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Papers (71)
2025
national-I 18 national-II 18 national-I_eol 18
2024
beijing 19 national-II 18 national-I_github 18
2023
national-A-science 20 national-B-science 15 national-II 7
2022
national-A-arts 13 national-A-science 11 national-B-arts 19 national-B-science 17 national-I 18 national-II 7
2021
national-A-arts 19 national-A-science 16 national-I 12 national-II 10
2020
national-I-arts 21 national-I-science 9 national-II-science 14 national-III-arts 22 national-III-science 16 shanghai 14
2019
beijing-science 17 national-I-arts 17 national-I-science 13 national-I-science_gkztc 18 national-II-arts 13 national-II-science 8 national-II-science_gkztc 14 national-III-arts 11 national-III-science 16 national-III-science_gkztc 15
2018
national-I-arts 14 national-I-science 13 national-II-arts 20 national-II-science 19 national-III-science 17
2017
national-I-arts 12 national-I-science 13 national-II-arts 14 national-II-science 9
2016
national-I 7
2015
anhui-arts 14 beijing-arts 18 beijing-science 16 chongqing-arts 16 chongqing-science 13 fujian-science 18 hubei-arts 18 hubei-science 15 hunan-arts 20 hunan-science 11 jiangsu 21 national-II-arts 16 national-II-science 16 shaanxi-science 18 sichuan-arts 16 sichuan-science 21 tianjin-arts 18 tianjin-science 17 zhejiang-arts 14 zhejiang-science 15
2011
shanghai-arts 21
2010
shanghai-arts 20 shanghai-science 12
2004
shanghai-science 17
Unknown
sample_questions 3 sample_questions_testmocks 6
2015 beijing-arts

18 maths questions

Q1 5 marks Inequalities Set Operations Using Inequality-Defined Sets View
If set $A = \{ x \mid -5 < x < 2 \}$, $B = \{ x \mid -3 < x < 3 \}$, then $A \cap B =$
Q2 5 marks Circles Circle Equation Derivation View
The equation of a circle with center $(1,1)$ and passing through the origin is
Q3 5 marks Function Transformations View
Which of the following functions is an even function?\n(A) $y = x ^ { 2 } \sin x$\n(B) $y = x ^ { 2 } \cos x$\n(C) $y = | \ln x |$\n(D) $y = 2 ^ { x }$
Q4 5 marks Data representation View
The number of senior, middle-aged, and young teachers at a certain school is shown in the table below. Using stratified sampling to investigate the physical condition of teachers, in the sample drawn, there are 320 young teachers. Then the number of senior teachers in the sample is\n\n\n
\n\nCategoryNumber of People
\n\nSenior Teachers900
\n\nMiddle-aged Teachers1800
\n\nYoung Teachers1600
\n\nTotal4300
\n\n
\n
Q6 5 marks Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Let $a, b$ be non-zero vectors. ``$a \cdot b = | a | | b |$'' is ``$a \parallel b$'' a
Q8 5 marks Measures of Location and Spread View
A certain car fills up its fuel tank every time it refuels. The table below records the situation at two consecutive refueling times. Note: ``Cumulative mileage'' refers to the total distance the car has traveled since leaving the factory. During this period, the average fuel consumption per 100 kilometers for this car is\n\n\n
\n\nRefueling DateRefueling Amount (liters)\begin{tabular}{ c }\nCumulative Mileage at
\nRefueling (kilometers)
\n
\n\hline\nMay 1, 2015 & 12 & 35000 \n\hline\nMay 15, 2015 & 48 & 35600 \n\hline\n\end{tabular}\n
Q9 5 marks Complex Numbers Arithmetic Identifying Real/Imaginary Parts or Components View
The real part of the complex number $i ( 1 + i )$ is
Q10 5 marks Exponential Functions Ordering and Comparing Surd or Numerical Values View
Among the three numbers $2 ^ { - 3 }$, $3 ^ { \frac { 1 } { 2 } }$, $\log _ { 2 } 5$, the largest is
Q11 5 marks Sine and Cosine Rules Find a side or angle using the sine rule View
In $\triangle ABC$, $a = 3$, $b = \sqrt { 6 }$, $\angle A = \frac { 2 \pi } { 3 }$, then $\angle B =$
Q12 5 marks Conic sections Eccentricity or Asymptote Computation View
Given that $(2,0)$ is a focus of the hyperbola $x ^ { 2 } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ $(b > 0)$, then $b =$
Q13 5 marks Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
As shown in the figure, the set of points consisting of $\triangle ABC$ and its interior is denoted as $D$. For any point $P(x, y)$ in $D$, the maximum value of $z = 2x + 3y$ is
Q14 5 marks Data representation View
267 students in the third year participated in the final examination. The ranking of Chinese, mathematics, and total scores of 37 students in a certain class in the entire grade is shown below. A, B, and C are three students in this class.\n\nBased on this examination score,\n(1) Between students A and B, the student whose Chinese score ranking is ahead of their total score ranking is\n(2) In the two subjects of Chinese and mathematics, the subject in which both students' score rankings are more advanced is
Q15 13 marks Harmonic Form View
Given the function $f ( x ) = \sin x - 2 \sqrt { 3 } \sin ^ { 2 } \frac { x } { 2 }$\n(I) Find the minimum positive period of $f ( x )$;\n(II) Find the minimum value of $f ( x )$ on the interval $\left[ 0 , \frac { 2 \pi } { 3 } \right]$.
Q16 13 marks Arithmetic Sequences and Series Multi-Part Structured Problem on AP View
Given that the arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } + a _ { 2 } = 10$ and $a _ { 4 } - a _ { 3 } = 2$.\n(I) Find the general term formula of $\left\{ a _ { n } \right\}$;\n(II) Let the geometric sequence $\left\{ b _ { n } \right\}$ satisfy $b _ { 2 } = a _ { 3 }$ and $b _ { 3 } = a _ { 7 }$. Question: Which term of the sequence $\left\{ a _ { n } \right\}$ is equal to $b _ { 6 }$?
Q17 13 marks Data representation Finite Equally-Likely Probability Computation View
A supermarket randomly selected 1000 customers and recorded their purchasing of four products: A, B, C, and D. The data is organized in the table below, where ``✓'' indicates purchase and ``×'' indicates no purchase.\n\n\n
\n\n\backslashbox{Number of Customers}{Product}ABCD
\n\n100✓×✓✓
\n\n217×✓×✓
\n\n200✓✓✓×
\n\n300✓×✓×
\n\n85✓×××
\n\n98×✓××
\n\n
\n\n\n(I) Estimate the probability that a customer purchases both B and C\n(II) Estimate the probability that a customer purchases exactly 3 of the four products A, B, C, and D\n(III) If a customer has purchased product A, which of products B, C, and D is the customer most likely to have purchased?
Q18 14 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
As shown in the figure, in the triangular pyramid $E - ABC$, plane $EAB \perp$ plane $ABC$, triangle $EAB$ is equilateral, $AC \perp BC$, and $AC = BC = \sqrt { 2 }$. $O$ and $M$ are the midpoints of $AB$ and $EA$ respectively.\n(1) Prove that $EB \parallel$ plane $MOC$.\n(2) Prove that plane $MOC \perp$ plane $EAB$.\n(3) Find the volume of the triangular pyramid $E - ABC$.
Q19 13 marks Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Let the function $f ( x ) = \frac { x ^ { 2 } } { 2 } - k \ln x$, $k > 0$\n(I) Find the monotonic intervals and extreme values of $f ( x )$;\n(II) Prove that if $f ( x )$ has a zero point, then $f ( x )$ has exactly one zero point on the interval $( 1 , \sqrt { e } )$.
Q20 14 marks Circles Intersection of Circles or Circle with Conic View
Given the ellipse $C : x ^ { 2 } + 3 y ^ { 2 } = 3$, a line passing through point $D ( 1,0 )$ but not through point $E ( 2,1 )$ intersects the ellipse $C$ at points $A$ and $B$. The line $AE$ intersects the line $x = 3$ at point $M$.\n(1) Find the eccentricity of the ellipse $C$;\n(II) If $AB$ is perpendicular to the $x$-axis, find the slope of line $BM$;\n(III) Determine the positional relationship between line $BM$ and line $DE$, and explain the reason.