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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
2015 sichuan-science

21 maths questions

Q1 Inequalities Set Operations Using Inequality-Defined Sets View
1. Let set $A = \{ \mathrm { x } / ( \mathrm { x } + 1 ) ( \mathrm { x } - 2 ) < 0 \}$, set $B = \{ \mathrm { x } / 1 < \mathrm { x } < 3 \}$, then $A \cup B =$
A. $\{ X / - 1 < X < 3 \}$ B. $\{ X / - 1 < X < 1 \}$ C. $\{ X / 1 < X < 2 \}$ D. $\{ X / 2 < X < 3 \}$
Q2 Complex Numbers Arithmetic Complex Division/Multiplication Simplification View
2. Let $i$ be the imaginary unit, then the complex number $i ^ { 2 } - \frac { 2 } { i } =$
A. $-i$ B. $-3i$
C. $i$ D. $3 i$
Q3 Trig Graphs & Exact Values View
3. Execute the program flowchart shown in the figure, the output value of $S$ is
A. $- \frac { \sqrt { 3 } } { 2 }$ B. $\frac { \sqrt { 3 } } { 2 }$ C. $- \frac { 1 } { 2 }$ D. $\frac { 1 } { 2 }$ [Figure]
Q4 Trig Graphs & Exact Values View
4. Among the following functions, the one with minimum positive period $\pi$ and whose graph is symmetric about the origin is
A. $\mathrm { y } = \cos \left( 2 x + \frac { \pi } { 2 } \right)$
B. $y = \sin \left( 2 x + \frac { \pi } { 2 } \right)$
C. $y = \sin 2 x + \cos 2 x$
D. $y = \sin x + \cos x$
Q5 Conic sections Eccentricity or Asymptote Computation View
5. A line passing through the right focus of the hyperbola $x ^ { 2 } - \frac { y ^ { 2 } } { 3 } = 1$ and perpendicular to the $x$-axis intersects the two asymptotes of the hyperbola at points $\mathrm { A }$ and $\mathrm { B }$, then $| A B | =$
(A) $\frac { 4 \sqrt { 3 } } { 3 }$
(B) $2 \sqrt { 3 }$
(C) $6$
(D) $4 \sqrt { 3 }$
Q6 Permutations & Arrangements Forming Numbers with Digit Constraints View
6. Using the digits $0, 1, 2, 3, 4, 5$ to form five-digit numbers with no repeated digits, the number of even numbers greater than $40000$ is
(A) $144$
(B) $120$
(C) $96$
(D) $72$
Q7 Vectors Introduction & 2D Dot Product Computation View
7. Let quadrilateral $A B C D$ be a parallelogram, $| \overrightarrow { A B } | = 6$, $| \overrightarrow { A D } | = 4$. If points $\mathrm { M }$ and $\mathrm { N }$ satisfy $\overrightarrow { B M } = 3 \overrightarrow { M C }$, $\overrightarrow { D N } = 2 \overrightarrow { N C }$, then $\overrightarrow { A M } \cdot \overrightarrow { N M } =$
(A) $20$
(B) $15$
(C) $9$
(D) $6$
Q8 Laws of Logarithms Compare or Order Logarithmic Values View
8. Let $a$ and $b$ be positive numbers not equal to $1$. Then ``$3 ^ { a } > 3 ^ { b } > 3$'' is ``$\log _ { a } 3 < \log _ { b } 3$'' a
(A) necessary and sufficient condition
(B) sufficient but not necessary condition
(C) necessary but not sufficient condition
(D) neither sufficient nor necessary condition
Q9 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
9. If the function $f ( x ) = \frac { 1 } { 2 } ( m - 2 ) x ^ { 2 } + ( n - 8 ) x + 1$ $(m \geq 0, n \geq 0)$ is monotonically decreasing on the interval $\left[ \frac { 1 } { 2 }, 2 \right]$, then the maximum value of $m n$ is
(A) $16$
(B) $18$
(C) $25$
(D) $\frac { 81 } { 2 }$
Q10 Circles Circle-Line Intersection and Point Conditions View
10. Let line $l$ intersect the parabola $y ^ { 2 } = 4 x$ at points $\mathrm { A }$ and $\mathrm { B }$, and be tangent to the circle $( x - 5 ) ^ { 2 } + y ^ { 2 } = r ^ { 2 }$ $(r > 0)$ at point $M$, where $M$ is the midpoint of segment $A B$. If there are exactly $4$ such lines $l$, then the range of $r$ is
(A) $( 1, 3 )$
(B) $( 1, 4 )$
(C) $( 2, 3 )$
(D) $( 2, 4 )$
II. Fill in the Blanks
Q11 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
11. In the expansion of $( 2 x - 1 ) ^ { 8 }$, the coefficient of the term containing $x^4$ is $\_\_\_\_$ (answer with a number).
Q12 Addition & Double Angle Formulae Simplification of Trigonometric Expressions with Specific Angles View
12. $\sin 15 ^ { \circ } + \sin 75 ^ { \circ } = \_\_\_\_$.
Q13 Exponential Equations & Modelling Threshold or Tipping-Point Calculation in Applied Exponential Models View
13. The shelf life $y$ (in hours) of a certain food and the storage temperature $x$ (in ${}^{\circ} \mathrm { C }$) satisfy the functional relationship $y = e ^ { k x + b }$ ($e = 2.718 \cdots$ is the base of the natural logarithm, $k$ and $b$ are constants). If the shelf life of this food at $0 ^ { \circ } \mathrm{C}$ is designed to be $192$ hours, and the shelf life at $22 ^ { \circ } \mathrm{C}$ is $45$ hours, then the shelf life of this food at $33 ^ { \circ } \mathrm{C}$ is $\_\_\_\_$ hours.
Q14 Vectors 3D & Lines MCQ: Angle Between Skew Lines View
14. As shown in the figure, quadrilaterals $ABCD$ and $ADPQ$ are both squares, and the planes they lie in are mutually perpendicular. A moving point $M$ is on segment $PQ$. $\mathrm { E }$ and $\mathrm { F }$ are the midpoints of $\mathrm { AB }$ and $\mathrm { BC }$ respectively. Let the angle between skew lines $EM$ and $AF$ be $\theta$, then the maximum value of $\cos \theta$ is $\_\_\_\_$. [Figure]
Q15 Curve Sketching Multi-Statement Verification (Remarks/Options) View
15. Given functions $f ( x ) = 2 ^ { x }$ and $g ( x ) = x ^ { 2 } + a x$ (where $a \in \mathbb{R}$). For unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, let $m = \frac { f \left( x _ { 1 } \right) - f \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$ and $n = \frac { g \left( x _ { 1 } \right) - g \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$.
The following propositions are given:
(1) For any unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, we have $m > 0$;
(2) For any $a$ and any unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, we have $n > 0$;
(3) For any $a$, there exist unequal real numbers $x _ { 1 }$ and $x _ { 2 }$ such that $m = n$;
(4) For any $a$, there exist unequal real numbers $x _ { 1 }$ and $x _ { 2 }$ such that $m = - n$. The true propositions are $\_\_\_\_$ (write out the numbers of all true propositions).
III. Solution Questions
Q16 Sequences and series, recurrence and convergence Closed-form expression derivation View
16. Let the sequence $\left\{ a _ { n } \right\}$ have the sum of the first $n$ terms $S _ { n } = 2 a _ { n } - a _ { 3 }$, and $a _ { 1 }$, $a _ { 2 } + 1$, $a _ { 3 }$ form an arithmetic sequence.
(1) Find the general term formula of the sequence $\left\{ a _ { n } \right\}$;
(2) Let $T _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ \frac { 1 } { a _ { n } } \right\}$. Find the minimum value of $n$ such that $\left| T _ { n } - 1 \right| < \frac { 1 } { 1000 }$.
Q17 Combinations & Selection Combinatorial Probability View
17. Two schools A and B organize student teams to participate in a debate competition. School A recommends $3$ male students and $2$ female students, while school B recommends $3$ male students and $4$ female students. The students recommended by both schools participate in training. Since the students' levels are comparable after training, $3$ people are randomly selected from the male students and $3$ people are randomly selected from the female students to form a representative team.
(1) Find the probability that at least $1$ student from school A is selected for the representative team.
(2) Before a certain competition, $4$ people are randomly selected from the $6$ team members to participate. Let $X$ denote the number of male students participating, find the probability distribution and mathematical expectation of $X$.
Q18 Vectors 3D & Lines Multi-Part 3D Geometry Problem View
18. A net of a cube and a schematic diagram of the cube are shown in the figure. In the cube, let $M$ be the midpoint of $BC$ and $N$ be the midpoint of $GH$.
(1) Mark the letters $F$, $G$, $H$ at the corresponding vertices of the cube (no explanation needed).
(2) Prove: line $MN \parallel$ plane $BDH$.
(3) Find the cosine of the dihedral angle $A - E G - M$. [Figure]
Q19 Addition & Double Angle Formulae Multi-Step Composite Problem Using Identities View
19. As shown in the figure, $A$, $B$, $C$, $D$ are the four interior angles of quadrilateral $ABCD$.
(1) Prove: $\tan \frac { A } { 2 } = \frac { 1 - \cos A } { \sin A }$;
(2) If $A + C = 180 ^ { \circ }$, $AB = 6$, $BC = 3$, $CD = 4$, $AD = 5$, find the value of $\tan \frac { A } { 2 } + \tan \frac { B } { 2 } + \tan \frac { C } { 2 } + \tan \frac { D } { 2 }$. [Figure]
Q20 Circles Circle-Related Locus Problems View
20. As shown in the figure, the ellipse $\mathrm { E } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ $(a > b > 0)$ has eccentricity $\frac { \sqrt { 2 } } { 2 }$. A moving line $l$ passing through point $\mathrm { P } ( 0, 1 )$ intersects the ellipse at points $\mathrm { A }$ and $\mathrm { B }$. When line $l$ is parallel to the $x$-axis, the length of the chord intercepted by line $l$ on ellipse $E$ is $2 \sqrt { 2 }$.
(1) Find the equation of ellipse $E$;
(2) In the rectangular coordinate system $xOy$, does there exist a fixed point $Q$ different from point $P$ such that $\frac { | Q A | } { | Q B | } = \frac { | P A | } { | P B | }$ always holds? If it exists, find the coordinates of point $Q$; if it does not exist, explain the reason. [Figure]
Q21 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
21. Given the function $f ( x ) = - 2 ( x + a ) \ln x + x ^ { 2 } - 2 a x - 2 a ^ { 2 } + a$, where $a > 0$.
(1) Let $g ( x )$ be the derivative of $f ( x )$. Discuss the monotonicity of $g ( x )$;
(2) Prove: there exists $a \in ( 0, 1 )$ such that $f ( x ) \geq 0$ holds on the interval $(1, + \infty)$, and $f ( x ) = 0$ has a unique solution in $(1, + \infty)$.