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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
2019 national-II-arts

13 maths questions

Q6 Function Transformations View
6. Let $f ( x )$ be an odd function, and when $x \geq 0$, $f ( x ) = \mathrm { e } ^ { x } - 1$. Then when $x < 0$, $f ( x ) =$
A. $\mathrm { e } ^ { - x } - 1$
B. $\mathrm { e } ^ { - x } + 1$
C. $- \mathrm { e } ^ { - x } - 1$
D. $- \mathrm { e } ^ { - x } + 1$
Q8 Trig Graphs & Exact Values View
8. If $x _ { 1 } = \frac { \pi } { 4 } , x _ { 2 } = \frac { 3 \pi } { 4 }$ are two adjacent extreme points of the function $f ( x ) = \sin \omega x ( \omega > 0 )$, then $\omega =$
A. 2
B. $\frac { 3 } { 2 }$
C. 1
D. $\frac { 1 } { 2 }$
Q9 Circles Circle Equation Derivation View
9. If the focus of the parabola $y ^ { 2 } = 2 p x ( p > 0 )$ is a focus of the ellipse $\frac { x ^ { 2 } } { 3 p } + \frac { y ^ { 2 } } { p } = 1$, then $p =$
A. 2
B. 3
C. 4
D. 8
Q10 Tangents, normals and gradients Find tangent line equation at a given point View
10. The equation of the tangent line to the curve $y = 2 \sin x + \cos x$ at the point $( \pi , - 1 )$ is
A. $x - y - \pi - 1 = 0$
B. $2 x - y - 2 \pi - 1 = 0$
C. $2 x + y - 2 \pi + 1 = 0$
D. $x + y - \pi + 1 = 0$
Q11 Addition & Double Angle Formulae Trigonometric Equation Solving via Identities View
11. Given $a \in \left( 0 , \frac { \pi } { 2 } \right) , 2 \sin 2 \alpha = \cos 2 \alpha + 1$, then $\sin \alpha =$
A. $\frac { 1 } { 5 }$
B. $\frac { \sqrt { 5 } } { 5 }$
C. $\frac { \sqrt { 3 } } { 3 }$
D. $\frac { 2 \sqrt { 5 } } { 5 }$
Q12 Conic sections Eccentricity or Asymptote Computation View
12. Let $F$ be the right focus of the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$, $O$ be the origin. The circle with diameter $O F$ and the circle $x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$ intersect at points $P$ and $Q$. If $| P Q | = | O F |$, then the eccentricity of $C$ is
A. $\sqrt { 2 }$
B. $\sqrt { 3 }$
C. 2
D. $\sqrt { 5 }$
II. Fill-in-the-Blank Questions: This section has 4 questions, 5 points each, 20 points total.
Q13 Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
13. If variables $x , y$ satisfy the constraint conditions $\left\{ \begin{array} { l } 2 x + 3 y - 6 \geq 0 , \\ x + y - 3 \leq 0 , \\ y - 2 \leq 0 , \end{array} \right.$ then the maximum value of $z = 3 x - y$ is $\_\_\_\_$ .
Q14 Measures of Location and Spread View
14. China's high-speed rail development is rapid and technologically advanced. According to statistics, among high-speed trains stopping at a certain station, 10 trains have an on-time rate of 0.97, 20 trains have an on-time rate of 0.98, and 10 trains have an on-time rate of 0.99. The estimated value of the average on-time rate for all trains stopping at this station is $\_\_\_\_$ .
Q15 Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
15. In $\triangle A B C$, the sides opposite to angles $A , B , C$ are $a , b , c$ respectively. Given $b \sin A + a \cos B = 0$, then $B =$ $\_\_\_\_$ .
Q17 12 marks Moments View
17. (12 points) As shown in the figure, the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ has a square base $A B C D$. Point $E$ is on edge $A A _ { 1 }$, and $B E \perp E C _ { 1 }$. [Figure]
(1) Prove: $B E \perp$ plane $E B _ { 1 } C _ { 1 }$;
(2) If $A E = A _ { 1 } E , A B = 3$, find the volume of the quadrangular pyramid $E - B B _ { 1 } C _ { 1 } C$.
Q18 12 marks Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
18. (12 points)
Given that $\left\{ a _ { n } \right\}$ is a geometric sequence with all positive terms, $a _ { 1 } = 2 , a _ { 3 } = 2 a _ { 2 } + 16$.
(1) Find the general term formula for $\left\{ a _ { n } \right\}$;
(2) Let $b _ { n } = \log _ { 2 } a _ { n }$, find the sum of the first $n$ terms of the sequence $\left\{ b _ { n } \right\}$.
Q19 12 marks Measures of Location and Spread View
19. (12 points) To understand the production situation of small and medium enterprises in the industry, a government department randomly surveyed 100 enterprises and obtained a frequency distribution table for the growth rate $y$ of production value in the first quarter compared to the previous year's first quarter.
Interval for $y$$[ - 0.20,0 )$$[ 0,0.20 )$$[ 0.20,0.40 )$$[ 0.40,0.60 )$$[ 0.60,0.80 )$
Number of enterprises22453147

(1) Estimate the proportion of enterprises with production value growth rate not less than $40\%$ and the proportion of enterprises with negative growth, respectively;
(2) Find the estimated values of the mean and standard deviation of the production value growth rate for this type of enterprise (use the midpoint of each interval as the representative value for data in that interval). (Accurate to 0.01)
Attachment: $\sqrt { 74 } \approx 8.602$ .
Q20 12 marks Circles Area and Geometric Measurement Involving Circles View
20. (12 points)
Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$, $P$ be a point on $C$, and $O$ be the origin.
(1) If $\triangle P O F _ { 2 }$ is an equilateral triangle, find the eccentricity of $C$;
(2) If there exists a point $P$ such that $P F _ { 1 } \perp P F _ { 2 }$ and the area of $\triangle F _ { 1 } P F _ { 2 }$ equals 16, find the value of $b$ and the range of values for $a$.