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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 chongqing-arts

20 maths questions

Q1 5 marks Probability Definitions Set Operations View
Given sets $A = \{ 1,2,3 \} , B = \{ 1,3 \}$, then $\mathrm { A } \cap \mathrm { B } =$
(A) $\{ 2 \}$
(B) $\{ 1,2 \}$
(C) $\{ 1,3 \}$
(D) $\{ 1,2,3 \}$
Q2 5 marks Proof Proof of Equivalence or Logical Relationship Between Conditions View
``$\mathrm { x } = 1$'' is ``$\mathrm { x } ^ { 2 } - 2 x + 1 = 0$'' a
(A) necessary and sufficient condition
(B) sufficient but not necessary condition
(C) necessary but not sufficient condition
(D) neither sufficient nor necessary condition
Q3 5 marks Laws of Logarithms 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
Q5 5 marks Volumes of Revolution Volume from Three-View or Cross-Section Diagram View
The three views of a certain solid are shown in the figure. The volume of this solid is
(A) $\frac { 1 } { 3 } + 2 \pi$
(B) $\frac { 13 \pi } { 6 }$
(C) $\frac { 7 \pi } { 3 }$
(D) $\frac { 5 \pi } { 2 }$
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) $\frac { 5 \pi } { 6 }$
Q8 5 marks Arithmetic Sequences and Series Flowchart or Algorithm Tracing Involving Sequences View
Executing the flowchart shown in figure (8), the output value of $s$ is
(A) $\frac { 3 } { 4 }$
(B) $\frac { 5 } { 6 }$
(C) $\frac { 11 } { 12 }$
(D) $\frac { 25 } { 24 }$
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 Solving quadratics and applications 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 Linear regression 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 Trig Graphs & Exact Values 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 $BC$.