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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 national-II-arts

16 maths questions

Q1 Probability Definitions Set Operations Using Inequality-Defined Sets View
1. Given sets $A = \{ x \mid - 1 < x < 2 \} , B = \{ x \mid 0 < x < 3 \}$, then $A \cup B =$
A. $( - 1,3 )$
B. $( - 1,0 )$
C. $( 0,2 )$
D. $( 2,3 )$
Q2 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
2. If $a$ is a real number and $\frac { 2 + a i } { 1 + i } = 3 + i$, then $a =$
A. $- 4$
B. $- 3$
C. $3$
D. $4$
Q3 Data representation View
3. Based on the bar chart of China's annual carbon dioxide emissions (in units of 10,000 tons) from 2004 to 2013 shown below, which of the following conclusions is incorrect? [Figure]
A. Year-on-year comparison shows that 2008 had the most significant effect in reducing carbon dioxide emissions
B. China's efforts to control carbon dioxide emissions showed results in 2007
C. Since 2006, China's annual carbon dioxide emissions have shown a decreasing trend
D. Since 2006, China's annual carbon dioxide emissions are positively correlated with the year
Q4 Vectors Introduction & 2D Dot Product Computation View
4. Given $\vec { a } = ( 0 , - 1 ) , \vec { b } = ( - 1,2 )$, then $( 2 \vec { a } + \vec { b } ) \cdot \vec { a } =$
A. $- 1$
B. $0$
C. $1$
D. $2$
Q5 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
5. Let $S _ { n }$ be the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 1 } + a _ { 3 } + a _ { 5 } = 3$, then $S _ { 5 } =$ [Figure] [Figure]
A. $5$
B. $7$
C. $9$
D. $11$
Q7 Circles Inscribed/Circumscribed Circle Computations View
7. Given three points $A ( 1,0 ) , B ( 0 , \sqrt { 3 } ) , C ( 2 , \sqrt { 3 } )$, the distance from the circumcenter of $\triangle A B C$ [Figure]
to the origin is
A. $\frac { 5 } { 3 }$
B. $\frac { \sqrt { 21 } } { 3 }$
C. $\frac { 2 \sqrt { 5 } } { 3 }$
D. $\frac { 4 } { 3 }$
Q9 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
9. Given a geometric sequence $\left\{ a _ { n } \right\}$ satisfying $a _ { 1 } = \frac { 1 } { 4 } , a _ { 3 } a _ { 5 } = 4 \left( a _ { 4 } - 1 \right)$, then $a _ { 2 } =$
A. $2$
B. $1$
C. $\frac { 1 } { 2 }$
D. $\frac { 1 } { 8 }$
Q11 Curve Sketching Identifying the Correct Graph of a Function View
11. As shown in the figure, the rectangle's edge, is the midpoint, point moves along the edge, and moves with, denote, express the sum of distances from the moving point to two points as a function of, then the graph of is approximately
[Figure]
A.
[Figure]
B.
[Figure]
C.
[Figure]
D.
Q12 Curve Sketching Solving inequalities involving modulus View
12. Let $f ( x ) = \ln ( 1 + | x | ) - \frac { 1 } { 1 + x ^ { 2 } }$. The range of $x$ for which $f ( x ) > f ( 2 x - 1 )$ holds is
A. $\left( \frac { 1 } { 3 } , 1 \right)$
B. $\left( - \infty , \frac { 1 } { 3 } \right) \cup ( 1 , + \infty )$
C. $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$
D. $\left( - \infty , - \frac { 1 } { 3 } \right) \cup \left( \frac { 1 } { 3 } , + \infty \right)$
II. Fill-in-the-Blank Questions: This section contains 4 questions, 5 points each, 20 points total
Q13 Exponential Functions Parameter Determination from Conditions View
13. Given that the graph of function $f ( x ) = a x ^ { 3 } - 2 x$ passes through the point $( - 1,4 )$, then $a = $ $\_\_\_\_$ .
Q14 Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
14. If $x , y$ satisfy the constraint conditions $\left\{ \begin{array} { c } x + y - 5 \leq 0 \\ 2 x - y - 1 \geq 0 \\ x - 2 y + 1 \leq 0 \end{array} \right.$, then the maximum value of $z = 2 x + y$ is $\_\_\_\_$ .
Q15 Conic sections Equation Determination from Geometric Conditions View
15. A hyperbola passes through the point $( 4 , \sqrt { 3 } )$ and has asymptote equations $y = \pm \frac { 1 } { 2 } x$. The standard equation of this hyperbola is $\_\_\_\_$ .
Q16 Tangents, normals and gradients Determine parameters from function or curve conditions View
16. The tangent line to the curve $y = x + \ln x$ at the point $( 1,1 )$ is tangent to the curve $y = a x ^ { 2 } + ( a + 2 ) x + 1$. Then $a = $ $\_\_\_\_$ .
III. Solution Questions
17 (This question is worth 12 points). In $\triangle A B C$, $D$ is a point on $BC$, $AD$ bisects $\angle B A C$, and $B D = 2 D C$.
(1) Find $\frac { \sin \angle B } { \sin \angle C }$; (II) If $\angle B A C = 60 ^ { \circ }$, find $\angle B$.
Q18 Sine and Cosine Rules Compute area of a triangle or related figure View
18. (This question is worth 12 points). A company randomly surveyed 40 users from regions $A$ and $B$ respectively to understand user satisfaction with its products. Based on user satisfaction scores, a frequency distribution histogram of user satisfaction scores in region $A$ and a frequency distribution table of user satisfaction scores in region $B$ were obtained.
Frequency Distribution Histogram of User Satisfaction Scores in Region A [Figure]
Satisfaction Score Interval\multicolumn{5}{|c|}{Frequency Distribution Table of User Satisfaction Scores in Region B}
Frequency$[ 50,60 )$$[ 60,70 )$$[ 70,80 )$$[ 80,90 )$$[ 90,100 ]$
2814106

(I) Draw the frequency distribution histogram of user satisfaction scores in region $B$ on the answer sheet, and compare the mean and dispersion of satisfaction scores in the two regions through this histogram (no need to calculate specific values, just provide conclusions); [Figure] (II) Based on user satisfaction scores, user satisfaction levels are divided into three categories:
Satisfaction ScoreBelow 7070 to 89At least 90
Satisfaction LevelDissatisfiedSatisfiedVery Satisfied

Estimate which region has a higher probability that a user's satisfaction level is dissatisfied, and explain the reasoning.
Q19 Vectors 3D & Lines Multi-Part 3D Geometry Problem View
19. (This question is worth 12 points). In the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, $A B = 16 , B C = 10 , A A _ { 1 } = 8$. Points $E$ and $F$ are on $A _ { 1 } B _ { 1 }$ and $D _ { 1 } C _ { 1 }$ respectively, with $A _ { 1 } E = D _ { 1 } F = 4$. A plane $\alpha$ passes through points $E$ and $F$ and intersects the faces of the rectangular prism, with the intersection lines forming a square. [Figure] (I) Draw this square in the figure (no need to explain the method or reasoning); (II) Find the ratio of the volumes of the two parts that plane $\alpha$ divides the rectangular prism into.
Q20 Circles Equation Determination from Geometric Conditions View
20. (This question is worth 12 points). The ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has eccentricity $\frac { \sqrt { 2 } } { 2 }$, and the point $( 2 , \sqrt { 2 } )$ lies