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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 beijing

21 maths questions

Q1 4 marks Probability Definitions Set Operations View
Given sets $M = \{ x \mid - 4 < x \leq 1 \} , N = \{ x \mid - 1 < x < 3 \}$, then $M \cup N =$ \_\_\_\_
Q2 4 marks Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
Given $\frac { Z } { \mathrm { i } } = \mathrm { i } - 1$, then $Z =$ \_\_\_\_
Q3 4 marks Circles Distance from Center to Line View
Find the distance from the center of the circle $x ^ { 2 } + y ^ { 2 } - 2 x + 6 y = 0$ to the line $x - y + 2 = 0$
Q4 4 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
The coefficient of $x ^ { 3 }$ in the binomial expansion of $( x - \sqrt { x } ) ^ { 4 }$ is \_\_\_\_
Q5 4 marks Vectors Introduction & 2D Vector Properties and Identities (Conceptual) View
Given vectors $\boldsymbol { a } , \boldsymbol { b }$, then ``$( \boldsymbol { a } + \boldsymbol { b } ) ( \boldsymbol { a } - \boldsymbol { b } ) = 0$'' is ``$\boldsymbol { a } = \boldsymbol { b }$ or $\boldsymbol { a } = - \boldsymbol { b }$'' a \_\_\_\_ condition.
Q6 4 marks Trig Graphs & Exact Values View
Given $f ( x ) = \sin \omega x , f \left( x _ { 1 } \right) = - 1 , f \left( x _ { 2 } \right) = 1 , \left| x _ { 1 } - x _ { 2 } \right| _ { \text {min} } = \frac { \pi } { 2 }$, then $\omega =$ \_\_\_\_
Q7 4 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
Let the water quality index be $d = \frac { S - 1 } { \ln n }$, and the larger $d$ is, the better the water quality. If $S$ remains constant and $d _ { 1 } = 2.1 , d _ { 2 } = 2.2$, then the relationship between $n _ { 1 }$ and $n_2$ is \_\_\_\_
Q8 4 marks Moments View
Given a quadrangular pyramid with a square base of side length 4, and lateral edges of lengths $4, 4, 2\sqrt{2}, 2\sqrt{2}$ respectively, find the height of the quadrangular pyramid.
Q9 4 marks Exponential Functions True/False or Multiple-Statement Verification View
Given that $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right)$ are points on $y = 2 ^ { x }$, which of the following is correct?
A. $\log _ { 2 } \frac { y _ { 1 } + y _ { 2 } } { 2 } > \frac { x _ { 1 } + x _ { 2 } } { 2 }$
B. $\log _ { 2 } \frac { y _ { 1 } + y _ { 2 } } { 2 } < \frac { x _ { 1 } + x _ { 2 } } { 2 }$
C. $\log _ { 2 } \frac { y _ { 1 } + y _ { 2 } } { 2 } > x _ { 1 } + x _ { 2 }$
D. $\log _ { 2 } \frac { y _ { 1 } + y _ { 2 } } { 2 } < x _ { 1 }$
Q10 4 marks Areas Between Curves Area Between Curves with Parametric or Implicit Region Definition View
If the set $\left\{ y \mid y = x + t \left( x ^ { 2 } - x \right) , 0 \leq t \leq 1, 1 \leq x \leq 2 \right\}$ represents a figure where the maximum distance between two points is $d$ and the area is $S$,
A. $d = 3 , S < 1$
B. $d = 3 , S > 1$
C. $d = \sqrt { 10 } , S < 1$
D. $d = \sqrt { 10 } , S > 1$
Q11 5 marks Conic sections Eccentricity or Asymptote Computation View
Given the parabola $y ^ { 2 } = 16 x$, the coordinates of the focus are \_\_\_\_.
Q12 5 marks Trig Graphs & Exact Values View
Given $\alpha \in \left[ \frac { \pi } { 6 } , \frac { \pi } { 3 } \right]$, and the terminal sides of $\alpha$ and $\beta$ are symmetric about the origin, then the maximum value of $\cos \beta$ is \_\_\_\_.
Q13 5 marks Conic sections Eccentricity or Asymptote Computation View
Given the hyperbola $\frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$, find the slopes of lines passing through $( 3,0 )$ that have only one intersection point with the hyperbola \_\_\_\_.
Q14 5 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
Given that the volumes of three cylinders form a geometric sequence with common ratio 10. The diameter of the first cylinder is 65 mm, the diameters of the second and third cylinders are 325 mm, and the height of the third cylinder is 230 mm. Find the heights of the first two cylinders respectively as \_\_\_\_.
Q15 5 marks Sequences and Series Recurrence Relations and Sequence Properties View
Given $M = \left\{ k \mid a _ { k } = b _ { k } \right\}$, where $a _ { n }, b _ { n }$ are not constant sequences and all terms are distinct. Which of the following is correct? \_\_\_
(1) If $a _ { n }, b _ { n }$ are both arithmetic sequences, then $M$ has at most one element;
(2) If $a _ { n }, b _ { n }$ are both geometric sequences, then $M$ has at most three elements;
(3) If $a _ { n }$ is an arithmetic sequence and $b _ { n }$ is a geometric sequence, then $M$ has at most three elements;
(4) If $a _ { n }$ is monotonically increasing and $b _ { n }$ is monotonically decreasing, then $M$ has at most one element.
Q16 Sine and Cosine Rules Compute area of a triangle or related figure View
In $\triangle ABC$, $a = 7$, $A$ is an obtuse angle, $\sin 2B = \frac { \sqrt { 3 } } { 7 } b \cos B$.
(1) Find $\angle A$;
(2) Choose one condition from conditions (1), (2), and (3) below as a given condition and find the area of $\triangle ABC$.
(1) $b = 7$; (2) $\cos B = \frac { 13 } { 14 }$; (3) $c \sin A = \frac { 5 } { 2 } \sqrt { 3 }$. Note: If conditions (1), (2), and (3) are solved separately, only the first solution will be graded.
Q17 Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
Given a quadrangular pyramid $P - ABCD$, where $AD \parallel BC$, $AB = BC = 1$, $AD = 3$, $DE = PE = 2$, $E$ is a point on $AD$, and $PE \perp AD$.
(1) If $F$ is the midpoint of $PE$, prove that $BF \parallel$ plane $PCD$.
(2) If $AB \perp$ plane $PED$, find the cosine of the dihedral angle between plane $PAB$ and plane $PCD$.
Q18 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
Given that the premium for a certain insurance is 0.4 ten thousand yuan. For the first 3 claims, each claim pays 0.8 ten thousand yuan; the 4th claim pays 0.6 ten thousand yuan.
Number of Claims01234
Number of Policies800100603010

A sample of 100 policies is drawn from the population. Using frequency to estimate probability:
(1) Find the probability that a randomly selected policy has at least 2 claims;
(2) (i) Gross profit is the difference between premium and claim amount. Let gross profit be $X$. Estimate the mathematical expectation of $X$;
(ii) If policies with no claims have their premium reduced by 4\% in the next insurance period, and policies with claims have their premium increased by 20\%, estimate the mathematical expectation of gross profit for the next insurance period.
Q19 Conic sections Chord Properties and Midpoint Problems View
Given the ellipse equation $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$. The foci and endpoints of the minor axis form a square with side length 2. A line $l$ passing through $( 0 , t ) ( t > \sqrt { 2 })$ intersects the ellipse at points $A, B$, and $C ( 0,1 )$. Connect $AC$ and it intersects the ellipse at $D$.
(1) Find the equation of the ellipse and its eccentricity;
(2) If the slope of line $BD$ is 0, find $t$.
Q20 Applied differentiation Tangent line computation and geometric consequences View
Given $f ( x ) = x + k \ln ( 1 + x )$, the tangent line to the curve at point $( t , f ( t ) ) ( t > 0 )$ is $l$.
(1) If the slope of tangent line $l$ is $k = - 1$, find the monotonic intervals of $f ( x )$;
(2) Prove that tangent line $l$ does not pass through $( 0,0 )$;
(3) Given $A ( t , f ( t ) ) , C ( 0 , f ( t ) ) , O ( 0,0 )$, where $t > 0$, and tangent line $l$ intersects the $y$-axis at point $B$. When $2 S _ { \triangle ACO } = 15 S _ { \triangle ABC }$, how many points $A$ satisfy the condition? (Reference data: $1.09 < \ln 3 < 1.10, 1.60 < \ln 5 < 1.61, 1.94 < \ln 7 < 1.95$.)
Q21 Proof Proof of Equivalence or Logical Relationship Between Conditions View
Let the set $M = \{ ( i , j , s , t ) \mid i \in \{ 1,2 \} , j \in \{ 3,4 \} , s \in \{ 5,6 \} , t \in \{ 7,8 \} \}$. For a given finite sequence $A$ and sequence $\Omega : \omega _ { 1 } , \omega _ { 2 } , \cdots , \omega _ { k } , \omega _ { k } = \left( i _ { k } , j _ { k } , s _ { k } , t _ { k } \right) \in M$, define transformation $T$: add 1 to columns $i _ { 1 } , j _ { 1 } , s _ { 1 } , t _ { 1 }$ of sequence $A$ to obtain sequence $T _ { 1 } ( A )$; add 1 to columns $i _ { 2 } , j _ { 2 } , s _ { 2 } , t _ { 2 }$ of sequence $T _ { 1 } ( A )$ to obtain sequence $T _ { 2 } T _ { 1 } ( A )$; repeat the above operations to obtain sequence $T _ { k } \cdots T _ { 2 } T _ { 1 } ( A )$, denoted as $\Omega ( A )$.
(3) If $a _ { 1 } + a _ { 3 } + a _ { 5 } + a _ { 7 }$ is even, prove that ``$\Omega ( A )$ is a constant sequence'' is a necessary and sufficient condition for ``$a _ { 1 } + a _ { 2 } = a _ { 3 } + a _ { 4 } = a _ { 5 } + a _ { 6 } = a _ { 7 } + a _ { 8 }$''.