gaokao

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
2020 national-II-science

17 maths questions

Q1 5 marks Inequalities Set Operations Using Inequality-Defined Sets View
Given sets $U = \{ - 2 , - 1,0,1,2,3 \} , A = \{ - 1,0,1 \} , B = \{ 1,2 \}$ , then $\complement _ { U } ( A \cup B ) =$
A． $\{ - 2,3 \}$
B． $\{ - 2,2,3 \}$
C． $\{ - 2 , - 1,0,3 \}$
D． $\{ - 2 , - 1,0,2,3 \}$
Q2 5 marks Addition & Double Angle Formulae Qualitative Reasoning about Double Angle Signs or Inequalities View
If $\alpha$ is an angle in the fourth quadrant, then
A． $\cos 2 \alpha > 0$
B． $\cos 2 \alpha \leqslant 0$
C． $\sin 2 \alpha > 0$
D． $\sin 2 \alpha < 0$
Q3 5 marks Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
During the COVID-19 pandemic prevention and control period, a supermarket opened online sales services and can complete 1200 orders per day. Due to a sharp increase in order volume, orders have accumulated. To solve this problem, many volunteers eagerly signed up to help with order fulfillment. It is known that the supermarket had 500 accumulated unfulfilled orders on a certain day, and the probability that the next day's new orders exceed 1600 is 0.05. Each volunteer can complete 50 orders per day. To ensure that the probability of completing accumulated orders and current day orders within two days is at least 0.95, the minimum number of volunteers needed is
A． 10 people
B． 18 people
C． 24 people
D． 32 people
Q4 5 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
The Circular Mound Altar at the Beijing Temple of Heaven is an ancient place for worshipping heaven, divided into three levels: upper, middle, and lower. At the center of the upper level is a circular stone slab (called the Heaven's Heart Stone), surrounded by 9 fan-shaped stone slabs forming the first ring, with each outer ring increasing by 9 slabs. On the next level, the first ring has 9 more slabs than the last ring of the upper level, and each outer ring also increases by 9 slabs. It is known that each level has the same number of rings, and the lower level has 729 more slabs than the middle level. The total number of fan-shaped stone slabs (excluding the Heaven's Heart Stone) in all three levels is
A． 3699 slabs
B． 3474 slabs
C． 3402 slabs
D． 3339 slabs
Q5 5 marks Circles Distance from Center to Line View
If a circle passing through point $(2,1)$ is tangent to both coordinate axes, then the distance from the center of the circle to the line $2 x - y - 3 = 0$ is
A．$\frac { \sqrt { 5 } } { 5 }$
B．$\frac { 2 \sqrt { 5 } } { 5 }$
C．$\frac { 3 \sqrt { 5 } } { 5 }$
D．$\frac { 4 \sqrt { 5 } } { 5 }$
Q6 5 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
In the sequence $\left\{ a _ { n } \right\}$ , $a _ { 1 } = 2 , a _ { m + n } = a _ { m } a _ { n }$ . If $a _ { k + 1 } + a _ { k + 2 } + \cdots + a _ { k + 10 } = 2 ^ { 15 } - 2 ^ { 5 }$ , then $k =$
A． 2
B． 3
C． 4
D． 5
Q8 5 marks Conic sections Eccentricity or Asymptote Computation View
Let $O$ be the origin of coordinates. The line $x = a$ intersects the two asymptotes of the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ at points $D$ and $E$ respectively. If the area of $\triangle O D E$ is 8, then the minimum value of the focal distance of $C$ is
A． 4
B． 8
C． 16
D． 32
Q9 5 marks Curve Sketching Function Properties from Symmetry or Parity View
Let the function $f ( x ) = \ln | 2 x + 1 | - \ln | 2 x - 1 |$ . Then $f ( x )$
A． is an even function and monotonically increasing on $\left( \frac { 1 } { 2 } , + \infty \right)$
B． is an odd function and monotonically decreasing on $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$
C． is an even function and monotonically increasing on $\left( - \infty , - \frac { 1 } { 2 } \right)$
D． is an odd function and monotonically decreasing on $\left( - \infty , - \frac { 1 } { 2 } \right)$
Q11 5 marks Inequalities Ordering and Sign Analysis from Inequality Constraints View
If $2 ^ { x } - 2 ^ { y } < 3 ^ { - x } - 3 ^ { - y }$ , then
A． $\ln ( y - x + 1 ) > 0$
B． $\ln ( y - x + 1 ) < 0$
C． $\ln | x - y | > 0$
D． $\ln | x - y | < 0$
Q12 5 marks Discrete Probability Distributions Combinatorial Counting in Probabilistic Context View
0-1 periodic sequences have important applications in communication technology. If a sequence $a _ { 1 } a _ { 2 } \cdots a _ { n } \cdots$ satisfies $a _ { i } \in \{ 0,1 \} ( i = 1,2 , \cdots )$ and there exists a positive integer $m$ such that $a _ { i + m } = a _ { i } ( i = 1,2 , \cdots )$ , then it is called a 0-1 periodic sequence, and the smallest positive integer $m$ satisfying $a _ { i + m } = a _ { i } ( i = 1,2 , \cdots )$ is called the period of this sequence. For a 0-1 sequence $a _ { 1 } a _ { 2 } \cdots a _ { n } \cdots$ with period $m$ , $C ( k ) = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } a _ { i } a _ { i + k } ( k = 1,2 , \cdots , m - 1 )$ is an important index describing its properties. Among the following 0-1 sequences with period 5, the one satisfying $C ( k ) \leqslant \frac { 1 } { 5 } ( k = 1,2,3,4 )$ is
A． $11010 \ldots$
B． $11011 \cdots$
C． $10001 \cdots$
D． $11001 \cdots$
Q13 5 marks Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Given unit vectors $\boldsymbol { a } , \boldsymbol { b }$ with an angle of $45 ^ { \circ }$ between them, and $k \boldsymbol { a } - \boldsymbol { b}$ is perpendicular to $\boldsymbol { a }$ , then $k = $ $\_\_\_\_$.
Q14 5 marks Combinations & Selection Distribution of Objects to Positions or Containers View
Four students participate in garbage classification publicity activities in 3 residential areas, with each student going to only 1 area and each area having at least 1 student assigned. The total number of different arrangement methods is $\_\_\_\_$.
Q15 5 marks Complex Numbers Argand & Loci Modulus Inequalities and Triangle Inequality Applications View
Let complex numbers $z _ { 1 } , z _ { 2 }$ satisfy $\left| z _ { 1 } \right| = \left| z _ { 2 } \right| = 2 , z _ { 1 } + z _ { 2 } = \sqrt { 3 } + \mathrm { i }$ , then $\left| z _ { 1 } - z _ { 2 } \right| = $ $\_\_\_\_$.
Q16 5 marks Proof True/False Justification View
Consider the following four propositions:
$p _ { 1 }$ : Three lines that are pairwise intersecting and do not pass through the same point must lie in the same plane.
$p _ { 2 }$ : Through any three points in space, there is exactly one plane.
$p _ { 3 }$ : If two lines in space do not intersect, then these two lines are parallel.
$p _ { 4 }$ : If line $l \subset$ plane $\alpha$ and line $m \perp$ plane $\alpha$ , then $m \perp l$ .
The sequence numbers of all true propositions among the following statements are $\_\_\_\_$.
(1) $p _ { 1 } \wedge p _ { 4 }$
(2) $p _ { 1 } \wedge p _ { 2 }$
(3) $\neg p _ { 2 } \vee p _ { 3 }$
(4) $\neg p _ { 3 } \vee \neg p _ { 4 }$
Q17 12 marks Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
In $\triangle A B C$ , $\sin ^ { 2 } A - \sin ^ { 2 } B - \sin ^ { 2 } C = \sin B \sin C$ .
(1) Find $A$ ;
(2) If $B C = 3$ , find the maximum value of the perimeter of $\triangle A B C$ .
Q18 12 marks Linear regression View
After treatment, the ecosystem in a desert region has improved significantly, and the number of wild animals has increased. To investigate the population of a certain wild animal species in this region, the area is divided into 200 plots of similar size. A simple random sample of 20 plots is selected as sample areas. The sample data obtained is $\left( x _ { i } , y _ { i } \right) ( i = 1,2 , \cdots , 20 )$ , where $x _ { i }$ and $y _ { i }$ represent the plant coverage area (in hectares) and the number of this wild animal species in the $i$-th sample area, respectively. The following calculations are obtained: $\sum _ { i = 1 } ^ { 20 } x _ { i } = 60 , \sum _ { i = 1 } ^ { 20 } y _ { i } = 1200 , \sum _ { i = 1 } ^ { 20 } \left( x _ { i } - \bar { x } \right) ^ { 2 } = 80 , \sum _ { i = 1 } ^ { 20 } \left( y _ { i } - \bar { y } \right) ^ { 2 } = 9000 , \sum _ { i = 1 } ^ { 20 } \left( x _ { i } - \bar { x } \right) \left( y _ { i } - \bar { y } \right) = 800$ .
(1) Find the estimated value of the population of this wild animal species in the region (the estimated value equals the average number of this wild animal species in the sample areas multiplied by the number of plots);
(2) Find the correlation coefficient of the sample $\left( x _ { i } , y _ { i } \right) ( i = 1,2 , \cdots , 20 )$ (accurate to 0.01);
(3) Based on current statistical data, there is great variation in plant coverage area among different plots. To improve the representativeness of the sample and obtain a more accurate estimate of the population of this wild animal species in the region, please suggest a more reasonable sampling method and explain your reasoning.
Q23 10 marks Inequalities Absolute Value Inequality View
Given the function $f ( x ) = \left| x - a ^ { 2 } \right| + | x - 2 a + 1 |$ .
(1) When $a = 2$, find the solution set of the inequality $f ( x ) \geqslant 4$;
(2) If $f ( x ) \geqslant 4$ for all $x$, find the range of values of $a$.