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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 shaanxi-science

17 maths questions

Q1 Probability Definitions Set Operations View
1. Let $M = \left\{ x \mid x ^ { 2 } = x \right\} , N = \{ x \mid \lg x \leq 0 \}$, then $M \bigcup N =$
A. $[ 0,1 ]$
B. $( 0,1 ]$
C. $[ 0,1 )$
D. $( - \infty , 1 ]$
Q2 Data representation View
2. A middle school has 110 teachers in the junior high division and 150 teachers in the senior high division. Their gender ratios are shown in the figure. The number of female teachers in the school is
A. 167
B. 137
C. 123
D. 93 [Figure]
Q3 Trig Graphs & Exact Values View
3. As shown in the figure, the water depth change curve of a certain port from 6 to 18 o'clock approximately satisfies the function $y = 3 \sin \left( \frac { \pi } { 6 } x + \varphi \right) + k$. Based on this function, the maximum water depth (in meters) during this period is
A. 5
B. 6
C. 8
D. 10 [Figure]
Q4 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
4. In the expansion of the binomial $( x + 1 ) ^ { n } \left( n \in N _ { + } \right)$, the coefficient of $x ^ { 2 }$ is 15. Then $n =$
A. 4
B. 5
C. 6
D. 7
Q7 Vectors Introduction & 2D Inequality or Proof Involving Vectors View
7. For any vectors $\vec { a } , \vec { b }$, which of the following relations always holds?
A. $| \vec { a } \bullet \vec { b } | \leq | \vec { a } | | \vec { b } |$
B. $| \vec { a } - \vec { b } | \leq | | \vec { a } | - | \vec { b } | |$
C. $( \vec { a } + \vec { b } ) ^ { 2 } = | \vec { a } + \vec { b } | ^ { 2 }$
D. $( \vec { a } + \vec { b } ) ( \vec { a } - \vec { b } ) = \vec { a } ^ { 2 } - \vec { b } ^ { 2 }$
Q9 Exponential Functions Ordering and Comparing Exponential Values View
9. Let $f ( x ) = \ln x , 0 < a < b$. If $p = f ( \sqrt { a b } ) , q = f \left( \frac { a + b } { 2 } \right)$, $r = \frac { 1 } { 2 } ( f ( a ) + f ( b ) )$, then the correct relation is
A. $q = r < p$
B. $q = r > p$
C. $p = r < q$
D. $p = r > q$
Q11 Geometric Probability View
11. Let the complex number $z = ( x - 1 ) + y i ( x , y \in R )$. If $| z | \leq 1$, the probability that $y \geq x$ is
A. $\frac { 3 } { 4 } + \frac { 1 } { 2 \pi }$
B. $\frac { 1 } { 4 } - \frac { 1 } { 2 \pi }$
C. $\frac { 1 } { 2 } - \frac { 1 } { \pi }$
D. $\frac { 1 } { 2 } + \frac { 1 } { \pi }$
Q12 Discriminant and conditions for roots Parameter range for specific root conditions (location/count) View
12. For the quadratic function $f ( x ) = a x ^ { 2 } + b x + c$ (where $a$ is a non-zero constant), four students each give a conclusion. Exactly one conclusion is wrong. The wrong conclusion is
A. $-1$ is a zero of $f ( x )$
B. $1$ is an extremum point of $f ( x )$
C. $3$ is an extremum value of $f ( x )$
D. The point $( 2,8 )$ lies on the curve $y = f ( x )$
II. Fill in the Blanks
Q13 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
13. A group of 1010 numbers with median 1010 form an arithmetic sequence with last term 2015. The first term of this sequence is $\_\_\_\_$
Q14 Conic sections Equation Determination from Geometric Conditions View
14. If the directrix of the parabola $y ^ { 2 } = 2 p x ( p > 0 )$ passes through a focus of the hyperbola $x ^ { 2 } - y ^ { 2 } = 1$, then $p = $ $\_\_\_\_$
Q15 Tangents, normals and gradients Normal or perpendicular line problems View
15. The tangent line to the curve $y = e ^ { x }$ at the point $( 0,1 )$ is perpendicular to the tangent line to the curve $y = \frac { 1 } { x } ( x > 0 )$ at point P. The coordinates of P are $\_\_\_\_$
Q16 Areas by integration View
16. As shown in the figure, a water channel with an isosceles trapezoidal cross-section has its boundary deformed into a parabolic shape due to silt deposition (shown by the dashed line in the figure). The ratio of the original maximum flow to the current maximum flow is $\_\_\_\_$ [Figure]
III. Solution Questions (This section has 6 questions totaling 70 points. Solutions must include explanations, proofs, and calculation steps.)
Q17 12 marks Trig Proofs Triangle Trigonometric Relation View
17. (12 points) In $\triangle \mathrm { ABC }$, the sides opposite to angles $\mathrm { A } , \mathrm { B }$, C are $a , b , c$ respectively. The vector $\vec { m } = ( a , \sqrt { 3 } b )$ is parallel to $\vec { n } = ( \cos \mathrm { A } , \sin \mathrm { B } )$. (I) Find A; (II) If $a = \sqrt { 7 } , b = 2$, find the area of $\triangle \mathrm { ABC }$.
Q18 12 marks Vectors 3D & Lines Dihedral Angle Computation View
18. (12 points) As shown in Figure 1, in right trapezoid ABCD, $\mathrm { AD } / / \mathrm { BC } , ~ \angle \mathrm { BAD } = \frac { \pi } { 2 } , \mathrm { AB } = \mathrm { BC } = 1$, $\mathrm { AD } = 2$, E is the midpoint of AD, and O is the intersection of AC and BE. Fold $\triangle \mathrm { ABE }$ along BE to the position $\Delta \mathrm { A } _ { 1 } \mathrm { BE }$ as shown in Figure 2.
[Figure]
Figure 1
[Figure]
Figure 2
(I) Prove that $\mathrm { CD } \perp$ plane $\mathrm { A } _ { 1 } \mathrm { OC }$; (II) If plane $\mathrm { A } _ { 1 } \mathrm { BE } \perp$ plane BCDE, find the cosine of the dihedral angle between plane $\mathrm { A } _ { 1 } \mathrm { BC }$ and plane $\mathrm { A } _ { 1 } \mathrm { CD }$.
Q19 12 marks Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View
19. (12 points) The one-way driving time T between a school's new and old campuses depends only on road conditions. A sample of 100 observations was collected with the following results:
T (minutes)25303540

Frequency20304010

(I) Find the probability distribution of T and the mathematical expectation ET; (II) Professor Liu drives from the old campus to the new campus for a 50-minute lecture, then immediately returns to the old campus. Find the probability that the total time from leaving the old campus to returning is no more than 120 minutes.
Q20 12 marks Circles Circle-Line Intersection and Point Conditions View
20. (12 points) The ellipse $\mathrm { E } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has semi-focal distance $c$. The distance from the origin O to the line passing through the points $( c , 0 )$ and $( 0 , b )$ is $\frac { 1 } { 2 } c$. (I) Find the eccentricity of ellipse E; (II) As shown in the figure, AB is a diameter of circle $\mathrm { M } : ( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = \frac { 5 } { 2 }$. If the ellipse E passes through points A and B, find the equation of ellipse E. [Figure]
Q21 12 marks Sequences and Series Functional Equations and Identities via Series View
21. (12 points) Let $f _ { n } ( x )$ be the sum of the terms of the geometric sequence $1 , x , x ^ { 2 } , \cdots , x ^ { n }$, where $x > 0$, $n \in \mathrm {~N} , ~ n \geq 2$. (I) Prove that the function $\mathrm { F } _ { n } ( x ) = f _ { n } ( x ) - 2$ has exactly one zero in $\left( \frac { 1 } { 2 } , 1 \right)$ (denoted as $x _ { n }$), and $x _ { n } = \frac { 1 } { 2 } + \frac { 1 } { 2 } x _ { n } ^ { n + 1 }$; (II) Consider an arithmetic sequence with the same first term, last term, and number of terms as the above geometric sequence, with sum $g _ { n } ( x )$. Compare the sizes of $f _ { n } ( x )$ and $g _ { n } ( x )$, and provide a proof.
Choose one of questions 22, 23, or 24 to answer. If you do more than one, only the first one will be graded. Mark the box number of your chosen question with a 2B pencil on the answer sheet.