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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 national-II-science

16 maths questions

Q1 5 marks Probability Definitions Set Operations View
Given sets $\mathrm { A } = \{ - 2 , - 1,0,2 \} , \mathrm { B } = \{ \mathrm { x } \mid ( \mathrm { x } - 1 ) ( \mathrm { x } + 2 ) < 0 \}$ , then $\mathrm { A } \cap \mathrm { B } =$
(A) $\{ - 1,0 \}$
(B) $\{ 0,1 \}$
(C) $\{ - 1,0,1 \}$
(D) $\{ 0,1,2 \}$
Q2 5 marks Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
If $a$ is a real number and $(2 + \mathrm { ai})(a - 2\mathrm{i}) = -4\mathrm{i}$, then $a =$
(A) $-1$
(B) $0$
(C) $1$
(D) $2$
Q3 5 marks Data representation View
Based on the bar chart of China's sulfur dioxide emissions (in units of 10,000 tons) from 2004 to 2013 shown below, which of the following conclusions is incorrect?
(A) Year-on-year comparison shows that 2008 had the most significant reduction in sulfur dioxide emissions
(B) China's treatment of sulfur dioxide emissions became evident in 2007
(C) Since 2006, China's annual sulfur dioxide emissions have shown a decreasing trend
(D) Since 2006, China's annual sulfur dioxide emissions are positively correlated with the year
Q4 5 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
A geometric sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 3 , a _ { 1 } + a _ { 3 } + a _ { 5 } = 21$ , then $a _ { 3 } + a _ { 5 } + a _ { 7 } =$
(A) $21$
(B) $42$
(C) $63$
(D) $84$
Q5 5 marks Function Transformations View
Let the function $\left\{ a _ { \mathrm { n } } \right\} =$ , then $f(-2) + f(0) =$
(A) $3$
(B) $6$
(C) $9$
(D) $12$
Q7 5 marks Circles Chord Length and Chord Properties View
The circle passing through three points $A ( 1,3 ) , B ( 4,2 ) , C ( 1,7 )$ intersects the $y$-axis at points $\mathrm { M }$ and $\mathrm { N }$. Then $| M N | =$
(A) $2 \sqrt { 6 }$
(B) $8$
(C) $4 \sqrt { 6 }$
(D) $10$
Q10 5 marks Curve Sketching Identifying the Correct Graph of a Function View
As shown in the figure, rectangle $ABCD$ has sides $\mathrm { AB } = 2 , \mathrm { BC } = 1$, and $O$ is the midpoint of $AB$. Point $P$ moves along edges $\mathrm { BC } , \mathrm { CD }$, and $DA$, with $\angle \mathrm { BOP } = \mathrm { x }$. The sum of distances from moving point $P$ to points $A$ and $B$ is expressed as a function of $x$, denoted $f ( x )$. The graph of $f ( x )$ is approximately
(A), (B), (C), or (D) [as shown in figures]
Q11 5 marks Conic sections Eccentricity or Asymptote Computation View
Points $A$ and $B$ are the left and right vertices of hyperbola $E$. Point $M$ is on $E$, and $\triangle A B M$ is an isosceles triangle with vertex angle $120 ^ { \circ }$. Then the eccentricity of $E$ is
(A) $\sqrt { 5 }$
(B) $2$
(C) $\sqrt { 3 }$
(D) $\sqrt { 2 }$
Q12 5 marks Curve Sketching Function Properties from Symmetry or Parity View
Let $\mathrm { f } ^ { \prime } ( \mathrm { x } )$ be the derivative of the odd function $f ( x ) ( x \in \mathbf { R } )$. Given $\mathrm { f } ( - 1 ) = 0$, and when $\mathrm { x } > 0$, $x f ^ { \prime } ( x ) - f ( x ) < 0$. Then the range of $x$ for which $f ( x ) > 0$ holds is
(A) $( - \infty , - 1 ) \cup ( 0,1 )$
(B) $( - 1,0 ) \cup ( 1 , + \infty )$
(C) $( - \infty , - 1 ) \cup ( - 1,0 )$
(D) $( 0,1 ) \cup ( 1 , + \infty )$
Q13 Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Let vectors $\mathrm { a }$ and $\mathrm { b }$ be non-parallel. If vector $\lambda a + b$ is parallel to $\boldsymbol { a } + 2 b$, then the real number $\lambda = $ $\_\_\_\_$.
Q14 Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
If $\mathrm { x } , \mathrm { y }$ satisfy the constraint conditions $\left\{ \begin{array} { l } x - y + 1 \geqslant 0 , \\ x - 2 y \leqslant 0 , \\ x + 2 y - 2 \leqslant 0 , \end{array} \right.$ then the maximum value of $z = x + y$ is $\_\_\_\_$ .
Q15 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
In the expansion of $( a + x ) ( 1 + x ) ^ { 4 }$, the sum of coefficients of odd-power terms of $x$ is 32. Then $a = $ $\_\_\_\_$ .
Q16 Sequences and series, recurrence and convergence Closed-form expression derivation View
Let $\mathrm { S } _ { \mathrm { n } }$ be the sum of the first $n$ terms of sequence $\left\{ \mathrm { a } _ { \mathrm { n } } \right\}$, and $a _ { 1 } = - 1 , a _ { \mathrm { n } + 1 } = S _ { n } S _ { n + 1 }$. Then $S _ { n } = $ $\_\_\_\_$ .
Q17 Sine and Cosine Rules Multi-step composite figure problem View
In $\triangle \mathrm { ABC }$, $D$ is a point on $BC$, $AD$ bisects $\angle \mathrm { BAC }$, and the area of $\triangle \mathrm { ABD }$ is 2 times the area of $\triangle \mathrm { ADC }$.
(I) Find $\frac { \sin \angle B } { \sin \angle C }$ ;
(II) If $A D = 1 , D C = \frac { \sqrt { 2 } } { 2 }$, find the lengths of $B D$ and $A C$.
Q18 Data representation View
A company conducted a survey of 20 users from regions A and B respectively to understand user satisfaction with its products. The satisfaction scores are as follows:
Region A: 62, 73, 81, 92, 95, 85, 74, 64, 53, 76, 78, 86, 95, 66, 97, 78, 88, 82, 76, 89
Region B: 73, 83, 62, 51, 91, 46, 53, 73, 64, 82, 93, 48, 65, 81, 74, 56, 54, 76, 65, 79
(I) Complete the stem-and-leaf plot for user satisfaction scores in both regions based on the two sets of data, and compare the mean and dispersion of satisfaction scores between the two regions through the stem-and-leaf plot (no need to calculate exact values, just draw conclusions);
(II) Based on user satisfaction scores, classify user satisfaction into three levels from low to high:
Satisfaction Score: Below 70, 70 to 89, At least 90 Satisfaction Level: Dissatisfied, Satisfied, Very Satisfied
Let event $C$: ``The satisfaction level of users in region A is higher than that of users in region B''. Assume the evaluation results from the two regions are independent. Based on the given data, using the frequency of event occurrence as the probability of the corresponding event, find the probability of event $C$.
Q19 12 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
As shown in the figure, in rectangular prism $\mathrm { ABCD } - \mathrm { A } _ { 1 } \mathrm { B } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$, we have $\mathrm { AB } = 16 , \mathrm { BC } = 10 , \mathrm { AA } _ { 1 } = 8$. Points $\mathrm { E }$ and $\mathrm { F }$ are on $\mathrm { A } _ { 1 } \mathrm { B } _ { 1 }$ and $\mathrm { D } _ { 1 } \mathrm { C } _ { 1 }$ respectively, with $\mathrm { A } _ { 1 } \mathrm { E } = \mathrm { D } _ { 1 } \mathrm { F }$. A plane $\alpha$ passes through points $E$ and $F$ and intersects the faces of the rectangular prism, with the intersection lines forming a square.
(I) Draw this square in the figure (no need to explain the method or reasoning)
(II) Find the sine of the angle between line