi } { 4 }$
B. $\frac { \pi } { 2 }$
C. $\frac { 3 \pi } { 4 }$
D. $\pi$
Q11 5 marks Sequences and Series Evaluation of a Finite or Infinite Sum View
Given that $f ( x )$ is an odd function with domain $( - \infty , + \infty )$ satisfying $f ( 1 - x ) = f ( 1 + x )$. If $f ( 1 ) = 2$, then $f ( 1 ) + f ( 2 ) + f ( 3 ) + \cdots + f ( 30 ) =$
A. $- 50$
B. $0$
C. $2$
D. $50$
Q12 5 marks Conic sections Eccentricity or Asymptote Computation View
Let $F _ { 1 } , F _ { 2 }$ be the left and right foci of ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$, $A$ is the left vertex of $C$. Point $P$ is on the line passing through $A$ with slope $\frac { \sqrt { 3 } } { 6 }$. $\triangle P F _ { 1 } F _ { 2 }$ is an isosceles triangle with $\angle F _ { 1 } F _ { 2 } P = 120 ^ { \circ }$, then the eccentricity of $C$ is
A. $\frac { 2 } { 3 }$
B. $\frac { 1 } { 2 }$
C. $\frac { 1 } { 3 }$
D. $\frac { 1 } { 4 }$
Q13 5 marks Tangents, normals and gradients Find tangent line equation at a given point View
The equation of the tangent line to the curve $y = 2 \ln ( x + 1 )$ at the point $( 0,0 )$ is $\_\_\_\_$.
Q14 5 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
If $x , y$ satisfy the constraints $\left\{ \begin{array} { l } x + 2 y - 5 \geqslant 0 , \\ x - 2 y + 3 \geqslant 0 , \\ x - 5 \leqslant 0 , \end{array} \right.$ then the maximum value of $z = x + y$ is $\_\_\_\_$.
Q15 5 marks Addition & Double Angle Formulae Trigonometric Equation Solving via Identities View
Given $\sin \alpha + \cos \beta = 1 , \cos \alpha + \sin \beta = 0$, then $\sin ( \alpha + \beta ) = \_\_\_\_$.
Q17 12 marks Arithmetic Sequences and Series Multi-Part Structured Problem on AP View
(12 points)
Let $S _ { n }$ be the sum of the first $n$ terms of arithmetic sequence $\{ a _ { n } \}$. Given $a _ { 1 } = - 7 , S _ { 3 } = - 15$.
(1) Find the general term formula of $\{ a _ { n } \}$;
(2) Find $S _ { n }$ and the minimum value of $S _ { n }$.
Q18 12 marks Linear regression View
(12 points)
The figure below is a line graph of environmental infrastructure investment $y$ (in units of 100 million yuan) from 2000 to 2016 in a certain region.
To forecast the environmental infrastructure investment for 2018 in this region, two linear regression models were established for $y$ and time variable $t$. Based on data from 2000 to 2016 (time variable $t$ takes values $1, 2, \ldots, 17$ respectively), Model (1) is established: $\hat { y } = - 30.4 + 13.5 t$; based on data from 2010 to 2016 (time variable $t$ takes values $1, 2, \ldots, 7$ respectively), Model (2) is established: $\hat { y } = 99 + 17.5 t$.
(1) Using each of these two models respectively, predict the environmental infrastructure investment for 2018 in this region;
(2) Which model do you think gives a more reliable prediction? Explain your reasoning.
Q19 12 marks Conic sections Circle-Conic Interaction with Tangency or Intersection View
(12 points)
Let the focus of parabola $C : y ^ { 2 } = 4 x$ be $F$. A line $l$ passing through $F$ with slope $k ( k > 0 )$ intersects $C$ at points $A$ and $B$. $| A B | = 8$.
(1) Find the equation of line $l$;
(2) Find the equation of the circle passing through points $A$ and $B$ and tangent to the directrix of $C$.
Q20 12 marks Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
(12 points)
As shown in the figure, in triangular pyramid $P - A B C$, $A B = B C = 2 \sqrt { 2 }$, $P A = P B = P C = A C = 4$, $O$ is the midpoint of $A C$.
(1) Prove that $P O \perp$ plane $A B C$;
(2) Point $M$ is on edge $B C$ such that the dihedral angle $M - P A - C$ is $30 ^ { \circ }$. Find the sine of the angle between $P C$ and plane $P A M$.