gaokao

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-arts

15 maths questions

Q1 Inequalities Set Operations Using Inequality-Defined Sets View
1. Given sets $\mathrm { P } = \left\{ x \mid x ^ { 2 } - 2 x \geq 3 \right\} , \mathrm { Q } = \{ x \mid 2 < x < 4 \}$ , then $\mathrm { P } \cap \mathrm { Q } =$ $\_\_\_\_$
Q7 Geometric Probability View
7. As shown in the figure, the angle between the oblique line segment AB and plane $\alpha$ is $60 ^ { \circ }$ , B is the foot of the oblique line, and the moving point P on plane $\alpha$ satisfies $\angle \mathrm { PAB } = 30 ^ { \circ }$ . Then the locus of point P is
A. a line
B. a parabola
C. an ellipse
D. one branch of a hyperbola
Q8 Laws of Logarithms Verify Truth of Logarithmic Statements View
8. Let real numbers $a , b , t$ satisfy $| a + 1 | = | \sin b | = t$
[Figure]
(Figure for Question 7)
A. If $t$ is determined, then $b ^ { 2 }$ is uniquely determined
B. If $t$ is determined, then $a ^ { 2 } + 2 a$ is uniquely determined
C. If $t$ is determined, then $\sin \frac { b } { 2 }$ is uniquely determined
D. If $t$ is determined, then $a ^ { 2 } + a$ is uniquely determined
II. Fill-in-the-Blank Questions (This section contains 7 questions. Multi-blank questions are worth 6 points each, single-blank questions are worth 4 points each, 36 points total.)
Q9 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
9. Calculate: $\log _ { 2 } \frac { \sqrt { 2 } } { 2 } =$ $\_\_\_\_$ , $2 ^ { \log _ { 2 } 3 + \log _ { 4 } 3 } =$ $\_\_\_\_$.
Q10 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
10. Given that $\left\{ a _ { n } \right\}$ is an arithmetic sequence with non-zero common difference $d$. If $a _ { 2 } , a _ { 3 } , a _ { 7 }$ form a geometric sequence, and $2 a _ { 1 } + a _ { 2 } = 1$ , then $a _ { 1 } =$ $\_\_\_\_$ , $d =$ $\_\_\_\_$.
Q11 Trig Graphs & Exact Values View
11. The function $f ( x ) = \sin ^ { 2 } x + \sin x \cos x + 1$ has minimum positive period $\_\_\_\_$ and minimum value $\_\_\_\_$.
Q12 Curve Sketching Range and Image Set Determination View
12. Given the function $f ( x ) = \left\{ \begin{array} { l } x ^ { 2 } , x \leq 1 \\ x + \frac { 6 } { x } - 6 , x > 1 \end{array} \right.$ , then $f [ f ( - 2 ) ] =$ $\_\_\_\_$ , and the minimum value of $f ( x )$ is $\_\_\_\_$.
Q13 Vectors Introduction & 2D Magnitude of Vector Expression View
13. Given that $\vec { e } _ { 1 } , \vec { e } _ { 2 }$ are unit vectors in the plane, and $\vec { e } _ { 1 } \cdot \vec { e } _ { 2 } = \frac { 1 } { 2 }$ . If the plane vector $\vec { b }$ satisfies $\vec { b } \cdot \vec { e } _ { 1 } = \vec { b } \cdot \vec { e } _ { 2 } = 1$ , then $| \vec { b } | =$ $\_\_\_\_$.
Q14 Inequalities Optimization Subject to an Algebraic Constraint View
14. Given that real numbers $x , y$ satisfy $x ^ { 2 } + y ^ { 2 } \leq 1$ , then the maximum value of $| 2 x + y - 4 | + | 6 - x - 3 y |$ is $\_\_\_\_$ .
Q15 Circles Circle Equation Derivation View
15. For the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ , the right focus $\mathrm { F } ( c , 0 )$ is symmetric to point Q with respect to the line $y = \frac { b } { c } x$ , and Q lies on the ellipse. Then the eccentricity of the ellipse is $\_\_\_\_$. III. Solution Questions (This section contains 5 questions, 74 points total. Solutions should include explanations, proofs, or calculation steps.)
Q16 14 marks Trig Graphs & Exact Values View
16. (14 points) In $\triangle A B C$ , the sides opposite to angles $\mathrm { A } , \mathrm { B }$ , C are $a , b , c$ respectively. Given that $\tan \left( \frac { \pi } { 4 } + \mathrm { A } \right) = 2$ .
(1) Find the value of $\frac { \sin 2 A } { \sin 2 A + \cos ^ { 2 } A }$ ;
(2) If $\mathrm { B } = \frac { \pi } { 4 } , a = 3$ , find the area of $\triangle A B C$ .
Q17 15 marks Arithmetic Sequences and Series Multi-Part Structured Problem on AP View
17. (15 points) Given sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$ satisfying $a _ { 1 } = 2 , b _ { 1 } = 1 , a _ { n + 1 } = 2 a _ { n } \left( \mathrm { n } \in \mathrm { N } ^ { * } \right)$ , $b _ { 1 } + \frac { 1 } { 2 } b _ { 2 } + \frac { 1 } { 3 } b _ { 3 } + \cdots + \frac { 1 } { n } b _ { n } = b _ { n + 1 } - 1 \left( \mathrm { n } \in \mathrm { N } ^ { * } \right)$ .
(1) Find $a _ { n }$ and $b _ { n }$ ;
(2) Let $T _ { n }$ denote the sum of the first n terms of the sequence $\left\{ a _ { n } b _ { n } \right\}$ . Find $T _ { n }$ .
Q18 15 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
18. (15 points) As shown in the figure, in the triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$ , $\angle \mathrm { ABC } = 90 ^ { \circ } , \mathrm { AB } = \mathrm { AC } = 2 , \mathrm { AA } _ { 1 } = 4$ , the projection of $A _ { 1 }$ on the base plane ABC is the midpoint of BC, and D is the midpoint of $B _ { 1 } C _ { 1 }$.
(1) Prove that $A _ { 1 } \mathrm { D } \perp$ plane $\mathrm { A } _ { 1 } \mathrm { BC }$ ;
(2) Find the sine of the angle between line $\mathrm { A } _ { 1 } \mathrm { B}$ and plane $\mathrm { BB } _ { 1 } \mathrm { C } C _ { 1 }$ . [Figure]
Q19 15 marks Circles Tangent Lines and Tangent Lengths View
19. (15 points) As shown in the figure, given the parabola $\mathrm { C } _ { 1 } : \mathrm { y } = \frac { 1 } { 4 } x ^ { 2 }$ , the circle $\mathrm { C } _ { 2 } : x ^ { 2 } + ( \mathrm { y } - 1 ) ^ { 2 } = 1$ , through point $\mathrm { P } ( \mathrm { t } , 0 ) ( \mathrm { t } > 0 )$ , draw lines $\mathrm { PA } , \mathrm { PB}$ not passing through the origin O that are tangent to the parabola $C _ { 1 }$ and circle $\mathrm { C } _ { 2 }$ respectively, with $\mathrm { A } , \mathrm { B}$ as the points of tangency.
(1) Find the coordinates of points $\mathrm { A } , \mathrm { B}$ ;
(2) Find the area of $\triangle \mathrm { PAB}$ . Note: If a line has exactly one common point with a parabola and is not parallel to the axis of symmetry of the parabola, then the line is tangent to the parabola, and the common point is called the point of tangency. [Figure]
Q20 15 marks Solving quadratics and applications Determining quadratic function from given conditions View
20. (15 points) Let the function $f ( x ) = x ^ { 2 } + a x + b , ( a , b \in R )$ .
(1) When $b = \frac { a ^ { 2 } } { 4 } + 1$ , find the expression for the minimum value $g ( a )$ of the function $f ( x )$ on $[ - 1,1 ]$ ;
(2) Given that the function $f ( x )$ has a zero on $[ - 1,1 ]$ , $0 \leq b - 2 a \leq 1$ , find the range of values for $b$ .