gaokao

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 sichuan-arts

16 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 Inequalities Sufficient/Necessary Conditions Between Inequality Conditions 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
Q5 Trig Graphs & Exact Values View
5. Among the following functions, the odd function with minimum positive period $\pi$ is
(A) $y = \cos \left( 2 x + \frac { \pi } { 2 } \right)$
(B) $y = \sin \left( x 2 + \frac { \pi } { 3 } \right)$
(C) $y = \sin z + \cos$ $y = \sin x +$
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
Q9 Inequalities Optimization Subject to an Algebraic Constraint View
9. Let real numbers $x , y$ satisfy $\left\{ \begin{array} { c } 2 x + y \leq 10 , \\ x + 2 y \leq 14 , \\ x + y \geq 6 , \end{array} \right.$ Then the maximum value of $x y$ is
(A) $\frac { 25 } { 2 }$
(B) $\frac { 49 } { 2 }$
(C) 12
(D) 16
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 Trig Proofs 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 Function Transformations 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 Permutations & Arrangements 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.
Q19 Trig Proofs 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 $x$