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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 hunan-arts

21 maths questions

Q1 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
1. Given $\frac { ( 1 - j ) ^ { 2 } } { z } = 1 + \mathrm { i }$ (where i is the imaginary unit), then the complex number $\mathrm { z } =$
Q2 Permutations & Arrangements Selection and Task Assignment View
2. In a marathon competition, the stem-and-leaf plot in Figure 1 shows the results (in minutes) of 35 athletes. [Figure]
If the athletes are numbered 1-35 according to their results from best to worst, and 7 people are selected using systematic sampling, then the number of athletes with results in the interval [139, 151] is
A. 3
B. 4
C. 5
D. 6
Q3 Inequalities Sufficient/Necessary Conditions Between Inequality Conditions View
3. Let $x \in R$. Then ``$x > 1$'' is ``$x ^ { 3 } > 1$'' a
A. sufficient but not necessary condition
B. necessary but not sufficient condition
C. necessary and sufficient condition
D. neither sufficient nor necessary condition
Q4 Completing the square and sketching Optimization on a constrained domain via completing the square View
4. If variables $x , y$ satisfy the constraint conditions $\left\{ \begin{array} { c } x + y \geq 1 , \\ y - x \leq 1 , \text { then } z = 2 x - y \text { has a minimum value of } \\ x \leq 1 , \end{array} \right.$
A. $-1$
B. 0
C. 1
D. 2
Q5 Arithmetic Sequences and Series Flowchart or Algorithm Tracing Involving Sequences View
5. Executing the flowchart shown in Figure 2, if the input is $n = 3$, then the output $S =$
A. $\frac { 6 } { 7 }$
B. $\frac { 3 } { 7 }$
C. $\frac { 8 } { 9 }$
D. $\frac { 4 } { 9 }$
[Figure]
Figure 2
Q6 Conic sections Eccentricity or Asymptote Computation View
6. If one asymptote of the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ passes through the point $( 3 , - 4 )$, then the eccentricity of this hyperbola is
A. $\frac { \sqrt { 7 } } { 3 }$
B. $\frac { 5 } { 4 }$
C. $\frac { 4 } { 3 }$
D. $\frac { 5 } { 3 }$
Q7 Indices and Surds Ratio and Proportion Problems View
7. If real numbers $\mathrm { a } , \mathrm { b }$ satisfy $\frac { 1 } { a } + \frac { 2 } { b } = \sqrt { a b }$, then the minimum value of $ab$ is
A. $\sqrt { 2 }$
B. 2
C. $2 \sqrt { 2 }$
D. 4
Q8 Function Transformations View
8. Let the function $f ( x ) = \ln ( 1 + x ) - \ln ( 1 - x )$. Then $f ( x )$ is
A. an odd function and increasing on $( 0,1 )$
B. an odd function and decreasing on $( 0,1 )$
C. an even function and increasing on $( 0,1 )$
D. an even function and decreasing on $( 0,1 )$
Q9 Vectors Introduction & 2D Optimization of a Vector Expression View
9. Points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ move on the circle $\chi ^ { 2 } + y ^ { 2 } = 1$, and $\mathrm { AB } \perp \mathrm { BC }$. If point P has coordinates $( 2,0 )$, then the maximum value of $| \overrightarrow { P A } + \overrightarrow { P B } + \overrightarrow { P C } |$ is
A. 6
B. 7
C. 8
D. 9
Q10 Volumes of Revolution Volume from Three-View or Cross-Section Diagram View
10. The three-view drawing of a workpiece is shown in Figure 3. The workpiece is to be machined by cutting into a rectangular parallelepiped with the largest possible volume, with one face of the new workpiece lying on a face of the original workpiece. Then the material utilization rate of the original workpiece is (material utilization rate = volume of new workpiece / volume of original workpiece) [Figure]
A. $\frac { 8 } { 9 \pi }$
B. $\frac { 8 } { 27 \pi }$
C. $\frac { 24 ( \sqrt { 2 } - 1 ) ^ { 3 } } { \pi }$
D. $\frac { 8 ( \sqrt { 2 } - 1 ) ^ { 3 } } { \pi }$
II. Fill-in-the-Blank Questions: This section has 5 questions, each worth 5 points, for a total of 25 points
Q11 Probability Definitions Set Operations View
11. Given the set $\mathrm { U } = \{ 1,2,3,4 \} , \mathrm { A } = \{ 1,3 \} , \mathrm { B } = \{ 1,3,4 \}$, then $\mathrm { A } \cup ( C \cup B ) =$ $\_\_\_\_$
Q12 Polar coordinates View
12. In the rectangular coordinate system xOyz, with the coordinate origin as the pole and the positive x-axis as the polar axis, if the polar equation of curve C is $\rho = 3 \sin \theta$, then the rectangular coordinate equation of curve C is $\_\_\_\_$
Q13 Circles Chord Length and Chord Properties View
13. If the line $3 x - 4 y + 5 = 0$ intersects the circle $x ^ { 2 } + y ^ { 2 } = r ^ { 2 } \quad ( r > 0 )$ at points $A$ and $B$, and $\angle A O B = 120 ^ { \circ }$ (where O is the coordinate origin), then $r =$ $\_\_\_\_$.
Q14 Curve Sketching Number of Solutions / Roots via Curve Analysis View
14. If the function $f ( x ) = \left| 2 ^ { x } - 2 \right| - b$ has two zeros, then the range of the real number $b$ is $\_\_\_\_$
Q15 Addition & Double Angle Formulae Function Analysis via Identity Transformation View
15. Given $w > 0$, the two closest intersection points of the graphs of $y = 2 \sin w x$ and $y = 2 \cos w x$ have a distance of $2 \sqrt { 3 }$. Then $w =$ $\_\_\_\_$.
III. Solution Questions: This section has 6 questions, for a total of 75 points. Solutions should include written explanations, proofs, or calculation steps.
Q16 Probability Definitions Finite Equally-Likely Probability Computation View
16. (This question is worth 12 points)
A shopping mall is holding a promotional lottery activity. After customers purchase goods of a certain amount, they can participate in the lottery. The lottery method is as follows: randomly draw 1 ball each from box A containing 2 red balls $\mathrm { A } _ { 1 } , \mathrm { A } _ { 2 }$ and 1 white ball B, and from box B containing 2 red balls $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 }$ and 2 white balls $\mathrm { b } _ { 1 } , \mathrm { b } _ { 2 }$. If both balls drawn are red, the customer wins; otherwise, the customer does not win. (I) List all possible outcomes of drawing balls using the ball labels. (II) Someone claims: Since there are more red balls than white balls in both boxes, the probability of winning is greater than the probability of not winning. Do you agree? Please explain your reasoning.
Q17 Trig Proofs Triangle Trigonometric Relation View
17. (This question is worth 12 points) Let the sides opposite to angles $A , B , C$ of $\triangle A B C$ be $a , b , c$ respectively, with $a = b \tan A$. (I) Prove that: $\sin \mathrm { B } = \cos \mathrm { A }$ (II) If $\sin C - \sin A \cos B = \frac { 3 } { 4 }$ and $B$ is an obtuse angle, find $A$, $B$, and $C$.
Q18 Moments View
18. (This question is worth 12 points) As shown in Figure 4, the right triangular prism $\mathrm { ABC } - \mathrm { A } _ { 1 } \mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ has an equilateral triangle base with side length 2. $\mathrm { E }$ and $\mathrm { F }$ are the midpoints of $\mathrm { BC }$ and $\mathrm { CC } _ { 1 }$ respectively. (I) Prove that: plane $\mathrm { AEF } \perp$ plane $\mathrm { B } _ { 1 } \mathrm { BCC } _ { 1 }$ (II) If the angle between line $A _ { 1 } C$ and plane $A _ { 1 } A B B _ { 1 }$ is $45 ^ { \circ }$, find the volume of the triangular pyramid F-AEC. [Figure]
Q19 Sequences and series, recurrence and convergence Closed-form expression derivation View
19. (This question is worth 13 points) Let $S _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given $a _ { 1 } = 1 , a _ { 2 } = 2$, and $a _ { n + 2 } = 3 S _ { n } - S _ { n + 1 } , n \in \mathbb { N } ^ { * }$. (I) Prove that: $a _ { n + 2 } = 3 a _ { n }$ (II) Find $\mathrm { S } _ { \mathrm { n } }$
Q20 Conic sections Equation Determination from Geometric Conditions View
20. (This question is worth 13 points) The focus F of the parabola $\mathrm { C } _ { 1 } : \mathrm { X } ^ { 2 } = 4 \mathrm { y }$ is also a focus of the ellipse $\mathrm { C } _ { 2 } : \frac { y ^ { 2 } } { a ^ { 2 } } + \frac { X ^ { 2 } } { b ^ { 2 } } = 1 ( \mathrm { a } > \mathrm { b } > 0 )$. The common chord of $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ has length $2 \sqrt { 6 }$. A line $l$ through point F intersects $\mathrm { C } _ { 1 }$ at points $\mathrm { A } , \mathrm { B }$ and intersects $\mathrm { C } _ { 2 }$ at points $\mathrm { C } , \mathrm { D }$, with $\overrightarrow { B D }$ and $\overrightarrow { A C }$ in the same direction.
(1) Find the equation of $\mathrm { C } _ { 2 }$;
(2) If $| \mathrm { AC } | = | \mathrm { BD } |$, find the slope of line $l$.
Q21 Differential equations Qualitative Analysis of DE Solutions View
21. (This question is worth 13 points) Given $\mathrm { a } > 0$, the function $\mathrm { f } ( \mathrm { x } ) = \mathrm { a } e ^ { x } \cos x$ for $\mathrm { x } \in [ 0 , + \infty )$. Let $x _ { n }$ denote the $n$-th (where $n \in \mathbb { N } ^ { * }$) extremum point of $f ( x )$ in increasing order. (I) Prove that: the sequence $\left\{ f \left( \mathrm { x } _ { \mathrm { n } } \right) \right\}$ is a geometric sequence; (II) If for all $n \in \mathbb { N } ^ { * }$, the inequality $x _ { n } \leq \left| f \left( x _ { n } \right) \right|$ always holds, find the range of $a$.