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
2015 chongqing-arts

16 maths questions

Q3 5 marks Composite & Inverse Functions Solve a Logarithmic Inequality View
The domain of the function $f ( \mathrm { x } ) = \log _ { 2 } \left( \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 3 \right)$ is
(A) $[ - 3,1 ]$
(B) $( - 3,1 )$
(C) $( - \infty , - 3 ] \cup [ 1 , + \infty )$
(D) $( - \infty , - 3 ) \cup ( 1 , + \infty )$
Q4 5 marks Measures of Location and Spread View
The stem-and-leaf plot of the average monthly temperatures (in $^\circ$C) in Chongqing in 2013 is shown below.
The median of this data set is
(A) 19
(B) 20
(C) 21.5
(D) 23
Q6 5 marks Addition & Double Angle Formulae Addition/Subtraction Formula Evaluation View
If $\tan a = \frac { 1 } { 3 } , \tan ( a + b ) = \frac { 1 } { 2 }$, then $\tan b =$
(A) $\frac { 1 } { 7 }$
(B) $\frac { 1 } { 6 }$
(C) $\frac { 5 } { 7 }$
(D) $\frac { 5 } { 6 }$
Q7 5 marks Vectors Introduction & 2D Angle or Cosine Between Vectors View
Given non-zero vectors $\vec { a } , \vec { b }$ satisfying $| \vec { b } | = 4 | \vec { a } |$ and $\vec { a } \perp ( 2 \vec { a } + \vec { b } )$, the angle between $\vec { a }$ and $\vec { b }$ is
(A) $\frac { \pi } { 3 }$
(B) $\frac { \pi } { 2 }$
(C) $\frac { 2 \pi } { 3 }$
(D) $\frac { 5 \pi } { 6 }$
Q9 5 marks Conic sections Eccentricity or Asymptote Computation View
For the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( \mathrm { a } > 0 , \mathrm {~b} > 0 )$, let $F$ be the right focus, and $\mathrm { A } _ { 1 } , \mathrm {~A} _ { 2 }$ be the left and right vertices respectively. A line through $F$ perpendicular to $A _ { 1 } A _ { 2 }$ intersects the hyperbola at points $B$ and $C$. If $A _ { 1 } B \perp A _ { 2 } C$, then the slope of the asymptotes of the hyperbola is
(A) $\pm \frac { 1 } { 2 }$
(B) $\pm \frac { \sqrt { 2 } } { 2 }$
(C) $\pm 1$
(D) $\pm \sqrt { 2 }$
Q10 5 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
If the system of inequalities $\left\{ \begin{array} { c } x + y - 2 \leq 0 \\ x + 2 y - 2 \geq 0 \\ x - y + 2 m \geq 0 \end{array} \right.$ represents a triangular region with area equal to $\frac { 4 } { 3 }$, then the value of $m$ is
(A) $-3$
(B) $1$
(C) $\frac { 4 } { 3 }$
(D) $3$
Q11 5 marks Complex Numbers Arithmetic Identifying Real/Imaginary Parts or Components View
The real part of the complex number $( 1 + 2 i ) i$ is $\_\_\_\_$ .
Q12 5 marks Circles Tangent Lines and Tangent Lengths View
If point $\mathrm { P } ( 1,2 )$ lies on a circle centered at the origin, then the equation of the tangent line to the circle at point $P$ is $\_\_\_\_$ .
Q13 5 marks Sine and Cosine Rules Find a side length using the cosine rule View
In $\triangle A B C$, let the sides opposite to angles $\mathrm { A } , \mathrm { B } , \mathrm { C }$ be $a , b , c$ respectively. Given $a = 2 , \cos C = - \frac { 1 } { 4 } , 3 \sin A = 2 \sin B$, then $\mathrm { c } =$ $\_\_\_\_$ .
Q14 5 marks Inequalities Optimization or extremal value of an expression via completing the square View
Let $a , b > 0 , a + b = 5$. The maximum value of $\sqrt { a + 1 } + \sqrt { b + 3 }$ is $\_\_\_\_$ .
Q15 5 marks Discriminant and conditions for roots Probability involving discriminant conditions View
A number $p$ is randomly chosen from the interval $[ 0,5 ]$. The probability that the equation $x ^ { 2 } + 2 p x + 3 p - 2 = 0$ has two negative roots is $\_\_\_\_$ .
Q16 13 marks Arithmetic Sequences and Series Multi-Part Structured Problem on AP View
An arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 3 } = 2$ and the sum of the first 3 terms $S _ { 3 } = \frac { 9 } { 2 }$ .
(I) Find the general term formula for $\left\{ a _ { n } \right\}$;
(II) A geometric sequence $\left\{ b _ { n } \right\}$ satisfies $b _ { 1 } = a _ { 1 } , b _ { 4 } = a _ { 15 }$. Find the sum of the first $n$ terms $T _ { n }$ of $\left\{ b _ { n } \right\}$.
Q17 13 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
With the development of China's economy, residents' savings deposits have increased year by year. The table below shows the year-end balance of urban and rural residents' RMB savings deposits in a certain region:
Year20102011201220132014
Time code $t$12345
\begin{tabular}{ l } Savings deposits $y$
(hundred billion yuan)
& 5 & 6 & 7 & 8 & 10 \hline \end{tabular}
(I) Find the regression equation $\hat { y } = \hat { b } t + \hat { a }$ for $y$ with respect to $t$.
(II) Use the regression equation to predict the RMB savings deposits in this region for 2015 ($t = 6$). Note: In the regression equation $\hat { y } = \hat { b } t + \hat { a }$,
$$\hat { b } = \frac { \sum _ { i = 1 } ^ { n } t _ { i } y _ { i } - n \overline { t } \overline { y } } { \sum _ { i = 1 } ^ { n } t _ { i } ^ { 2 } - n \bar { t } ^ { 2 } } , \hat { \mathrm { a } } = \overline { \mathrm { y } } - \hat { \mathrm { b } } \overline { \mathrm { t } }$$
Q18 13 marks Addition & Double Angle Formulae Function Analysis via Identity Transformation View
Given the function $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { 2 } \sin 2 \mathrm { x } - \sqrt { 3 } \cos ^ { 2 } x$ .
(I) Find the minimum positive period and minimum value of $\mathrm { f } ( \mathrm { x } )$;
(II) The graph of function $\mathrm { f } ( \mathrm { x } )$ is transformed by stretching each point's horizontal coordinate to twice the original length while keeping the vertical coordinate unchanged, resulting in the graph of function $\mathrm { g } ( \mathrm { x } )$. When $\mathrm { x } \in \left[ \frac { \pi } { 2 } , \pi \right]$, find the range of $\mathrm { g } ( \mathrm { x } )$.
Q19 12 marks Stationary points and optimisation Determine parameters from given extremum conditions View
Given the function $\mathrm { f } ( \mathrm { x } ) = \mathrm { a } x ^ { 3 } + x ^ { 2 } ( \mathrm { a } \in \mathrm { R } )$ has an extremum at $\mathrm { x } = - \frac { 4 } { 3 }$ .
(I) Determine the value of $a$;
(II) Let $\mathrm { g } ( \mathrm { x } ) = \mathrm { f } ( \mathrm { x } ) e ^ { x }$. Discuss the monotonicity of $\mathrm { g } ( \mathrm { x } )$.
Q20 12 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
As shown in the figure for question (20), in the triangular pyramid $\mathrm { P } - \mathrm { ABC }$, plane $\mathrm { PAC } \perp$ plane $\mathrm { ABC }$, $\angle \mathrm { ABC } = \frac { \pi } { 2 }$. Points $D$ and $E$ lie on segment $AC$ with $\mathrm { AD } = \mathrm { DE } = \mathrm { EC } = 2$, $\mathrm { PD } = \mathrm { PC } = 4$. Point $F$ lies on segment $AB$ with $\mathrm { EF } \parallel \mathrm { BC }$ .
(I) Prove that $\mathrm { AB } \perp$ plane $PFE$ .
(II) If the volume of the quadrangular pyramid $\mathrm { P } - \mathrm { DFBC }$ is 7, find the length of segment $BC$.