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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 zhejiang-science

15 maths questions

Q1 Inequalities Set Operations Using Inequality-Defined Sets View
1. Given sets $P = \left\{ x \mid x ^ { 2 } - 2 x \geq 0 \right\} , Q = \{ x \mid 1 < x \leq 2 \}$ , then $\left( \complement _ { \mathbb{R} } P \right) \cap Q =$
A. $[0,1)$
B. $( 0,2 ]$
C. $( 1,2 )$
D. $[ 1,2 ]$
Q3 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
3. Given that $\left\{ a _ { n } \right\}$ is an arithmetic sequence with non-zero common difference $d$, and the sum of the first $n$ terms is $S _ { n }$. If $a _ { 3 } , a _ { 4 } , a _ { 8 }$ form a geometric sequence, then
A. $a _ { 1 } d > 0 , d S _ { n } > 0$
B. $a _ { 1 } d < 0 , d S _ { n } < 0$
C. $a _ { 1 } d > 0 , d S _ { n } < 0$
D. $a _ { 1 } d < 0 , d S _ { n } > 0$ [Figure]
Q5 Circles Area and Geometric Measurement Involving Circles View
5. As shown in the figure, let $F$ be the focus of the parabola $y ^ { 2 } = 4 x$. A line not passing through the focus contains three distinct points $A , B , C$, where points $A , B$ are on the parabola and point $C$ is on the $y$-axis. Then the ratio of the areas of $\triangle BCF$ and $\triangle ACF$ is
A. $\frac { | B F | - 1 } { | A F | - 1 }$
B. $\frac { | B F | ^ { 2 } - 1 } { | A F | ^ { 2 } - 1 }$
C. $\frac { | B F | + 1 } { | A F | + 1 }$
D. $\frac { | B F | ^ { 2 } + 1 } { | A F | ^ { 2 } + 1 }$
Q7 Composite & Inverse Functions Identifying Whether a Relation Defines a Function View
7. There exists a function $f ( x )$ satisfying, for all $x \in \mathbb{R}$,
A. $f ( \sin 2 x ) = \sin x$
B. $f ( \sin 2 x ) = x ^ { 2 } + x$
C. $f \left( x ^ { 2 } + 1 \right) = | x + 1 |$
D. $f \left( x ^ { 2 } + 2 x \right) = | x + 1 |$
Q9 Conic sections Eccentricity or Asymptote Computation View
9. The focal distance of the hyperbola $\frac { x ^ { 2 } } { 2 } - y ^ { 2 } = 1$ is $\_\_\_\_$ , and the equations of the asymptotes are $\_\_\_\_$ .
Q10 Curve Sketching Range and Image Set Determination View
10. Given the function $f ( x ) = \left\{ \begin{array} { l } x + \frac { 2 } { x } - 1 , x \geq 1 \\ \lg \left( x ^ { 2 } + 1 \right) , x < 1 \end{array} \right.$ , then $f ( f ( - 3 ) ) =$ $\_\_\_\_$ , and the minimum value of $f ( x )$ is $\_\_\_\_$ .
Q11 Trig Graphs & Exact Values View
11. The minimum positive period of the function $f ( x ) = \sin ^ { 2 } x + \sin x \cos x + 1$ is $\_\_\_\_$ , and the decreasing interval is $\_\_\_\_$ .
Q12 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
12. If $a = \log _ { 2 } 3$ , then $2 ^ { a } + 2 ^ { - a } =$ $\_\_\_\_$ .
Q13 Vectors 3D & Lines MCQ: Angle Between Skew Lines View
13. As shown in the figure, in the triangular pyramid $A - B C D$, $AB = AC = BD = CD = 3$ , $AD = BC = 2$ , and $M , N$ are the midpoints of $AD , BC$ respectively. Then the cosine of the angle between the skew lines $AN$ and $CM$ is $\_\_\_\_$ .
Q15 Vectors Introduction & 2D Expressing a Vector as a Linear Combination View
15. Given that $\vec { e } _ { 1 } , \vec { e } _ { 2 }$ are unit vectors in space with $\vec { e } _ { 1 } \cdot \vec { e } _ { 2 } = \frac { 1 } { 2 }$ . If the space vector $\vec { b }$ satisfies $\vec { b } \cdot \vec { e } _ { 1 } = 2 , \vec { b } \cdot \vec { e } _ { 2 } = \frac { 5 } { 2 }$ , and for all $x , y \in \mathbb{R}$ , $\left| \vec { b } - \left( x \vec { e } _ { 1 } + y \vec { e } _ { 2 } \right) \right| \geq \left| \vec { b } - \left( x _ { 0 } \vec { e } _ { 1 } + y _ { 0 } \vec { e } _ { 2 } \right) \right| = 1$ ( $x _ { 0 } , y _ { 0 } \in \mathbb{R}$ ), then $x _ { 0 } =$ $\_\_\_\_$ , $y _ { 0 } =$ $\_\_\_\_$ , $| \vec { b } | =$ $\_\_\_\_$ . III. Solution Questions: This section contains 5 questions, for a total of 74 points. Solutions should include explanations, proofs, or calculation steps.
Q16 Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
16. (This question is worth 14 points) In $\triangle ABC$, the sides opposite to angles $A , B , C$ are $a , b , c$ respectively. Given that $A = \frac { \pi } { 4 }$ and $b ^ { 2 } - a ^ { 2 } = \frac { 1 } { 2 } c ^ { 2 }$ . (I) Find the value of $\tan C$; (II) If the area of $\triangle ABC$ is 7, find the value of $b$.
Q17 Vectors 3D & Lines Multi-Part 3D Geometry Problem View
17. (This question is worth 15 points) As shown in the figure, in the triangular prism $ABC - A _ { 1 } B _ { 1 } C _ { 1 }$, $\angle BAC = 90 ^ { \circ }$ , $AB = AC = 2$ , $A _ { 1 } A = 4$ , the projection of $A _ { 1 }$ on the base plane $ABC$ is the midpoint of $BC$, and $D$ is the midpoint of $B _ { 1 } C _ { 1 }$ . (I) Prove that $A _ { 1 } D \perp$ plane $A _ { 1 } B C _ { 1 }$ ; (II) Find the cosine of the plane angle of the dihedral angle $A _ { 1 } - BD - B _ { 1 }$ . [Figure]
Q18 Completing the square and sketching Determining coefficients from given conditions on function values or geometry View
18. (This question is worth 15 points)
Given the function $f ( x ) = x ^ { 2 } + ax + b$ ( $a , b \in \mathbb{R}$ ), let $M ( a , b )$ denote the maximum value of $| f ( x ) |$ on the interval $[ - 1 , 1 ]$ . (I) Prove that when $| a | \geq 2$ , $M ( a , b ) \geq 2$ ; (II) When $a , b$ satisfy $M ( a , b ) \leq 2$ , find the maximum value of $| a | + | b |$ .
Q19 Circles Area and Geometric Measurement Involving Circles View
19. (This question is worth 15 points) Two distinct points $A , B$ on the ellipse $\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$ are symmetric about the line $y = mx + \frac { 1 } { 2 }$ . (I) Find the range of the real number $m$; (II) Find the maximum value of the area of $\triangle AOB$ (where $O$ is the origin). [Figure]
Q20 Sequences and series, recurrence and convergence Convergence proof and limit determination View
20. (This question is worth 15 points) Given that the sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = \frac { 1 } { 2 }$ and $a _ { n + 1 } = a _ { n } - a _ { n } ^ { 2 }$ ( $n \in \mathbb{N