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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
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2022
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2021
national-A-arts 19 national-A-science 18 national-I 14 national-II 12
2020
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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
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2017
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2016
national-I 7
2015
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Unknown
sample_questions 7
2015 tianjin-arts

17 maths questions

Q1 Inequalities Set Operations Using Inequality-Defined Sets View
1. Given the universal set $U = \{ 1,2,3,4,5,6 \}$, set $A = \{ 2,3,4 \}$, set $B = \{ 1,3,4,6 \}$, then set $A \cap C _ { U } B =$
(A) $\{ 3 \}$
(B) $\{ 2,5 \}$
(C) $\{ 1,4,6 \}$
(D) $\{ 2,3,5 \}$
Q2 Inequalities Linear Programming (Optimize Objective over Linear Constraints) View
2. Let variables $x , y$ satisfy the constraint conditions $x - 2y \geq 0$, $x \leq 2$, $y \geq 0$. Then the maximum value of the objective function $z = 3x + y$ is
(A) 7
(B) 8
(C) 9
(D) 14
Q4 Inequalities Sufficient/Necessary Conditions Between Inequality Conditions View
4. Let $x \in \mathbb{R}$. Then ``$1 < x < 2$'' is ``$|x - 2| < 1$'' a
(A) sufficient but not necessary condition
(B) necessary but not sufficient condition
(C) necessary and sufficient condition
(D) neither sufficient nor necessary condition
Q5 Conic sections Equation Determination from Geometric Conditions View
5. Given the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with one focus at $F ( 2,0 )$, and the asymptote of the hyperbola is tangent to the circle $( x - 2 ) ^ { 2 } + y ^ { 2 } = 3$, then the equation of the hyperbola is
(A) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 13 } = 1$
(B) $\frac { x ^ { 2 } } { 13 } - \frac { y ^ { 2 } } { 9 } = 1$
(C) $\frac { x ^ { 2 } } { 3 } - y ^ { 2 } = 1$
Q7 Exponential Functions Ordering and Comparing Exponential Values View
7. Given the function $f ( x ) = 2 ^ { | x - m | } - 1$ defined on $\mathbb{R}$ (where $m$ is a real number), let $a = f \left( \log _ { 0.5 } 3 \right)$, $b = f \left( \log _ { 2 } 5 \right)$, $c = f ( 2m )$. Then the size relationship of $a, b, c$ is
(A) $a < b < c$
(B) $c < a < b$
(C) $a < c < b$
(D)
Q8 Curve Sketching Number of Solutions / Roots via Curve Analysis View
8. Given the function $f ( x ) = \begin{cases} 2 - | x | , & x \leq 2 \\ ( x - 2 ) ^ { 2 } , & x > 2 \end{cases}$, and function $g ( x ) = 3 - f ( 2 - x )$, then the number of intersections of the graphs of $y = f(x)$ and $y = g(x)$ is
(A) 2
(B) 3
(C) 4
(D) 5
II. Fill-in-the-Blank Questions: This section has 6 questions, each worth 5 points, for a total of 30 points.
Q9 Complex Numbers Arithmetic Complex Division/Multiplication Simplification View
9. $i$ is the imaginary unit. Calculate $\frac { 1 - 2 i } { 2 + i }$ and the result is $\_\_\_\_$.
Q11 Chain Rule Straightforward Polynomial or Basic Differentiation View
11. Given the function $f ( x ) = a x \ln x , x \in ( 0 , + \infty )$, where $a$ is a real number, and $f ^ { \prime } ( x )$ is the derivative of $f ( x )$. If $f ^ { \prime } ( 1 ) = 3$, then the value of $a$ is $\_\_\_\_$.
Q12 Laws of Logarithms Optimize a Logarithmic Expression View
12. Given $a > 0 , b > 0 , ab = 8$, then when $a$ equals $\_\_\_\_$, $\log _ { 2 } a \cdot \log _ { 2 } ( 2 b )$ attains its maximum value.
Q13 Vectors Introduction & 2D Dot Product Computation View
13. In isosceles trapezoid $ABCD$, $AB \parallel DC$, $AB = 2$, $BC = 1$, $\angle ABC = 60 ^ { \circ }$. Points $E$ and $F$ are on segments $BC$ and $CD$ respectively, with $\overrightarrow { BE } = \frac { 2 } { 3 } \overrightarrow { BC }$, $\overrightarrow { DF } = \frac { 1 } { 6 } \overrightarrow { DC }$. Then the value of $\overrightarrow { AE } \cdot \overrightarrow { AF }$ is $\_\_\_\_$.
Q14 Trig Graphs & Exact Values View
14. Given the function $f ( x ) = \sin \omega x + \cos \omega x ( \omega > 0 ) , x \in \mathbb{R}$. If the function $f ( x )$ is monotonically increasing on the interval $( - \omega , \omega )$, and the graph of $f ( x )$ is symmetric about the line $x = \omega$, then the value of $\omega$ is $\_\_\_\_$.
III. Solution Questions: This section has 6 questions, for a total of 80 points.
Q15 13 marks Permutations & Arrangements Probability via Permutation Counting View
15. (13 points) Three table tennis associations have 27, 9, and 18 members respectively. Using stratified sampling, 6 athletes are selected from these three associations to participate in a competition. (I) Find the number of athletes to be selected from each of the three associations respectively; (II) The 6 selected athletes are numbered $A _ { 1 } , A _ { 2 } , A _ { 3 } , A _ { 4 } , A _ { 5 } , A _ { 6 }$ respectively. Two athletes are randomly selected from these 6 athletes to participate in a doubles match.
(i) List all possible outcomes using the given numbering;
(ii) Let event $A$ be ``at least one of the two athletes numbered $A _ { 5 }$ and $A _ { 6 }$ is selected''. Find the probability of event $A$ occurring.
Q16 13 marks Sine and Cosine Rules Find a side length using the cosine rule View
16. (13 points) In $\triangle ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. Given that the area of $\triangle ABC$ is $3 \sqrt { 15 }$, $b - c = 2$, $\cos A = - \frac { 1 } { 4 }$. (I) Find the values of $a$ and $\sin C$; (II) Find the value of $\cos \left( 2 A + \frac { \pi } { 6 } \right)$.
Q17 13 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
17. (13 points) As shown in the figure, $AA _ { 1 } \perp$ plane $ABC$, $BB _ { 1 } \parallel AA _ { 1 }$, $AB = AC = 3$, $BC = 2 \sqrt { 5 }$, $AA _ { 1 } = \sqrt { 7 }$, $BB _ { 1 } = 2 \sqrt { 7 }$. Points $E$ and $F$ are the midpoints of $BC$ and $A _ { 1 } C$ respectively. (I) Prove that $EF \parallel$ plane $A _ { 1 } B _ { 1 } BA$; (II) Prove that plane $AEA _ { 1 } \perp$ plane $BCB _ { 1 }$. (III) Find the angle between line $A _ { 1 } B _ { 1 }$ and plane $BCB _ { 1 }$. [Figure]
Q18 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
18. Given that $\{ a _ { n } \}$ is a geometric sequence with all positive terms, $\{ b _ { n } \}$ is an arithmetic sequence, and $a _ { 1 } = b _ { 1 } = 1$, $b _ { 2 } + b _ { 3 } = 2 a _ { 3 }$, $a _ { 5 } - 3 b _ { 2 } = 7$.
(1) Find the general term formulas for $\{ a _ { n } \}$ and $\{ b _ { n } \}$;
(2) Let $c _ { n } = a _ { n } b _ { n } , n \in \mathbb{N} ^ { * }$. Find the sum of the first $n$ terms of the sequence $\{ c _ { n } \}$.
Q19 Circles Optimization on a Circle View
19. Given the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ with upper vertex $B$, left focus $F$, and eccentricity $\frac { \sqrt { 5 } } { 5 }$.
(1) Find the slope of line $BF$;
(2) Let line $BF$ intersect the ellipse at point $P$ (where $P$ is different from $B$). A line passing through $B$ and perpendicular to $BF$ intersects the ellipse at point $Q$ (where $Q$ is different from $B$). Line $PQ$ intersects the $x$-axis at point $M$, and $|PM| = l|MQ|$.
1) Find the value of $l$;
2) If $|PM| \sin \angle BQP = \frac { 7 \sqrt { 5 } } { 9 }$, find the equation of the ellipse.
Q20 Stationary points and optimisation Prove an inequality using calculus-based optimisation View
20. Given the function $f ( x ) = 4 x - x ^ { 4 } , x \in \mathbb{R}$.
(1) Find the monotonicity of $f ( x )$;
(2) Let $P$ be the intersection point of the curve $y = f ( x )$ and the positive $x$-axis. The tangent line to the curve at point $P$ is $y = g ( x )$. Prove that for any positive real number $x$, we have $f ( x ) \leq g ( x )$;
(3) If the equation $f ( x ) = a$ (where $a$ is a real number) has two positive real roots $x _ { 1 } , x _ { 2 }$ with $x _ { 1 } < x _ { 2 }$, prove that $x _ { 2 } - x _ { 1 } < - \frac { a } { 3 } + 4 ^ { \frac { 1 } { 3 } }$.