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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
2024 national-II

19 maths questions

Q1 5 marks Complex Numbers Arithmetic Modulus Computation View
Given $z = - 1 - \mathrm { i }$, then $| z | =$
A. 0
B. 1
C. $\sqrt { 2 }$
D. 2
Q2 5 marks Proof True/False Justification View
Given proposition $p : \forall x \in \mathbf { R } , | x + 1 | > 1$; proposition $q : \exists x > 0 , x ^ { 3 } = x$, then
A. Both $p$ and $q$ are true propositions
B. Both $\neg p$ and $q$ are true propositions
C. Both $p$ and $\neg q$ are true propositions
D. Both $\neg p$ and $\neg q$ are true propositions
Q3 5 marks Vectors Introduction & 2D Magnitude of Vector Expression View
Given vectors $\vec { a } , \vec { b }$ satisfying $| \vec { a } | = 1 , | \vec { a } + 2 \vec { b } | = 2$, and $( \vec { b } - 2 \vec { a } ) \perp \vec { b }$, then $| \vec { b } | =$
A. $\frac { 1 } { 2 }$
B. $\frac { \sqrt { 2 } } { 2 }$
C. $\frac { \sqrt { 3 } } { 2 }$
D. 1
Q4 5 marks Measures of Location and Spread View
An agricultural research department planted a new type of rice on 100 rice paddies of equal area and obtained the yield per mu (unit: kg) for each paddy, with partial data organized in the table below
Yield per mu[900, 950)[950, 1000)[1000, 1050)[1100, 1150)[1150, 1200)
Frequency612182410

Based on the data in the table, the correct conclusion is
A. The median yield per mu of the 100 paddies is less than 1050 kg
B. The proportion of paddies with yield per mu below 1100 kg among the 100 paddies exceeds $80 \%$
C. The range of yield per mu of the 100 paddies is between 200 kg and 300 kg
D. The mean yield per mu of the 100 paddies is between 900 kg and 1000 kg
Q5 5 marks Circles Circle-Related Locus Problems View
Given curve $C : x ^ { 2 } + y ^ { 2 } = 16 ( y > 0 )$, from any point $P$ on $C$, draw a perpendicular segment $P P ^ { \prime }$ to the $x$-axis, where $P ^ { \prime }$ is the foot of the perpendicular. The locus of the midpoint of segment $P P ^ { \prime }$ is
A. $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1 \quad ( y > 0 )$
B. $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 8 } = 1 \quad ( y > 0 )$
C. $\frac { y ^ { 2 } } { 16 } + \frac { x ^ { 2 } } { 4 } = 1 \quad ( y > 0 )$
D. $\frac { y ^ { 2 } } { 16 } + \frac { x ^ { 2 } } { 8 } = 1 \quad ( y > 0 )$
Q6 5 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View
Let $f ( x ) = a ( x + 1 ) ^ { 2 } - 1 , g ( x ) = \cos x + 2 a x$. When $x \in ( - 1,1 )$, the curves $y = f ( x )$ and $y = g ( x )$ have exactly one intersection point. Then $a =$
A. $- 1$
B. $\frac { 1 } { 2 }$
C. 1
D. 2
Q7 5 marks SUVAT in 2D & Gravity View
Given a regular triangular frustum $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$ with volume $\frac { 52 } { 3 }$, $A B = 6$, $A _ { 1 } B _ { 1 } = 2$, then the tangent of the angle between $A _ { 1 } A$ and plane $A B C$ is ( )
A. $\frac { 1 } { 2 }$
B. 1
C. 2
D. 3
Q8 5 marks Applied differentiation Inequality proof via function study View
Let $f ( x ) = ( x + a ) \ln ( x + b )$. If $f ( x ) \geq 0$, then the minimum value of $a ^ { 2 } + b ^ { 2 }$ is
A. $\frac { 1 } { 8 }$
B. $\frac { 1 } { 4 }$
C. $\frac { 1 } { 2 }$
D. 1
Q9 6 marks Trig Graphs & Exact Values View
For functions $f ( x ) = \sin 2 x$ and $g ( x ) = \sin \left( 2 x - \frac { \pi } { 4 } \right)$, the correct statements are
A. $f ( x )$ and $g ( x )$ have the same zeros
B. $f ( x )$ and $g ( x )$ have the same maximum value
C. $f ( x )$ and $g ( x )$ have the same minimum positive period
D. The graphs of $f ( x )$ and $g ( x )$ have the same axes of symmetry
Q10 6 marks Conic sections Circle-Conic Interaction with Tangency or Intersection View
The directrix of parabola $C : y ^ { 2 } = 4 x$ is $l$. Let $P$ be a moving point on $C$. Draw a tangent line to circle $\odot A : x ^ { 2 } + ( y - 4 ) ^ { 2 } = 1$ through $P$, with $Q$ as the point of tangency. Draw a perpendicular from $P$ to line $l$, with $B$ as the foot of the perpendicular. Then
A. Line $l$ is tangent to $\odot A$
B. When $P , A , B$ are collinear, $| P Q | = \sqrt { 15 }$
C. When $| P B | = 2$, $P A \perp A B$
D. There are exactly 2 points $P$ satisfying $| P A | = | P B |$
Q11 6 marks Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $f ( x ) = 2 x ^ { 3 } - 3 a x ^ { 2 } + 1$. Then
A. When $a > 1$, $f ( x )$ has three zeros
B. When $a < 0$, $x = 0$ is a local maximum point of $f ( x )$
C. There exist $a , b$ such that $x = b$ is an axis of symmetry of the curve $y = f ( x )$
D. There exists $a$ such that the point $( 1 , f ( 1 ) )$ is a center of symmetry of the curve $y = f ( x )$
Q12 5 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 3 } + a _ { 4 } = 7$ and $3 a _ { 2 } + a _ { 5 } = 5$, then $S _ { 10 } =$ $\_\_\_\_$ .
Q13 5 marks Addition & Double Angle Formulae Addition/Subtraction Formula Evaluation View
Given that $\alpha$ is an angle in the first quadrant, $\beta$ is an angle in the third quadrant, $\tan \alpha + \tan \beta = 4$, $\tan \alpha \tan \beta = \sqrt { 2 } + 1$, then $\sin ( \alpha + \beta ) =$ $\_\_\_\_$ .
Q14 5 marks Combinations & Selection Distribution of Objects to Positions or Containers View
In the $4 \times 4$ grid table shown in the figure, select 4 squares such that each row and each column contains exactly one selected square. The total number of ways to do this is $\_\_\_\_$. Among all selections satisfying the above requirement, the maximum sum of the 4 numbers in the selected squares is $\_\_\_\_$.
11213140
12223342
13223343
15243444
Q15 Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
Let the interior angles $A , B , C$ of $\triangle A B C$ and their opposite sides $a , b , c$ satisfy $\sin A + \sqrt { 3 } \cos A = 2$.
(1) Find $A$.
(2) If $a = 2$ and $\sqrt { 2 } b \sin C = c \sin 2 B$, find the perimeter of $\triangle A B C$.
Q16 Applied differentiation Tangent line computation and geometric consequences View
Given function $f ( x ) = \mathrm { e } ^ { x } - a x - a ^ { 3 }$.
(1) When $a = 1$, find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 1 , f ( 1 ) )$;
(2) If $f ( x )$ has a local minimum value that is negative, find the range of values for $a$.
Q17 Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
As shown in the figure, in planar quadrilateral $A B C D$, $A B = 8$, $C D = 3$, $A D = 5 \sqrt { 3 }$, $\angle A D C = 90 ^ { \circ }$, $\angle B A D = 30 ^ { \circ }$. Points $E$ and $F$ satisfy $\overrightarrow { A E } = \frac { 2 } { 5 } \overrightarrow { A D }$ and $\overrightarrow { A F } = \frac { 1 } { 2 } \overrightarrow { A B }$. Fold $\triangle A E F$ along $E F$ to $\triangle P E F$ such that $P C = 4 \sqrt { 3 }$.
(1) Prove: $E F \perp P D$;
(2) Find the sine of the dihedral angle between plane $P C D$ and plane $P B F$.
Q18 Discrete Probability Distributions Binomial Distribution Identification and Application View
A shooting competition is divided into two stages. Each participating team consists of two members. The specific rules are as follows: In the first stage, one team member shoots 3 times. If all 3 shots miss, the team is eliminated with a score of 0. If at least one shot is made, the team advances to the second stage, where the other team member shoots 3 times, earning 5 points for each made shot and 0 points for each missed shot. The team's final score is the total points from the second stage. A participating team consists of members A and B. Let the probability that A makes each shot be $p$, and the probability that B makes each shot be $q$. Each shot is independent.
(1) If $p = 0.4$ and $q = 0.5$, with A participating in the first stage, find the probability that the team's score is at least 5 points.
(2) Assume $0 < p < q$.
(i) To maximize the probability that the team's score is 15 points, who should participate in the first stage?
(ii) To maximize the expected value of the team's score, who should participate in the first stage?
Q19 Conic sections Locus and Trajectory Derivation View
Given hyperbola $C : x ^ { 2 } - y ^ { 2 } = m ( m > 0 )$, point $P _ { 1 } ( 5,4 )$ is on $C$, and $k$ is a constant with $0 < k < 1$. Through $P_n$ on the right branch of $C$, draw a line with slope $k$; this line intersects $C$ at another point $Q_n$ on the left branch. The reflection of $Q_n$ across the $y$-axis gives $P_{n+1}$ on the right branch.
(1) Find the coordinates of $P_2$ when $k = \frac{1}{2}$.
(2) Prove that $\{x_n - y_n\}$ is a geometric sequence.
(3) Prove that $S_n = \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1})$ is a constant independent of $n$.