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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-III-arts

14 maths questions

Q5 Trig Graphs & Exact Values View
5. The number of zeros of the function $f ( x ) = 2 \sin x - \sin 2 x$ on $[ 0,2 \pi ]$ is
A. 2
B. 3
C. 4
D. 5
Q6 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
6. A geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms has the sum of its first 4 terms equal to 15, and $a _ { 5 } = 3 a _ { 3 } + 4 a _ { 1 }$ . Then $a _ { 3 } =$
A. 16
B. 8
C. 4
D. 2
Q7 Tangents, normals and gradients Determine unknown parameters from tangent conditions View
7. The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a e )$ has equation $y = 2 x + b$ . Then
A. $a = \mathrm { e } , b = - 1$
B. $a = \mathrm { e } , b = 1$
C. $a = \mathrm { e } ^ { - 1 } , b = 1$
D. $a = \mathrm { e } ^ { - 1 } , b = - 1$
Q8 Vectors 3D & Lines MCQ: Relationship Between Two Lines View
8. As shown in the figure, point $N$ is the center of square $A B C D$ , $\triangle E C D$ is an equilateral triangle, plane $E C D \perp$ plane $A B C D$ , and $M$ is the midpoint of segment $E D$ . Then [Figure]
A. $B M = E N$ , and lines $B M$ and $E N$ are intersecting lines
B. $B M \neq E N$ , and lines $B M$ and $E N$ are intersecting lines
C. $B M = E N$ , and lines $B M$ and $E N$ are skew lines
D. $B M \neq E N$ , and lines $B M$ and $E N$ are skew lines
Q9 Fixed Point Iteration View
9. Executing the flowchart below, if the input $\varepsilon$ is 0.01 , then the output value of $s$ equals [Figure]
A. $2 - \frac { 1 } { 2 ^ { 4 } }$
B. $2 - \frac { 1 } { 2 ^ { 5 } }$
C. $2 - \frac { 1 } { 2 ^ { 6 } }$
D. $2 - \frac { 1 } { 2 ^ { 7 } }$
Q10 Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
10. Let $F$ be a focus of the hyperbola $C : \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 5 } = 1$ . Point $P$ is on $C$ , $O$ is the origin. If $| O P | = | O F |$ , then the area of $\triangle O P F$ is
A. $\frac { 3 } { 2 }$
B. $\frac { 5 } { 2 }$
C. $\frac { 7 } { 2 }$
D. $\frac { 9 } { 2 }$
Q11 Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
11. Let the plane region represented by the system of inequalities $\left\{ \begin{array} { l } x + y \geq 6 , \\ 2 x - y \geq 0 \end{array} \right.$ be $D$ . Proposition $p : \exists ( x , y ) \in D , 2 x + y \geq 9$ ; Proposition $q : \forall ( x , y ) \in D , 2 x + y \leq 12$ . Four propositions are given below:
(1) $p \vee q$
(2) $\neg p \vee q$
(3) $p \wedge \neg q$
(4) $\neg p \wedge \neg q$
The numbers of all true propositions among these four are
A. (1)(3)
B. (1)(2)
C. (2)(3)
D. (3)(4)
Q12 Function Transformations View
12. Let $f ( x )$ be an even function with domain $\mathbf { R }$ , and monotonically decreasing on $( 0 , + \infty )$ . Then
A. $f \left( \log _ { 3 } \frac { 1 } { 4 } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right)$
B. $f \left( \log _ { 3 } \frac { 1 } { 4 } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right)$
C. $f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$
D. $f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$ II. Fill-in-the-Blank Questions: This section has 4 questions, each worth 5 points, for a total of 20 points.
Q13 Vectors Introduction & 2D Angle or Cosine Between Vectors View
13. Given vectors $\boldsymbol { a } = ( 2,2 ) , \boldsymbol { b } = ( - 8,6 )$ , then $\cos \langle \boldsymbol { a } , \boldsymbol { b } \rangle =$
Q14 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
14. Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$ . If $a _ { 3 } = 5 , a _ { 7 } = 13$ , then $S _ { 10 } =$ $\_\_\_\_$ .
Q15 Circles Inscribed/Circumscribed Circle Computations View
15. Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$ . Let $M$ be a point on $C$ in the first quadrant. If $\triangle M F _ { 1 } F _ { 2 }$ is an isosceles triangle, then the coordinates of $M$ are $\_\_\_\_$
Q17 12 marks Data representation View
17. (12 points) To understand the retention levels of two types of ions in mice, the following experiment was conducted: 200 mice were randomly divided into groups $A$ and $B$, with 100 mice in each group. Group $A$ mice were given a solution of ion type 1, and group $B$ mice were given a solution of ion type 2. Each mouse was given the same volume of solution with the same molar concentration. After a period of time, the percentage of ions retained in the mice's bodies was measured using a scientific method. Based on the experimental data, the following histograms were obtained: [Figure] [Figure]
Let $C$ be the event: ``the retention percentage of ion type 2 in the body is at least 5.5.'' Based on the histogram, the estimated value of $P ( C )$ is 0.70.
(1) Find the values of $a$ and $b$ in the histogram for ion type 2's retention percentage.
(2) Estimate the mean retention percentages for ions of type 1 and type 2 respectively (use the midpoint of each interval as the representative value for data in that interval).
Q18 12 marks Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
18. (12 points) In $\triangle A B C$ , the sides opposite to angles $A , B , C$ are $a , b , c$ respectively. Given that $a \sin \frac { A + C } { 2 } = b \sin A$ .
(1) Find $B$ .
(2) If $\triangle A B C$ is an acute triangle and $c = 1$ , find the range of the area of $\triangle A B C$ .
Q19 12 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
19. (12 points) Figure 1 is a planar figure composed of rectangle $A D E B$ , right triangle $A B C$ , and rhombus $B F G C$ , where $A B = 1 , B E = B F = 2$ , $\angle F B C = 60 ^ { \circ }$ . Fold it along $A B$ and $B C$ so that $B E$ and $B F$ coincide, and connect $D G$ , as shown in Figure 2.
(1) Prove: In Figure 2, points $A , C , G , D$ are coplanar, and plane $A B C \perp$ plane $B C G E$ .
(2) Therefore, from the known condition we have $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - a ) ^ { 2 } \geq \frac { ( 2 + a ) ^ { 2 } } { 3 }$ , equality holds if and only if $x = \frac { 4 - a } { 3 } , y = \frac { 1 - a } { 3 } , z = \frac { 2 a - 2 } { 3 }$ . Thus the minimum value of $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - a ) ^ { 2 }$ is $\frac { ( 2 + a ) ^ { 2 } } { 3 }$ .
From the given condition we have $\frac { ( 2 + a ) ^ { 2 } } { 3 } \geq \frac { 1 } { 3 }$ , solving gives $a \leq - 3$ or $a \geq - 1$ .