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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-arts

15 maths questions

Q2 Vectors Introduction & 2D Perpendicularity or Parallel Condition View
2. Vector ${ } ^ { \mathbf { 1 } } = ( 2,4 )$ is collinear with vector ${ } ^ { \mathbf { 1 } } = ( x , 6 )$. Then the real number $x =$
(A) 2
(B) 3
(C) 4
(D) 6
Q4 Independent Events View
4. Let $a , b$ be positive real numbers. Then ``$a > b > 1$'' is a \_\_\_\_ condition for ``$\log _ { 2 } a > \log _ { 2 } b > 0$''
(A) necessary and sufficient condition
(B) sufficient but not necessary condition
(C) necessary but not sufficient condition
(D) neither sufficient nor necessary condition
Q7 Conic sections Eccentricity or Asymptote Computation View
7. 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 $A$ and $B$. Then $| A B | =$ [Figure]
(A) $\frac { 4 \sqrt { 3 } } { 3 }$
(B) $2 \sqrt { 3 }$
(C) 6
(D) $4 \sqrt { 3 }$
Q8 Exponential Equations & Modelling Threshold or Tipping-Point Calculation in Applied Exponential Models View
8. The shelf life $y$ (in hours) of a certain food and storage temperature $x$ (in ${ } ^ { \circ } \mathrm { C }$ ) satisfy the functional relationship $y = e ^ { k x + b }$ (where $e = 2.718 \ldots$ is the base of natural logarithm, and $k , b$ are constants). If the shelf life of this food at $0 { } ^ { \circ } \mathrm { C }$ is 192 hours and at $22 { } ^ { \circ } \mathrm { C }$ is 48 hours, then the shelf life at $33 { } ^ { \circ } \mathrm { C }$ is
(A) 16 hours
(B) 20 hours
(C) 24 hours
(D) 28 hours
Q10 Circles Circle-Line Intersection and Point Conditions View
10. Let line $l$ intersect the parabola $y ^ { 2 } = 4 x$ at points $A , B$, and be tangent to the circle $( x - 5 ) ^ { 2 } + y ^ { 2 } = r ^ { 2 } ( r > 0 )$ at point $M$. If $M$ is the midpoint of segment $A B$, and 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 Complex Numbers Arithmetic Complex Division/Multiplication Simplification View
11. Let $i$ be the imaginary unit. Then $i - \frac { 1 } { i } =$ \_\_\_\_.
Q12 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
12. The value of $\lg 0.01 + \log _ { 2 } 16$ is \_\_\_\_.
Q13 Addition & Double Angle Formulae Trigonometric Equation Solving via Identities View
13. Given $\sin \alpha + 2 \cos \alpha = 0$, the value of $2 \sin \alpha \cos \alpha - \cos ^ { 2 } \alpha$ is \_\_\_\_.
Q15 Curve Sketching Multi-Statement Verification (Remarks/Options) View
15. Given functions $f ( x ) = 2 ^ { x } , g ( x ) = \hat { x } ^ { 2 } + a _ { 2 }$ (where $a \in R$). For unequal real numbers $x _ { 1 } , x _ { 2 }$, let $m = \frac { f \left( x _ { 1 } \right) - f \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } } , n = \frac { g \left( x _ { 1 } \right) - g \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$. Consider the following propositions:
(1) For any unequal real numbers $x _ { 1 } , x _ { 2 }$, we have $m > 0$; (2) For any $a$ and any unequal real numbers $x _ { 1 } , x _ { 2 }$, we have $n > 0$;
(3) For any $a$, there exist unequal real numbers $x _ { 1 } , x _ { 2 }$ such that $m = n$; (4) For any $a$, there exist unequal real numbers $x _ { 1 } , x _ { 2 }$ such that $m = - n$. The true propositions are \_\_\_\_ (write the numbers of all true propositions).
III. Solution Questions:
Q16 Sequences and series, recurrence and convergence Closed-form expression derivation View
16. (This question is worth 12 points) Let the sequence $\left\{ a _ { n } \right\} ( n = 1,2,3 \ldots )$ have the sum of the first $n$ terms $S _ { n } = 2 a _ { n } - a _ { 1 }$, and $a _ { 1 } , a _ { 1 } + 1 , a _ { 3 }$ 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 $T _ { n }$.
Q17 Probability Definitions Combinatorial Counting (Non-Probability) View
17. (This question is worth 12 points)
A minibus has 5 seats numbered $1,2,3,4,5$. Passengers $P _ { 1 } , P _ { 2 } , P _ { 3 } , P _ { 4 } , P _ { 5 }$ have assigned seat numbers $1,2,3,4,5$ respectively. They board in order of increasing seat numbers. Passenger $P _ { 1 }$ did not sit in seat 1 due to health reasons. The driver then requires the remaining passengers to be seated according to the following rule: if their own seat is empty, they must sit in their own seat; if their own seat is occupied, they can choose any of the remaining empty seats. (1) If passenger $P _ { 1 }$ sits in seat 3 and other passengers are seated according to the rule, there are 4 possible seating arrangements. The table below shows two of them. Please fill in the remaining two arrangements (enter the seat numbers where passengers sit in the blank cells);
(2) If passenger
Passenger$P _ { 1 }$$P _ { 2 }$$P _ { 3 }$$P _ { 4 }$$P _ { 5 }$
\multirow{3}{*}{Seat Number}32145
32451

$P _ { 1 }$ sits in seat 2, and other passengers are seated according to the rule, find the probability that passenger $P _ { 5 }$ sits in seat 5.
Q18 Vectors 3D & Lines Multi-Part 3D Geometry Problem View
18. (This question is worth 12 points)
A net of a cube and a schematic diagram of the cube are shown in the figure. (1) Mark the letters $F , G , H$ at the corresponding vertices of the cube (no explanation needed); (2) Determine the positional relationship between plane $B E G$ and plane $A C H$, and prove your conclusion; (3) Prove: line $D F \perp$ plane $B E G$. [Figure] [Figure]
Q19 Addition & Double Angle Formulae Multi-Step Composite Problem Using Identities View
19. (This question is worth 12 points)
Let $A , B , C$ be the interior angles of $\triangle A B C$. $\tan A , \tan B$ are the two real roots of the equation $x ^ { 2 } + \sqrt { 3 } p x - p + 1 = 0 ( p \in R )$. (1) Find the size of $C$; (2) If $A B = 3 , A C = \sqrt { 6 }$, find the value of $p$.
Q20 Circles Optimization on a Circle View
20. (This question is worth 13 points) 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 $\frac { \sqrt { 2 } } { 2 }$. The point $( 0,1 )$ is on the minor axis $CD$, and $\overline { P C } \overline { P D } = - 1$. (I) Find the equation of ellipse E; (II) Let O be the origin. A moving line through point P intersects the ellipse at points A and B. Does there exist a constant $\lambda$ such that $\overline { O A } \overline { O B } + \lambda \overline { P A } \overline { P B }$ is a constant? If it exists, find the value of $\lambda$; if it does not exist, explain why. [Figure]
Q21 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
21. (This question is worth 14 points) Given the function $f ( x ) = - 2 \ln x + x ^ { 2 } - 2 a x + a ^ { 2 }$, 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 for all $x$