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
2020 shanghai

17 maths questions

Q1 4 marks Inequalities Set Operations Using Inequality-Defined Sets View
Given sets $A = \{ 1,2,4 \} , B = \{ 2,3,4 \}$, find $A \cap B =$ $\_\_\_\_$
Q2 4 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
$\lim _ { n \rightarrow \infty } \frac { n + 1 } { 3 n - 1 } =$ $\_\_\_\_$
Q3 4 marks Complex Numbers Arithmetic Modulus Computation View
Given that the complex number $z$ satisfies $z = 1 - 2 i$ ($i$ is the imaginary unit), find $| z | =$ $\_\_\_\_$
Q4 4 marks Matrices Determinant and Rank Computation View
Given the determinant $\left| \begin{array} { l l l } 1 & a & c \\ 2 & d & b \\ 3 & 0 & 0 \end{array} \right| = 6$, find the determinant $\left| \begin{array} { l l } a & c \\ d & b \end{array} \right| =$ $\_\_\_\_$
Q5 4 marks Composite & Inverse Functions Find or Apply an Inverse Function Formula View
Given $f ( x ) = x ^ { 3 }$, find $f ^ { -1 } ( x ) =$ $\_\_\_\_$
Q6 4 marks Measures of Location and Spread View
Given that the median of $a, b, 1, 2$ is 3 and the mean is 4, find $ab =$ $\_\_\_\_$
Q7 5 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
Given $\left\{ \begin{array} { l } x + y \geq 2 \\ y \geq 0 \\ x + 2 y - 3 \leq 0 \end{array} \right.$, find the maximum value of $z = y - 2 x$ as $\_\_\_\_$
Q8 5 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Given that $\left\{ a _ { n } \right\}$ is an arithmetic sequence with non-zero common difference, and $a _ { 1 } + a _ { 10 } = a _ { 9 }$, find $\frac { a _ { 1 } + a _ { 2 } + \cdots a _ { 9 } } { a _ { 10 } } =$ $\_\_\_\_$
Q9 5 marks Combinations & Selection Distribution of Objects to Positions or Containers View
From 6 people, select 4 to work on duty, each person works for 1 day. The first day needs 1 person, the second day needs 1 person, the third day needs 2 people. There are $\_\_\_\_$ ways to arrange them.
Q10 5 marks Conic sections Chord Properties and Midpoint Problems View
For the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$, a line $l$ passes through the right focus $F$ and intersects the ellipse at points $P$ and $Q$, with $P$ in the second quadrant. Given $Q \left( x _ { Q } , y _ { Q } \right)$ and $Q ^ { \prime } \left( x _ { Q } ^ { \prime } , y _ { Q } ^ { \prime } \right)$ both on the ellipse, with $y _ { Q } + y _ { Q } ^ { \prime } = 0$ and $F Q ^ { \prime } \perp P Q$, find the equation of line $l$ as $\_\_\_\_$
Q11 5 marks Curve Sketching Range and Image Set Determination View
Let $a \in \mathbb{R}$. If there exists a function $f ( x )$ with domain $\mathbb{R}$ that satisfies both ``for any $x _ { 0 } \in \mathbb{R}$, the value of $f \left( x _ { 0 } \right)$ is either $x _ { 0 } ^ { 2 }$ or $x _ { 0 }$'' and ``the equation $f ( x ) = a$ has no real solutions'', find the range of $a$ as $\_\_\_\_$
Q12 5 marks Vectors Introduction & 2D Geometric Property Identification via Vectors View
Given vectors $\overrightarrow { a _ { 1 } } , \overrightarrow { a _ { 2 } } , \overrightarrow { b _ { 1 } } , \overrightarrow { b _ { 2 } } , \ldots , \overrightarrow { b _ { k } } \left( k \in \mathbb{N} ^ { * } \right)$ that are pairwise non-parallel in the plane, satisfying $\left| \overrightarrow { a _ { 1 } } - \overrightarrow { a _ { 2 } } \right| = 1$ and $\left| \overrightarrow { a _ { i } } - \overrightarrow { b _ { j } } \right| \in \{ 1,2 \}$ (where $i = 1,2$ and $j = 1,2 , \ldots , k$), find the maximum value of $k$ as $\_\_\_\_$
Q13 5 marks Inequalities Identify Always-True Inequality from Options View
Which of the following inequalities always holds? ( )
A. $a ^ { 2 } + b ^ { 2 } \leq 2 a b$
B. $a ^ { 2 } + b ^ { 2 } \geq - 2 a b$
C. $a + b \geq - 2 \sqrt { | a b | }$
D. $a + b \leq 2 \sqrt { | a b | }$
Q14 5 marks Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View
Given that the equation of line $l$ is $3 x - 4 y + 1 = 0$, which of the following is a parametric equation of $l$? ( )
A. $\left\{ \begin{array} { l } x = 4 + 3 t \\ y = 3 - 4 t \end{array} \right.$
B. $\left\{ \begin{array} { l } x = 4 + 3 t \\ y = 3 + 4 t \end{array} \right.$
C. $\left\{ \begin{array} { l } x = 1 - 4 t \\ y = 1 + 3 t \end{array} \right.$
D. $\left\{ \begin{array} { l } x = 1 + 4 t \\ y = 1 + 3 t \end{array} \right.$
Q15 5 marks Vectors 3D & Lines MCQ: Perpendicularity or Parallelism of Lines and Planes View
In a cube $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ with edge length 10, $P$ is a point on the left face $A D D _ { 1 } A _ { 1 }$. Given that the distance from point $P$ to $A _ { 1 } D _ { 1 }$ is 3 and the distance from point $P$ to $A A _ { 1 }$ is 2, a line through point $P$ parallel to $A _ { 1 } C$ intersects the cube at points $P$ and $Q$. On which face of the cube is point $Q$ located? ( )
A. $A A _ { 1 } B _ { 1 } B$
B. $B B _ { 1 } C _ { 1 } C$
C. $C C _ { 1 } D _ { 1 } D$
D. $A B C D$
Q16 5 marks Curve Sketching Function Properties from Symmetry or Parity View
If there exists $a \in \mathbb{R}$ with $a \neq 0$ such that for all $x \in \mathbb{R}$, the inequality $f ( x + a ) < f ( x ) + f ( a )$ always holds, then function $f ( x )$ is said to have property $P$. Given: $q _ { 1 }$: $f ( x )$ is monotonically decreasing and $f ( x ) > 0$ always holds; $q _ { 2 }$: $f ( x )$ is monotonically increasing and there exists $x _ { 0 } < 0$ such that $f \left( x _ { 0 } \right) = 0$. Which is a sufficient condition for $f ( x )$ to have property $P$? ( )
A. Only $q _ { 1 }$
B. Only $q _ { 2 }$
C. Both $q _ { 1 }$ and $q _ { 2 }$
D. Neither $q _ { 1 }$ nor $q _ { 2 }$
Q17 Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
A square $ABCD$ with side length 1 is rotated around $BC$ to form a cylinder.
(1) Find the surface area of the cylinder;
(2) The square $ABCD$ is rotated counterclockwise by $\frac { \pi } { 2 }$ around $BC$ to position $A _ { 1 } B C D _ { 1 }$. Find the angle between $A D _ { 1 }$ and plane $ABCD$.