Mathematics examination for science students

This paper is divided into two parts, the first volume and the second volume, with a total of 4 pages. The perfect score is 150. Examination time 120 minutes. When the exam is over, be sure to hand in your test paper and answer sheet together.

Precautions:

1. Before answering questions, candidates must fill in the name, admission ticket number, county and department on the answer sheet and the position specified in the test paper with a black ink pen with a diameter of 0.5 mm. ..

2. After choosing the answers for each small question in Book 1, use 2B pencil to blacken the answer label of the corresponding question on the answer sheet; If you need to change it, clean it with an eraser, and then choose to apply other answer labels. The answer can't be answered on the paper.

3. Volume 2 must be answered with a 0.5 mm black pen, and the answer must be written in the corresponding position in the designated area of each topic on the answer sheet, not on the test paper; If you need to change, cross out the original answer first, and then write a new answer; Correction fluid, adhesive tape and correction tape cannot be used. Answers that do not answer according to the above requirements are invalid.

Please fill in the answers directly. The answer should be written in words, proof process or calculus steps.

Reference formula:

The volume formula of the cone is V=Sh, where s is the bottom area of the cone and h is the height of the cone.

If events A and B are mutually exclusive, then p (a+b) = p (a)+p (b); If events A and B are independent, then P (AB) = P (A) P (B).

The first volume (***60 points)

1. Multiple-choice question: This topic is entitled *** 12, with 5 points for each question and 60 points for each question. Only one of the four options given in each small question meets the requirements of the topic.

1 If the complex number x satisfies z(2-i)= 1 1+7i(i is an imaginary unit), then z is

A 3+5i B 3-5i C -3+5i D -3-5i

2 given the complete set ={0, 1, 2,3,4}, set A = {1, 2,3}, and b = {2,4}, then (CuA)B is

A { 1，2，4} B {2，3，4}

C {0，2，4} D {0，2，3，4}

3 let a > 0 a ≠ 1, then "function f(x)= a3 is a decreasing function on R" and "function g(x)=(2-a) is a increasing function on R".

Sufficient and unnecessary conditions b Necessary and insufficient conditions

C necessary and sufficient condition d is neither sufficient nor necessary.

(4) 32 people were selected from 960 people by systematic sampling method for questionnaire survey, so the random numbers were 1, 2, ..., 960. After grouping, among the 32 people selected by simple random sampling method, those with numbers falling within the interval were given questionnaire A, those with numbers falling within the interval were given questionnaire B, and the rest were given questionnaire C.,

(A)7 (B) 9 (C) 10 (D) 15

(5) The value range of the objective function z=3x-y is

(1)

(4)

(6) Perform the following program chart. If input a=4, the output value of n is

2(B)3(C)4(D)5

(7) If,, then sin=

(A)(B)(C)(D)

(8) The function f(x) defined on R satisfies f(x+6)=f(x), when -3 ≤ x

335(B)338(C) 1678(D)20 12

(9) The image of the function is roughly [Source: www.shulihua.net].

[Source: Xue. Part. Net]

(10) It is known that the eccentricity of ellipse c: is 0. Hyperbolic x? -Really? The asymptote of = 1 has four intersections with the radius, and the area of the quadrilateral with these four focuses as the vertex is 16, then the equation of ellipse c is

(1 1) There are 16 different cards, including 4 red cards, 4 yellow cards, 4 blue cards and 3 green cards. The required cards cannot be of the same color, and the maximum number of red cards is 1. The number of different cards is

232 (B)252 (C)472 (D)484

(12) Let the function (x)=, g(x)=ax2+bx If the image of y=f(x) and the image of y=g(x) have only two different common points A (x 1, y 1), B (x2

A. Being a

B. Being a

C. when a>0, x 1+x2 < 0, y1+y2 <; 0[ Source: www.shulihua.net]

D. when a>0, x1+x2 > 0，y 1+y2 & gt； 0

Volume II (***90 points)

2. Fill-in-the-blank question: This big question has four small questions, each with 4 points, *** 16 points.

(13) If the solution set of the inequality is, then the real number k = _ _ _ _ _ _ _ _

(14) as shown in the figure, if the side length of the cube ABCD-a 1 b1d1is1,e and f are line segments AA 1 and b/kloc-respectively.

(15) let a > 0. If the area of a closed figure surrounded by curves and straight lines x = a and y=0 is a, then a = _ _ _ _ _

(16) As shown in the figure, in the plane rectangular coordinate system xOy, the initial position of the center of the unit circle is (0, 1), the position of a point P on the circle is (0,0), and the circle rolls forward on the X axis. When the circle rolls to the center of (2, 1), the coordinate of is _ _ _ _ _ _ _ _ _.

Third, answer: This big question is ***6 small questions, ***74 points.

(17) (the full score of this small question is 12)

It is known that the maximum value of vector m=(sinx, 1) and function f (x) = m n is 6.

(i) Find one;

(2) Shift the image with function y=f(x) to the left by one unit, and then shorten the abscissa of each point of the obtained image to the original multiple, and keep the ordinate unchanged to obtain the image with function y=g(x). Find the range of g(x) on.

(18) (the full score of this small question is 12)

In the geometry shown in the figure, the quadrilateral ABCD is an isosceles trapezoid, ab∨CD, ∠ DAB = 60, FC⊥ plane ABCD, AE⊥BD, CB = CD = CF

(i) Verification: BD⊥ planar aed；;

(ii) Find the cosine of dihedral angle f-bd-c. ..

(19) (the full score of this small question is 12)

First, at targets A and B, a shooter shoots at a target once, and the hit probability is: 65438+ hit 0, miss 0; Shoot the target B twice, and the probability of each hit is: 2 points for each hit, and 0 points for no hit. The results of each shot by the shooter are independent of each other. Suppose the shooter has completed the above three shots.

(i) Find out the probability that the shooter only hits once;

(2) Find the distribution table of the shooter's total score X and the mathematical expectation EX [Source: Xue Ke. com]

(20) (The full score of this small question is 12)

In arithmetic progression {an}, a3+a4+a5=84 and a5=73.

(i) Find the general term formula of the sequence {an};

(ii) For any m ∈ n 𔲁, the number of terms in the sequence {an} falling into the interval (9n, 92n) is denoted as bm, and the first m terms and Sn of the sequence {bn} are found. [Source: www.shulihua.net www.shulihua.net]

(2 1) (the full score of this small question is 13)

In the plane rectangular coordinate system xOy, f is the focus of parabola C: x2 = 2py (p > 0), m is any point on parabola C in the first quadrant, the center of the circle passing through m, f and o is q, and the distance from point q to the directrix of parabola C is.

(1) Find the equation of parabola c;

(ii) Is there a point m that makes the straight line MQ tangent to the parabola c? If it exists, find the coordinates of point m; If it does not exist, explain the reasons;

(3) If the abscissa of point M is, straight line L:Y = KX+ has two different intersections with parabola C, and L has two different intersections with circle Q, D and E, find the minimum value when ≤k≤2.

22 (the full score of this small question is 13) [Source: www.shulihua.net]

It is known that the function f(x) = (k is constant, c = 2.7 1828 ... is the base of natural logarithm), and the tangent of the curve y= f(x) at the point (1, f( 1)) is parallel to the X axis.

(i) find the value of k;

(ii) Find the monotone interval of f(x);

(iii) let g(x)=(x2+x), where is the derivative function of f(x), and prove that for any x > 0, g (x) < 1+e-2.