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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
2017 national-I-science

14 maths questions

Q1 5 marks Exponential Functions Exponential Equation Solving View
Given sets $A = \{ x \mid x < 1 \} , B = \left\{ x \mid 3 ^ { x } < 1 \right\}$, then
A. $A \cap B = \{ x \mid x < 0 \}$
B. $A \cup B = \mathbf { R }$
C. $A \cup B = \{ x \mid x > 1 \}$
D. $A \cap B = \varnothing$
Q2 5 marks Geometric Probability View
As shown in the figure, the pattern inside square $ABCD$ comes from the ancient Chinese Tai Chi diagram with central symmetry about the center. If a point is randomly selected inside the square, the probability that this point is in the shaded region is
A. $\frac { 1 } { 4 }$
B. $\frac { \pi } { 8 }$
C. $\frac { 1 } { 2 }$
D. $\frac { \pi } { 4 }$
Q3 5 marks Complex Numbers Arithmetic True/False or Property Verification Statements View
Consider the following four propositions
$P _ { 1 }$: If complex number $z$ satisfies $\frac { 1 } { z } \in \mathbf { R }$, then $z \in \mathbf { R }$;
$p _ { 2 }$: If complex number $z$ satisfies $z ^ { 2 } \in \mathbf { R }$, then $z \in \mathbf { R }$;
$p _ { 3 }$: If complex numbers $z _ { 1 }$, $z _ { 2 }$ satisfy $z _ { 1 } z _ { 2 } \in \mathbf { R }$, then $z _ { 1 } = \overline { z _ { 2 } }$;
$p _ { 4 }$: If $z + \bar { z } \in \mathbf { R }$, then $z \in \mathbf { R }$.
The true propositions are
A. $p _ { 1 } , p _ { 3 }$
B. $p _ { 1 } , p _ { 4 }$
C. $p _ { 2 } , p _ { 3 }$
D. $p _ { 2 } , p _ { 4 }$
Q4 5 marks Arithmetic Sequences and Series Find Common Difference from Given Conditions View
Let $S$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 4 } + a _ { 5 } = 24$ and $S _ { 6 } = 48$, then the common difference of $\left\{ a _ { n } \right\}$ is
A. 1
B. 2
C. 4
D. 8
Q5 5 marks Curve Sketching Range and Image Set Determination View
The range of the function $f(x) = \sqrt{x^2 - 2x - 3}$ is
A. $[-2, 2]$
B. $[-1, 1]$
C. $[0, 4]$
D. $[1, 3]$
Q9 5 marks Function Transformations View
Given curves $C _ { 1 }: y = \cos x$ and $C _ { 2 }: y = \sin \left( 2 x + \frac { \pi } { 6 } \right)$, which of the following conclusions is correct?
A. Stretch the horizontal coordinates of points on $C _ { 1 }$ to twice the original length while keeping the vertical coordinates unchanged, then shift the resulting curve to the right by $\frac { \pi } { 6 }$ units to obtain curve $C _ { 2 }$
B. Stretch the horizontal coordinates of points on $C _ { 1 }$ to twice the original length while keeping the vertical coordinates unchanged, then shift the resulting curve to the left by $\frac { \pi } { 12 }$ units to obtain curve $C _ { 2 }$
C. Compress the horizontal coordinates of points on $C _ { 1 }$ to half the original length while keeping the vertical coordinates unchanged, then shift the resulting curve to the right by $\frac { \pi } { 6 }$ units to obtain curve $C _ { 2 }$
D. Compress the horizontal coordinates of points on $C _ { 1 }$ to half the original length while keeping the vertical coordinates unchanged, then shift the resulting curve to the left by $\frac { \pi } { 12 }$ units to obtain curve $C _ { 2 }$
Q10 5 marks Conic sections Focal Chord and Parabola Segment Relations View
Let $F$ be the focus of the parabola $C : y ^ { 2 } = 4 x$. Two perpendicular lines $l _ { 1 }$ and $l _ { 2 }$ pass through $F$. Line $l _ { 1 }$ intersects $C$ at points $A$ and $B$, and line $l _ { 2 }$ intersects $C$ at points $D$ and $E$. The minimum value of $|AB| + |DE|$ is
A. 16
B. 14
C. 12
D. 10
Q11 5 marks Exponential Functions Ordering and Comparing Exponential Values View
Let $x, y, z$ be positive numbers such that $2 ^ { x } = 3 ^ { y } = 5 ^ { z }$, then
A. $2x < 3y < 5z$
B. $5z < 2x < 3y$
C. $3y < 5z < 2x$
D. $3y < 2x < 5z$
Q12 5 marks Geometric Sequences and Series Find a Threshold Index (Algorithm or Calculation) View
The answer to a problem is: The sequence is $1, 1, 2, 2^0, 2^1$, and the next three terms are $2^0, 2^1, 2^2$, and so on. The first term is $2^0$. Find the smallest positive integer $N$ such that $N > 100$ and the sum of the first $N$ terms of this sequence is an integer power of 2.
A. 440
B. 330
C. 220
D. 110
Q13 5 marks Vectors Introduction & 2D Magnitude of Vector Expression View
Given vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ with an angle of $60 ^ { \circ }$ between them, $| \boldsymbol{a} | = 2$, and $| \boldsymbol{b} | = 1$, then $| \boldsymbol{a} + 2 \boldsymbol{b} | = $ \_\_\_\_
Q14 5 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
Let $x$ and $y$ satisfy the constraints $\left\{ \begin{array} { l } x + 2 y \leqslant 1, \\ 2 x + y \geqslant - 1, \\ x - y \leqslant 0, \end{array} \right.$ then the minimum value of $z = 3 x - 2 y$ is \_\_\_\_
Q15 5 marks Conic sections Eccentricity or Asymptote Computation View
Let circle $A$ have center at $A$ and intersect one asymptote of hyperbola $C$ at points $M$ and $N$. If $\angle M A N = 60 ^ { \circ }$, then the eccentricity of $C$ is \_\_\_\_
Q16 5 marks Volumes of Revolution Volume of a 3D Geometric Solid (Pyramid/Tetrahedron) View
As shown in the figure, a circular piece of paper has center $O$ and radius $5$ cm. An equilateral triangle $ABC$ on this paper has center at $O$. Points $D, E, F$ are on circle $O$. Triangles $DBC, ECA, FAB$ are isosceles triangles with $BC, CA, AB$ as their bases respectively. After cutting along the dashed lines and folding triangles $DBC, ECA, FAB$ along $BC, CA, AB$ respectively so that $D, E, F$ coincide, a triangular pyramid is formed. As the side length of $\triangle ABC$ varies, the maximum volume (in $\mathrm { cm } ^ { 3 }$) of the resulting triangular pyramid is \_\_\_\_
Q17 12 marks Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
In triangle $ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. Given that the area of $\triangle ABC$ is $\frac { a ^ { 2 } } { 3 \sin A }$.
(1) Find $\sin B \sin C$;
(2) If $b + c = 2$, find the range of values of $a$.