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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-arts

13 maths questions

Q1 5 marks Inequalities Set Operations Using Inequality-Defined Sets View
Given sets $A = \{ x \mid x < 2 \}$, $B = \{ x \mid 3 - 2x > 0 \}$, then
A. $A \cap B = \{ x \mid x \leq \frac{3}{2} \}$
B. $A \cap B = \varnothing$
C. $A \cup B = \left\{ x \left\lvert \, x < \frac{3}{2} \right. \right\}$
D. $A \cup B = \mathbf{R}$
Q2 5 marks Measures of Location and Spread View
To evaluate the planting effectiveness of a crop, $n$ plots of experimental land were selected. The per-acre yields (in kg) of these $n$ plots are $x_1, x_2, \cdots, x_n$ respectively. Among the following indicators, which can be used to evaluate the stability of this crop's per-acre yield?
A. Mean of $x_1, x_2, \cdots, x_n$
B. Median of $x_1, x_2, \cdots, x_n$
C. Maximum value of $x_1, x_2, \cdots, x_n$
D. Standard deviation of $x_1, x_2, \cdots, x_n$
Q3 5 marks Complex Numbers Arithmetic Complex Division/Multiplication Simplification View
Among the following expressions, which has a result that is a pure imaginary number?
A. $i(1 + i)^2$
B. $i^2(1 - i)$
C. $(1 + i)^2$
D. $i(1 + i)$
Q4 5 marks Geometric Probability View
As shown in the figure, the pattern inside square $ABCD$ comes from the ancient Chinese Tai Chi diagram. The black and white parts in the inscribed circle of the square are symmetric with respect to the center of the square. If a point is randomly selected from the square, the probability that it falls in the black part is
A. $\frac{1}{4}$
B. $\frac{\pi}{8}$
C. $\frac{1}{2}$
D. $\frac{\pi}{4}$
Q5 5 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
Given that $F$ is the right focus of the hyperbola $C: x^2 - \frac{y^2}{3} = 1$, $P$ is a point on $C$, and $PF$ is perpendicular to the $x$-axis. Point $A$ has coordinates $(1, 3)$. Then the area of $\triangle APF$ is
A. $\frac{3}{2}$
B. $\frac{1}{2}$
C. $\frac{2}{3}$
D. $\frac{3}{4}$
Q7 5 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
Let $x, y$ satisfy the constraint conditions $\left\{ \begin{array}{l} x - y \geq 1 \\ y \geq 0 \end{array} \right.$. Then the maximum value of $z = x + y$ is
A. 0
B. 1
C. 2
D. 3
Q8 5 marks Trig Graphs & Exact Values View
The partial graph of the function $y = \frac{\sin 2x}{1 - \cos x}$ is approximately (see options A, B, C, D in the figures provided).
Q9 5 marks Laws of Logarithms Analyze a Logarithmic Function (Limits, Monotonicity, Zeros, Extrema) View
Given the function $f(x) = \ln x + \ln(2 - x)$, then
A. $f(x)$ is monotonically increasing on $(0, 2)$
B. $f(x)$ is monotonically decreasing on $(0, 2)$
C. $f(x)$ is increasing on $(0, 1)$ and decreasing on $(1, 2)$
D. $f(x)$ is decreasing on $(0, 1)$ and increasing on $(1, 2)$
Q19 12 marks Bivariate data View
(12 points)
To monitor the production process of a production line for a certain component, an inspector randomly selects one component every 30 minutes and measures its size (in cm). Below are the sizes of 16 components randomly selected by the inspector in one day:
Sampling Order12345678
Component Size9.9510.129.969.9610.019.929.9810.04
Sampling Order910111213141516
Component Size10.269.9110.1310.029.2210.0410.059.95

$\sqrt{\sum_{i=1}^{16}(i - 8.5)^2} \approx 18.439$, $\sum_{i=1}^{16}(x_i - \bar{x})(i - 8.5) = -2.78$, where $x_i$ is the size of the $i$-th component sampled, $i = 1, 2, \cdots, 16$.
(1) Find the correlation coefficient $r$ of $(x_i, i)$ $(i = 1, 2, \cdots, 16)$, and determine whether it can be concluded that the size of components produced on this day does not systematically increase or decrease as the production process progresses.
(2) Among the components sampled in one day, if a component with size outside $(\bar{x} - 3s, \bar{x} + 3s)$ appears, it is considered that the production line may have experienced an abnormal situation on this day, and the production process needs to be checked.
(i) Based on the sampling results of this day, is it necessary to check the production process?
(ii) Data outside $(\bar{x} - 3s, \bar{x} + 3s)$ are called outliers. Remove the outliers and estimate the mean and standard deviation of the component sizes produced by this production line on this day. (Round to 0.01)
$$\text{Attachment: For a sample }(x_i, y_i) (i = 1, 2, \cdots, n), \text{ the correlation coefficient is } r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2}\sqrt{\sum_{i=1}^{n}(y_i - \bar{y})^2}}.$$
$$\sqrt{0.008} \approx 0.09.$$
Q20 12 marks Tangents, normals and gradients Find tangent line equation at a given point View
(12 points)
Let $A$ and $B$ be two points on the curve $C: y = \frac{x^2}{4}$, and the sum of the $x$-coordinates of $A$ and $B$ is 4.
(1) Find the slope of line $AB$;
(2) Find the equation of line $AB$.
Q21 12 marks Applied differentiation Full function study (variation table, limits, asymptotes) View
(12 points)
Given the function $f(x) = e^x(e^x - a) - a^2x$.
(1) Discuss the monotonicity of $f(x)$;
(2) If $f(x) \geq 0$, find the range of values for $a$.
Q22 10 marks Parametric curves and Cartesian conversion View
[Optional 4-4: Coordinate Systems and Parametric Equations] (10 points)
In the rectangular coordinate system $xOy$, the parametric equation of curve $C$ is $\left\{\begin{array}{l} x = 3\cos\theta \\ y = \sin\theta \end{array}\right.$ ($\theta$ is the parameter), and the parametric equation of line $l$ is $\left\{\begin{array}{l} x = a + 4t \\ y = 1 - t \end{array}\right.$ ($t$ is the parameter).
(1) If $a = -1$, find the coordinates of the intersection points of $C$ and $l$.
(2) If the maximum distance from a point on $C$ to line $l$ is $\sqrt{17}$, find $a$.
Q23 10 marks Inequalities Quadratic Inequality Holding for All x (or a Restricted Domain) View
[Optional 4-5: Inequalities] (10 points)
Given functions $f(x) = -x^2 + ax + 4$ and $g(x) = |x + 1| + |x - 1|$.
(2) If the solution set of the inequality $f(x) \geq g(x)$ contains $[-1, 1]$, find the range of values for $a$.