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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-science

13 maths questions

Q1 Probability Definitions Proof of Set Membership, Containment, or Structural Property View
1. Given sets $A = \{ 1,2,3 \} , B = \{ 2,3 \}$, then
A. $\mathrm { A } = \mathrm { B }$
B. $\mathrm { A } \cap \mathrm { B } = \varnothing$
C. $A \subset B$
D. $B \subset A$
Q2 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
2. In the arithmetic sequence $\left\{ a _ { n } \right\}$, if $a _ { 2 } = 4 , a _ { 4 } = 2$, then $a _ { 6 } =$
A. $-1$
B. $0$
C. $1$
D. $6$
Q3 Data representation View
3. The stem-and-leaf plot below shows the average monthly temperatures (${ } ^ { \circ } C$) in Chongqing in 2013:
The median of this data set is
A. $19$
B. $20$
C. $21.5$
D. $23$
4. ``$\mathrm { x } > 1$'' is ``$\log _ { \frac { 1 } { 2 } } ( \mathrm { x } + 2 ) < 0$'' a
A. necessary and sufficient condition
B. sufficient but not necessary condition
C. necessary but not sufficient condition
D. neither sufficient nor necessary condition
Q6 Vectors Introduction & 2D Angle or Cosine Between Vectors View
6. If non-zero vectors $\mathbf { a } , \mathbf { b }$ satisfy $| \mathbf { a } | = \frac { 2 \sqrt { 2 } } { 3 } | \mathbf { b } |$ and $(\mathbf{a}-\mathbf{b})\perp(3\mathbf{a}+2\mathbf{b})$, then the angle between $\mathbf { a }$ and $\mathbf { b }$ is
A. $\frac { \pi } { 4 }$
B. $\frac { \pi } { 2 }$
C. $\frac { 3 \pi } { 4 }$
D. $\pi$
Q8 Circles Tangent Lines and Tangent Lengths View
8. Given that the line $l$: $x + a y - 1 = 0 ( a \in R )$ is an axis of symmetry of the circle $C$: $x ^ { 2 } + y ^ { 2 } - 4 x - 2 y + 1 = 0$. A tangent line to circle $C$ is drawn from point $\mathrm { A } ( - 4 , \mathrm { a } )$, with tangent point $B$. Then $| \mathrm { AB } | =$
A. $2$
B. $4 \sqrt { 2 }$
C. $6$
D. $2 \sqrt { 10 }$
Q9 Addition & Double Angle Formulae Simplification of Trigonometric Expressions with Specific Angles View
9. If $\tan \alpha = 2 \tan \frac { \pi } { 5 }$, then $\frac { \cos \left( \alpha - \frac { 3 \pi } { 10 } \right) } { \sin \left( \alpha - \frac { \pi } { 5 } \right) } =$
A. $1$
B. $2$
C. $3$
D. $4$
Q10 Conic sections Eccentricity or Asymptote Computation View
10. Let the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( \mathrm { a } > 0 , \mathrm {~b} > 0 )$ have right focus $F$. A line through $F$ perpendicular to $AF$ intersects the hyperbola at points $\mathrm { B }$ and $\mathrm { C }$. Lines through $\mathrm { B }$ and $\mathrm { C }$ perpendicular to $\mathrm { AC }$ and $\mathrm { AB }$ respectively intersect at point $D$. If the distance from $D$ to line $BC$ is less than $a + \sqrt { a ^ { 2 } + b ^ { 2 } }$, then the range of the slope of the asymptotes of the hyperbola is
A. $( - 1,0 ) \cup ( 0,1 )$
B. $( - \infty , - 1 ) \cup \left( 1 , + \infty \right)$
C. $( - \sqrt { 2 } , 0 ) \cup ( 0 , \sqrt { 2 } )$
D. $( - \infty , - \sqrt { 2 } ) \cup ( \sqrt { 2 } , + \infty )$
II. Fill-in-the-Blank Questions: This section contains 6 questions. Candidates answer 5 of them, each worth 5 points, for a total of 25 points. Write your answers in the corresponding positions on the answer sheet.
Q11 Complex Numbers Arithmetic Modulus Computation View
11. If the modulus of the complex number $\mathrm { a } + \mathrm { bi } ( \mathrm { a } , \mathrm { b } \in \mathrm { R } )$ is $\sqrt { 3 }$, then $( \mathrm { a } + \mathrm { bi } ) ( \mathrm { a } - \mathrm { bi } ) = $ $\_\_\_\_$ .
Q12 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
12. The coefficient of $x ^ { 8 }$ in the expansion of $\left( x ^ { 3 } + \frac { 1 } { 2 \sqrt { x } } \right) ^ { 5 }$ is $\_\_\_\_$ (answer with numerals).
Q13 Sine and Cosine Rules Multi-step composite figure problem View
13. In $\triangle \mathrm { ABC }$, $\mathrm { B } = 120 ^ { \circ } , \mathrm { AB } = \sqrt { 2 }$, and the angle bisector from $A$ is $\mathrm { AD } = \sqrt { 3 }$, then $\mathrm { AC } = $ $\_\_\_\_$ . Note for Candidates: Questions (14), (15), and (16) are optional. Please choose any two to answer. If all three are answered, only the first two will be graded.
Q14 Circles Chord Length and Chord Properties View
14. As shown in question (14), chords $\mathrm { AB }$ and $\mathrm { CD }$ of circle $O$ intersect at point $E$. A tangent line to circle $O$ is drawn through point $A$ and intersects the extension of $DC$ at point $P$. If $P A = 6 , A E = 9 , P C = 3 , C E : E D = 2 : 1$, then $B E = $ $\_\_\_\_$ . [Figure]
Q15 Polar coordinates View
15. The parametric equation of line $l$ is $\left\{ \begin{array} { c } x = - 1 + t \\ y = 1 + t \end{array} \right.$ (where $t$ is the parameter). With the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C$ is $\rho ^ { 2 } \cos 2 \theta = 4 \left( \rho > 0 , \frac { 3 \pi } { 4 } < \theta < \frac { 5 \pi } { 4 } \right)$. The polar coordinates of the intersection point of line $l$ and curve $C$ are $\_\_\_\_$ .
Q16 Modulus function Optimisation of sums of absolute values View
16. If the minimum value of the function $f ( x ) = | x + 1 | + 2 | x - a |$ is 5, then the real number $a = $ $\_\_\_\_$ . III. Solution Questions: This section contains 6 questions, for a total of 75 points. Show your work, proofs, or calculation steps. (17) (This question is worth 13 points: part (I) is worth 5 points, part (II) is worth 8 points) Eating zongzi on Dragon Boat Festival is a traditional custom in China. A plate contains 10 zongzi: 2 with red bean paste, 3 with meat, and 5 plain. The three types of zongzi look identical. Three zongzi are randomly selected. (I) Find the probability that one zongzi of each type is selected. (II) Let $X$ denote the number of red bean paste zongzi selected. Find the probability distribution and mathematical expectation of $X$. (18) (This question is worth 13 points: part (I) is worth 7 points, part (II) is worth 6 points) Given the function $f ( x ) = \sin \left( \frac { \pi } { 2 } - x \right) \sin x - \sqrt { 3 } \cos ^ { 2 } x$ (I) Find the minimum positive period and maximum value of $f ( x )$. (II) Discuss the monotonicity of $f ( x )$ on $\left[ \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } \right]$. (19) (This question is worth 13 points: part (I) is worth 4 points, part (II) is worth 9 points) As shown in question (19), in the triangular pyramid $P - A B C$, $P C \perp$ plane $A B C$, $P C = 3$, $\angle A C B = \frac { \pi } { 2 }$. Points $D$ and $E$ are on segments $A B$ and $B C$ respectively, with $C D = D E = \sqrt { 2 }$, $C E = 2 E B = 2$. (I) Prove that $D E \perp$ plane $P C D$. (II) Find the cosine of the dihedral angle $A - P D - C$. (20) (This question is worth 12 points: part (I) is worth 7 points, part (II) is worth 5 points) Let the function $f ( x ) = \frac { 3 x ^ { 2 } + a x } { e ^ { x } } ( a \in R )$. (I) If $f ( x )$ has an extremum at $x = 0$, determine the value of $a$ and find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 1 , f ( 1 ) )$. (II) If $f ( x )$ is decreasing on $[ 3 , + \infty )$, find the range of values for $a$. (21) (This question is worth 12 points: part (I) is worth 5 points, part (II) is worth 7 points)
[Figure]
Question (19) Figure
As shown in question (21), the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ {