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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-science_gkztc

14 maths questions

Q4 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View
4. The coefficient of $x ^ { 3 }$ in the expansion of $\left( 1 + 2 x ^ { 2 } \right) ( 1 + x ) ^ { 4 }$ is
A. 12
B. 16
C. 20
D. 24
Q5 Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
5. 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
Q6 Tangents, normals and gradients Determine unknown parameters from tangent conditions View
6. The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a \mathrm { e } )$ has equation $y = 2 x + b$ . Then
A. $a = \mathrm { e } , \quad b = - 1$
B. $a = \mathrm { e } , b = 1$
C. $a = \mathrm { e } ^ { - 1 } , b = 1$
D. $a = \mathrm { e } ^ { - 1 } , b = - 1$
Q7 Curve Sketching Identifying the Correct Graph of a Function View
7. The graph of the function $y = \frac { 2 x ^ { 3 } } { 2 ^ { x } + 2 ^ { - x } }$ on $[ - 6,6 ]$ is approximately
A. [Figure]
B. [Figure]
C. [Figure]
D. [Figure]
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. For the hyperbola $C : \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 2 } = 1$ with right focus $F$ , if point $P$ is on one of the asymptotes of $C$ , $O$ is the origin, and $| P O | = | P F |$ , then the area of $\triangle P F O$ is
A. $\frac { 3 \sqrt { 2 } } { 4 }$
B. $\frac { 3 \sqrt { 2 } } { 2 }$
C. $2 \sqrt { 2 }$
D. $3 \sqrt { 2 }$
Q11 Function Transformations View
11. Let $f ( x )$ be an even function with domain $\mathbf { R }$ that is 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. $f \left( 2 ^ { - \frac { 2 } { 3 } } \right) > f \left( 2 ^ { - \frac { 3 } { 2 } } \right) > f \left( \log _ { 3 } \frac { 1 } { 4 } \right)$
Q12 Trig Graphs & Exact Values View
12. Let $f ( x ) = \sin \left( \omega x + \frac { \pi } { 5 } \right) ( \omega > 0 )$ . Given that $f ( x )$ has exactly 5 zeros on $[ 0,2 \pi ]$ , consider the following four conclusions:
(1) $f ( x )$ has exactly 3 local maximum points on $( 0,2 \pi )$
(2) $f ( x )$ has exactly 2 local minimum points on $( 0,2 \pi )$
(3) $f ( x )$ is monotonically increasing on $\left( 0 , \frac { \pi } { 10 } \right)$
(4) The range of $\omega$ is $\left[ \frac { 12 } { 5 } , \frac { 29 } { 10 } \right)$ The numbers of all correct conclusions are
A. (1)(4)
B. (2)(3)
C. (1)(2)(3)
D. (1)(3)(4)
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 that $\boldsymbol { a } , \boldsymbol { b }$ are unit vectors and $\boldsymbol { a } \cdot \boldsymbol { b } = 0$ , if $\boldsymbol { c } = 2 \boldsymbol { a } - \sqrt { 5 } \boldsymbol { b }$ , then $\cos \langle \boldsymbol { a } , \boldsymbol { c } \rangle =$ $\_\_\_\_$ .
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 $a _ { 1 } \neq 0$ and $a _ { 2 } = 3 a _ { 1 }$ , then $\frac { S _ { 10 } } { S _ { 5 } } =$ $\_\_\_\_$ .
Q15 Circles Inscribed/Circumscribed Circle Computations View
15. Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$ , and 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 residual 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 甲, and group B mice were given a solution of ion type 乙. Each mouse was given the same volume of solution with the same molar concentration. After a period of time, a scientific method was used to measure the percentage of ions remaining in the mice's bodies. Based on the experimental data, the following histograms were obtained: [Figure]
Let $C$ be the event: ``the residual percentage of ion 乙 in the body is not less than 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 the residual percentage of ion 乙;
(2) Estimate the mean residual percentage for ions 甲 and 乙 respectively (use the midpoint of each interval as the representative value for data in that interval).
Q18 12 marks Sine and Cosine Rules Compute area of a triangle or related figure View
18. (12 points) In $\triangle A B C$ , the sides opposite to angles $A , B , C$ are $a , b , c$ respectively. Given that When $t = 0$, $S = 3$; when $t = \pm 1$, $S = 4\sqrt{2}$.
Therefore, the area of quadrilateral $ADBE$ is $3$ or $4\sqrt{2}$.