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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
2023 national-A-science

18 maths questions

Q1 5 marks Probability Definitions Set Operations View
Let $A = \{ x \mid x = 3k + 1 , k \in Z \} , B = \{ x \mid x = 3k + 2 , k \in Z \} , U$ be the set of integers, then $C_{U}(A \bigcap B) =$
A. $\{ x \mid x = 3k , k \in Z \}$
B. $\{ x \mid x = 3k - 1 , k \in Z \}$
C. $\{ x \mid x = 3k - 1 , k \in \mathrm{Z} \}$
D. $\varnothing$
Q2 5 marks Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
If the complex number $( a + i )( 1 - ai ) = 2$ , then $a =$
A. $-1$
B. $0$
C. $1$
D. $2$
Q4 5 marks Vectors Introduction & 2D Angle or Cosine Between Vectors View
Vectors $|\boldsymbol{a}| = |\boldsymbol{b}| = 1 , |\boldsymbol{c}| = \sqrt{2}$ and $\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c} = 0$ , then $\cos \langle \boldsymbol{a} - \boldsymbol{b} , \boldsymbol{b} - \boldsymbol{c} \rangle =$
A. $-\frac{1}{5}$
B. $-\frac{2}{5}$
C. $\frac{2}{5}$
D. $\frac{4}{5}$
Q5 5 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
In the sequence $\left\{ a_{n} \right\}$ , let $S_{n}$ be the sum of the first $n$ terms of $\left\{ a_{n} \right\}$ , $S_{5} = 5S_{3} - 4$ , then $S_{4} =$
A. $7$
B. $9$
C. $15$
D. $20$
Q6 5 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
50 people registered for the soccer club, 60 people registered for the table tennis club, and 70 people registered for either the soccer or table tennis club. If a person is known to have registered for the soccer club, the probability that they also registered for the table tennis club is
A. $0.8$
B. $0.4$
C. $0.2$
D. $0.1$
Q7 5 marks Trig Proofs Trigonometric Equation Constraint Deduction View
``$\sin^{2} \alpha + \sin^{2} \beta = 1$'' is ``$\cos \alpha + \cos \beta = 0$'' a
A. sufficient but not necessary condition
B. necessary but not sufficient condition
C. necessary and sufficient condition
D. neither sufficient nor necessary condition
Q8 5 marks Circles Chord Length and Chord Properties View
The eccentricity of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \ (a > 0 , b > 0)$ is $\sqrt{5}$ . One of its asymptotes intersects the circle $(x - 2)^{2} + (y - 3)^{2} = 1$ at points $A , B$ , then $|AB| =$
A. $\frac{1}{5}$
B. $\frac{\sqrt{5}}{5}$
C. $\frac{2\sqrt{5}}{5}$
D. $\frac{4\sqrt{5}}{5}$
Q9 5 marks Combinations & Selection Selection with Group/Category Constraints View
Five volunteers participate in community service over Saturday and Sunday. Each day, two people are randomly selected from them to participate. The number of ways to select such that exactly one person participates on both days is
A. $120$
B. $60$
C. $40$
D. $30$
Q10 5 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View
Let $f(x)$ be the function obtained by shifting $y = \cos\left(2x + \frac{\pi}{4}\right)$ to the left by $\frac{\pi}{6}$ units. The number of intersection points of $y = f(x)$ and $y = \frac{1}{2}x - \frac{1}{2}$ is
A. $1$
B. $2$
C. $3$
D. $4$
Q11 5 marks Sine and Cosine Rules Compute area of a triangle or related figure View
In the quadrangular pyramid $P - ABCD$ , the base $ABCD$ is a square with $AB = 4$ , $PC = PD = 3$ , $\angle PCA = 45^{\circ}$ , then the area of $\triangle PBC$ is
A. $2\sqrt{2}$
B. $3\sqrt{2}$
C. $4\sqrt{2}$
D. $5\sqrt{2}$
Q12 5 marks Conic sections Focal Distance and Point-on-Conic Metric Computation View
The ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{6} = 1$ has foci $F_{1} , F_{2}$ and center $O$ . Let $P$ be a point on the ellipse. If $\cos \angle F_{1}PF_{2} = \frac{3}{5}$ , then $|OP| =$
A. $\frac{2}{5}$
B. $\frac{\sqrt{30}}{2}$
C. $\frac{3}{5}$
D. $\frac{\sqrt{35}}{2}$
Q13 5 marks Function Transformations View
If $y = (x - 1)^{2} + ax + \sin\left(x + \frac{\pi}{2}\right)$ is an even function, then $a =$ $\_\_\_\_$ .
Q14 5 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
Let $x , y$ satisfy the constraints $\left\{ \begin{array}{l} -2x + 3y \leqslant 3 \\ 3x - 2y \leqslant 3 \\ x + y = 1 \end{array} \right.$ . Let $z = 3x + 2y$ . The maximum value of $z$ is $\_\_\_\_$ .
Q16 5 marks Sine and Cosine Rules Multi-step composite figure problem View
In $\triangle ABC$ , $\angle BAC = 60^{\circ} , AB = 2 , BC = \sqrt{6}$ . $AD$ bisects $\angle BAC$ and intersects $BC$ at point $D$ . Then $AD =$ $\_\_\_\_$ .
Q17 12 marks Sequences and series, recurrence and convergence Closed-form expression derivation View
In the sequence $\left\{ a_{n} \right\}$ , $a_{2} = 1$ . Let $S_{n}$ be the sum of the first $n$ terms of $\left\{ a_{n} \right\}$ . $2S_{n} = na_{n}$ .
(1) Find the general term formula for $\left\{ a_{n} \right\}$ ;
(2) Find the sum $T_{n}$ of the first $n$ terms of the sequence $\left\{ \frac{a_{n} + 1}{2^{n}} \right\}$ .
Q20 12 marks Conic sections Focal Chord and Parabola Segment Relations View
The line $x - 2y + 1 = 0$ intersects the parabola $y^{2} = 2px \ (p > 0)$ at points $A , B$ with $AB = 4\sqrt{15}$ .
(1) Find the value of $p$ ;
(2) Let $F$ be the focus of $y^{2} = 2px$ . Let $M , N$ be two points on the parabola such that $\overrightarrow{MF} \perp \overrightarrow{NF}$ . Find the minimum area of $\triangle MNF$ .
Q21 12 marks Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Given $f(x) = ax - \frac{\sin x}{\cos^{2} x} , \quad x \in \left(0 , \frac{\pi}{2}\right)$ ,
(1) When $a = 8$ , discuss the monotonicity of $f(x)$ ;
(2) If $f(x) < \sin 2x$ , find the range of values for $a$ .
Q23 10 marks Inequalities Absolute Value Inequality View
[Elective 4-5: Inequalities]
Given $f(x) = 2|x - a| - a , \ a > 0$ .
(1) Solve the inequality $f(x) < x$ ;
(2) If the area enclosed by $y = f(x)$ and the coordinate axes is 2 , find $a$ .