gaokao

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
2019 national-III-science

16 maths questions

Q1 5 marks Probability Definitions Set Operations Using Inequality-Defined Sets View
Given sets $A = \{ - 1,0,1,2 \} , B = \left\{ x \mid x ^ { 2 } \leqslant 1 \right\}$ , then $A \cap B =$
A. $\{ - 1,0,1 \}$
B. $\{ 0,1 \}$
C. $\{ - 1,1 \}$
D. $\{ 0,1,2 \}$
Q2 5 marks Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
If $z ( 1 + \mathrm { i } ) = 2 \mathrm { i }$ , then $z =$
A. $- 1 - \mathrm { i }$
B. $- 1 + \mathrm { i }$
C. $1 - \mathrm { i }$
D. $1 + \mathrm { i }$
Q4 5 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View
The coefficient of $x ^ { 3 }$ in the expansion of $\left( 1 + 2 x ^ { 2 } \right) ( 1 + x ) ^ { 4 }$ is
A. 12
B. 16
C. 20
D. 24
Q5 5 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
A geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms has the sum of its first 4 terms equal to 15, and $a _ { 5 } = 3 a _ { 3 } + 4 a _ { 1 }$ . Then $a _ { 3 } =$
A. 16
B. 8
C. 4
D. 2
Q6 5 marks Differentiating Transcendental Functions Determine unknown parameters from tangent conditions View
The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a \mathrm { e } )$ has equation $y = 2 x + b$ . Then
A. $a = \mathrm { e } , b = - 1$
B. $a = \mathrm { e } , b = 1$
C. $a = \mathrm { e } ^ { - 1 } , b = 1$
D. $a = \mathrm { e } ^ { - 1 } , b = - 1$
Q7 5 marks Curve Sketching Identifying the Correct Graph of a Function View
The graph of the function $y = \frac { 2 x ^ { 3 } } { 2 ^ { x } + 2 ^ { - x } }$ on $[ - 6,6 ]$ is approximately
A. [graph A]
B. [graph B]
C. [graph C]
D. [graph D]
Q8 5 marks Vectors 3D & Lines Coplanarity and Relative Position of Planes View
As shown in the figure, point $N$ is the center of square $ABCD$, $\triangle ECD$ is an equilateral triangle, plane $ECD \perp$ plane $ABCD$, and $M$ is the midpoint of segment $ED$. Then
A. $BM = EN$, and lines $BM$ and $EN$ are intersecting lines
B. $BM \neq EN$, and lines $BM$ and $EN$ are intersecting lines
C. $BM = EN$, and lines $BM$ and $EN$ are skew lines
D. $BM \neq EN$, and lines $BM$ and $EN$ are skew lines
Q9 5 marks Arithmetic Sequences and Series Algorithmic/Computational Implementation for Sequences and Series View
Executing the flowchart on the right, if the input $\varepsilon$ is 0.01, then the output value of $s$ equals
A. $2 - \frac { 1 } { 2 ^ { 4 } }$
B. $2 - \frac { 1 } { 2 ^ { 5 } }$
C. $2 - \frac { 1 } { 2 ^ { 6 } }$
D. $2 - \frac { 1 } { 2 ^ { 7 } }$
Q10 5 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
The right focus of the hyperbola $C : \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 2 } = 1$ is $F$. Point $P$ is on one of the asymptotes of $C$, and $O$ is the origin. If $| PO | = | PF |$, then the area of $\triangle PFO$ is
A. $\frac { 3 \sqrt { 2 } } { 4 }$
B. $\frac { 3 \sqrt { 2 } } { 2 }$
C. $2 \sqrt { 2 }$
D. $3 \sqrt { 2 }$
Q11 5 marks Curve Sketching Compare or Order Logarithmic Values View
Let $f ( x )$ be an even function with domain $\mathbf { R }$ that is monotonically decreasing on $( 0 , + \infty )$. Then
A. $f \left( \log _ { 3 } \frac { 1 } { 4 } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right)$
B. $f \left( \log _ { 3 } \frac { 1 } { 4 } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right)$
C. $f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$
D. $f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$
Q12 5 marks Trig Graphs & Exact Values Count zeros or intersection points involving trigonometric curves View
Let the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 5 } \right) ( \omega > 0 )$. It is known that $f ( x )$ has exactly 5 zeros on $[ 0,2 \pi ]$. The following are four conclusions:
(1) $f ( x )$ has exactly 3 local maximum points on $( 0,2 \pi )$
(2) $f ( x )$ has exactly 2 local minimum points on $( 0,2 \pi )$
(3) $f ( x )$ is monotonically increasing on $\left( 0 , \frac { \pi } { 10 } \right)$
(4) The range of $\omega$ is $\left[ \frac { 12 } { 5 } , \frac { 29 } { 10 } \right)$
The numbers of all correct conclusions are
A. (1)(4)
B. (2)(3)
C. (1)(2)(3)
D. (1)(3)(4)
Q13 5 marks Vectors Introduction & 2D Angle or Cosine Between Vectors View
Given that $\boldsymbol { a } , \boldsymbol { b }$ are unit vectors and $\boldsymbol { a } \cdot \boldsymbol { b } = 0$ , if $\boldsymbol { c } = 2 \boldsymbol { a } - \sqrt { 5 } \boldsymbol { b }$ , then $\cos \langle \boldsymbol { a } , \boldsymbol { c } \rangle =$ \_\_\_\_\_\_.
Q14 5 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 1 } \neq 0 , a _ { 2 } = 3 a _ { 1 }$ , then $\frac { S _ { 10 } } { S _ { 5 } } =$ \_\_\_\_\_\_.
Q15 5 marks Circles Triangle or Quadrilateral Area and Perimeter with Foci View
Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$ , and $M$ be a point on $C$ in the first quadrant. If $\triangle M F _ { 1 } F _ { 2 }$ is an isosceles triangle, then the coordinates of $M$ are \_\_\_\_\_\_.
Q16 5 marks Volumes of Revolution Volume by Displacement or Composite Solid View
Students engage in labor practice at a factory using 3D printing technology to create models. As shown in the figure, the model is a rectangular prism $ABCD - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ with a square pyramid $O - EFGH$ removed, where $O$ is the center of the rectangular prism, and $E , F , G , H$ are the midpoints of the respective edges. $AB = BC = 6 \mathrm{~cm} , AA _ { 1 } = 4 \mathrm{~cm}$. The density of the 3D printing material is $0.9 \mathrm{~g} / \mathrm{cm} ^ { 3 }$. Disregarding printing losses, the mass of material needed to create this model is \_\_\_\_\_\_.
Q23 10 marks Completing the square and sketching Optimization Subject to an Algebraic Constraint View
Let $x, y, z \in \mathbf{R}$ and $x + y + z = 1$.
(1) Find the minimum value of $(x-1)^2 + (y+1)^2 + (z+1)^2$;
(2) If $(x-2)^2 + (y-1)^2 + (z-a)^2 \geqslant \frac{1}{3}$ holds for all $x, y, z$ satisfying $x + y + z = 1$, prove that $a \leqslant -3$ or $a \geqslant -1$.