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
2004 shanghai-science

17 maths questions

Q4 Standard trigonometric equations Evaluate trigonometric expression given a constraint View
4. Use a 2B pencil to answer multiple-choice questions, and use a black pen, marker, or ballpoint pen to answer non-multiple-choice questions.
I. Fill-in-the-Blank Questions (Total Score: 48 points, 4 points each)
1. If $\operatorname { tg } \alpha = \frac { 1 } { 2 }$, then $\operatorname { tg } \left( \alpha + \frac { \pi } { 4 } \right) =$ $\_\_\_\_$.
2. A parabola has vertex at $(2,0)$ and directrix $x = -1$. Its focus is at $\_\_\_\_$.
3. Let $A = \left\{ 5 , \log _ { 2 } ( a + 3 ) \right\}$ and $B = \{ a , b \}$. If $A \cap B = \{ 2 \}$, then $A \cup B =$ $\_\_\_\_$.
4. For a geometric sequence $\left\{ a _ { n } \right\} ( n \in \mathbb{N} )$ with common ratio $q = - \frac { 1 } { 2 }$, if $\lim _ { n \rightarrow \infty } \left( a _ { 1 } + a _ { 3 } + a _ { 5 } + \cdots + a _ { 2 n - 1 } \right) = \frac { 8 } { 3 }$, then $a _ { 1 } =$ $\_\_\_\_$.
Q5 Curve Sketching Function Properties from Symmetry or Parity View
5. An odd function $f ( x )$ has domain $[ - 5,5 ]$. When $x \in [ 0,5 ]$, the graph of $f ( x )$ is shown in the figure on the right. The solution set of the inequality $f ( x ) < 0$ is $\_\_\_\_$.
Q6 Vectors Introduction & 2D Vector Word Problem / Physical Application View
6. Given point $A ( 1 , - 2 )$, if vector $\overrightarrow { AB }$ is in the same direction as $\vec { a } = \{ 2,3 \}$, [Figure] and $| \overrightarrow { AB } | = 2 \sqrt { 13 }$, then the coordinates of point B are $\_\_\_\_$.
Q7 Polar coordinates View
7. In the polar coordinate system, the distance from point $M \left( 4 , \frac { \pi } { 3 } \right)$ to the line $l : \rho ( 2 \cos \theta + \sin \theta ) = 4$ is $d =$ $\_\_\_\_$.
Q8 Circles Circle Equation Derivation View
8. Circle C has its center on the line $2 x - y - 7 = 0$ and intersects the y-axis at two points $A ( 0 , - 4 )$ and $B ( 0 , - 2 )$. The equation of circle C is $\_\_\_\_$.
Q9 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
9. In the expansion of the binomial $( x + 1 ) ^ { 10 }$, if a term is chosen at random, the probability that the coefficient of that term is odd is $\_\_\_\_$. (Express the result as a fraction)
Q10 Modulus function Differentiability of functions involving modulus View
10. If the function $f ( x ) = a | x - b | + 2$ is increasing on $[ 0 , + \infty )$, then the range of real numbers $a$ and $b$ is $\_\_\_\_$.
Q12 Geometric Sequences and Series True/False or Multiple-Statement Verification View
12. A set of quantities that can uniquely determine a sequence is called the ``fundamental quantities'' of that sequence. For an infinite geometric sequence $\left\{ a _ { n } \right\}$ with common ratio $q$, among the following four groups of quantities, which group(s) can definitely serve as the ``fundamental quantities'' of the sequence? (Write all group numbers that satisfy the requirement)
(1) $S _ { 1 }$ and $S _ { 2 }$;
(2) $a _ { 2 }$ and $S _ { 3 }$;
(3) $a _ { 1 }$ and $a _ { n }$;
(4) $q$ and $a _ { n }$.
Here $n$ is an integer greater than 1, and $S _ { n }$ is the sum of the first $n$ terms of $\left\{ a _ { n } \right\}$. II. Multiple-Choice Questions (Total Score: 16 points, 4 points each)
Q14 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
14. Given that $y = f ( x )$ is a function with period $2 \pi$, and when $x \in [ 0,2 \pi )$, $f ( x ) = \sin \frac { x } { 2 }$, the solution set of $f ( x ) = \frac { 1 } { 2 }$ is
A. $\left\{ x \left\lvert \, x = 2 k \pi + \frac { \pi } { 3 } \right. , k \in \mathbb{Z} \right\}$.
B. $\left\{ x \left\lvert \, x = 2 k \pi + \frac { 5 \pi } { 3 } \right. , k \in \mathbb{Z} \right\}$.
C. $\left\{ x \left\lvert \, x = 2 k \pi \pm \frac { \pi } { 3 } \right. , k \in \mathbb{Z} \right\}$.
D. $\left\{ x \left\lvert \, x = 2 k \pi + \frac { \pi } { 3 } + ( - 1 ) ^ { k } \frac{\pi}{3} \right. , k \in \mathbb{Z} \right\}$.
Q15 Function Transformations View
15. If the graph of function $y = f ( x )$ can be obtained by rotating the graph of function $y = \lg ( x + 1 )$ counterclockwise by $\frac { \pi } { 2 }$ around the origin O, then $f ( x ) =$
A. $10 ^ { - x } - 1$.
B. $10 ^ { x } - 1$.
C. $1 - 10 ^ { - x }$.
D. $1 - 10 ^ { x }$.
Q16 Data representation View
16. The following table shows the top 5 industries in job applications and recruitment for the first quarter of 2004 in a certain region:
IndustryComputerMachineryMarketingLogisticsTrade
Job Applications2158302002501546767457065280

IndustryComputerMarketingMachineryConstructionChemical
Job Openings124620102935891157651670436

If the employment situation of an industry is measured by the ratio of job applications to job openings in that industry, then based on the data in the table, the employment situation is definitely
A. Better in the computer industry than in the chemical industry.
B. Better in the construction industry than in the logistics industry.
C. Most tight in the machinery industry.
D. More tight in the marketing industry than in the trade industry.
III. Solution Questions (Total Score: 86 points)
Q17 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
17. (Total Score: 12 points) Given that the complex number $z _ { 1 }$ satisfies $( 1 + i ) z _ { 1 } = - 1 + 5 i$, and $z _ { 2 } = a - 2 - i$, where $i$ is the imaginary unit and $a \in \mathbb{R}$. If $\left| z _ { 1 } - \overline { z _ { 2 } } \right| < \left| z _ { 1 } \right|$, find the range of $a$.
Q18 Applied differentiation Geometric or applied optimisation problem View
18. (Total Score: 12 points)
A unit uses wood to make a frame as shown in the figure. The lower part of the frame is a rectangle with sides $x$ and $y$ (in meters). The upper part is an isosceles right triangle. The total area enclosed by the frame is required to be $8 \text{ m}^2$. Find the values of $x$ and $y$ (accurate to 0.001 m) that minimize the amount of material used. [Figure]
Q19 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
19. (Total Score: 14 points) Subquestion 1 is worth 6 points, Subquestion 2 is worth 8 points.
Let the domain of function $f ( x ) = \sqrt { 2 - \frac { x + 3 } { x + 1 } }$ be $A$, and the domain of $g ( x ) = \lg [ ( x - a - 1 ) ( 2 a - x ) ]$ (where $a < 1$) be $B$.
(1) Find $A$;
(2) If $B \subseteq A$, find the range of real number $a$.
Q20 Proof Number of Solutions / Roots via Curve Analysis View
20. (Total Score: 14 points) Subquestion 1 is worth 6 points, Subquestion 2 is worth 8 points
A quadratic function $y = f _ { 1 } ( x )$ has its vertex at the origin and passes through the point $(1,1)$. A reciprocal function $y = f _ { 2 } ( x )$ has its graph intersecting the line $y = x$ at two points with distance 8 between them. Let $f ( x ) = f _ { 1 } ( x ) + f _ { 2 } ( x )$.
(1) Find the expression for function $f ( x )$;
(2) Prove: When $a > 3$, the equation $f ( x ) = f ( a )$ in $x$ has three real solutions.
Q21 Proof Direct Proof of a Stated Identity or Equality View
21. (Total Score: 16 points) Subquestion 1 is worth 4 points, Subquestion 2 is worth 6 points, Subquestion 3 is worth 6 points As shown in the figure, $P - ABC$ is a regular triangular pyramid with base edge length 1. Points $D$, $E$, $F$ are on edges $PA$, $PB$, $PC$ respectively. The cross-section $DEF$ is parallel to the base $ABC$, and the sum of edge lengths of the frustum $DEF - ABC$ equals the sum of edge lengths of the pyramid $P - ABC$. (The sum of edge lengths is the sum of the lengths of all edges of a polyhedron)
(1) Prove: $P - ABC$ is a regular tetrahedron;
(2) If $PD = \frac { 1 } { 2 } PA$, find the dihedral angle $D - BC - A$. (Express the result using inverse trigonometric functions)
(3) Let the volume of the frustum $DEF - ABC$ be $V$. Does there exist a right parallelepiped with all edges equal and volume $V$ such that it has the same sum of edge lengths as the frustum $DEF - ABC$? If it exists, construct such a parallelepiped explicitly and provide a proof; if it does not exist, explain why. [Figure]
Q22 Proof Optimization on Conics View
22. (Total Score: 18 points) Subquestion 1 is worth 6 points, Subquestion 2 is worth 8 points, Subquestion 3 is worth 4 points. Let $P _ { 1 } ( x _ { 1 } , y _ { 1 } )$, $P _ { 2 } ( x _ { 2 } , y _ { 2 } )$, $\cdots$, $P _ { n } ( x _ { n } , y _ { n } )$ ($n \geq 3$, $n \in \mathbb{N}$) be points on a conic section C, and let $a _ { 1 } = \left| OP _ { 1 } \right| ^ { 2 From $\left\{ \begin{array} { l } \frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 25 } = 1 \\ x _ { 3 } ^ { 2 } + y _ { 3 } ^ { 2 } = 70 \end{array} \right.$, we obtain $\left\{ \begin{array} { l } x _ { 3 } ^ { 2 } = 60 \\ y _ { 3 } ^ { 2 } = 10 \end{array} \right.$ $\therefore$ The coordinates of point $P_{3}$ can be $( 2 \sqrt { 15 } , \sqrt { 10 } )$.
(2) Solution 1: The minimum distance from the origin $O$ to each point on the conic curve $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ $(a > b > 0)$ is $b$, and the maximum distance is $a$. $\because a _ { 1 } = \left| O P _ { 1 } \right| ^ { 2 } = a ^ { 2 }$, $\therefore d < 0$, and $a _ { n } = \left| O P _ { n } \right| ^ { 2 } = a ^ { 2 } + ( n - 1 ) d \geq b ^ { 2 }$, $\therefore \frac { b ^ { 2 } - a ^ { 2 } } { n - 1 } \leq d < 0$. $\because n \geq 3$, $\frac { n ( n - 1 ) } { 2 } > 0$, $\therefore S _ { n } = n a ^ { 2 } + \frac { n ( n - 1 ) } { 2 } d$ is increasing on $\left[ \frac { b ^ { 2 } - a ^ { 2 } } { n - 1 } , 0 \right)$, thus the minimum value of $S _ { n }$ is $n a ^ { 2 } + \frac { n ( n - 1 ) } { 2 } \cdot \frac { b ^ { 2 } - a ^ { 2 } } { n - 1 } = \frac { n \left( a ^ { 2 } + b ^ { 2 } \right) } { 2 }$. Solution 2: For each natural number $k$ $(2 \leq k \leq n)$, from $\left\{ \begin{array} { l } x _ { k } ^ { 2 } + y _ { k } ^ { 2 } = a ^ { 2 } + ( k - 1 ) d \\ \frac { x _ { k } ^ { 2 } } { a ^ { 2 } } + \frac { y _ { k } ^ { 2 } } { b ^ { 2 } } = 1 \end{array} \right.$, we solve to get $y _ { k } ^ { 2 } = \frac { - b ^ { 2 } ( k - 1 ) d } { a ^ { 2 } - b ^ { 2 } }$ $\because 0 < y _ { k } ^ { 2 } \leq b ^ { 2 }$, we obtain $\frac { b ^ { 2 } - a ^ { 2 } } { k - 1 } \leq d < 0$, $\therefore \frac { b ^ { 2 } - a ^ { 2 } } { n - 1 } \leq d < 0$. The rest is the same as Solution 1.
(3) Solution 1: If the hyperbola $C: \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, and point $P _ { 1 } ( a , 0 )$, then for a given $n$, the necessary and sufficient condition for the existence of points $P _ { 1 } , P _ { 2 } , \cdots , P _ { n }$ is $d > 0$. $\because$ The distance from the origin $O$ to each point on the hyperbola $C$ is $h \in [ |a| , +\infty )$, and $\left| O P _ { 1 } \right| = a ^ { 2 }$, $\therefore$ Points $P _ { 1 } , P _ { 2 } , \cdots , P _ { n }$ exist if and only if $\left| O P _ { n } \right| ^ { 2 } > \left| O P _ { 1 } \right| ^ { 2 }$, i.e., $d > 0$. Solution 2: If the parabola $C : y ^ { 2 } = 2px$, and point $P _ { 1 } ( 0 , 0 )$, then for a given $n$, the necessary and sufficient condition for the existence of points $P _ { 1 } , P _ { 2 } , \cdots , P _ { n }$ is $d > 0$. The reasoning is the same as above. Solution 3: If the circle $C: ( x - a ) ^ { 2 } + y ^ { 2 } = a ^ { 2 }$ $(a \neq 0)$, and $P _ { 1 } ( 0 , 0 )$, then for a given $n$, the necessary and sufficient condition for the existence of points $P _ { 1 } , P _ { 2 } , \cdots , P _ { n }$ is $0 < d \leq \frac { 4 a ^ { 2 } } { n - 1 }$. $\because$ The minimum distance from the origin $O$ to each point on the circle $C$ is $0$, and the maximum distance is $2 |a|$, and $\left| O P _ { 1 } \right| ^ { 2 } = 0$, $\therefore d > 0$ and $\left| O P _ { n } \right| ^ { 2 } = ( n - 1 ) d \leq 4 a ^ { 2 }$, i.e., $0 < d \leq \frac { 4 a ^ { 2 } } { n - 1 }$.