Mathematics (science, engineering, agriculture and medicine)

volume one

1. Multiple choice questions: (This big question is *** 12, with 5 points for each small question. One of the four options given in each question meets the requirements of the topic.

(1) known set, then

(A) (B)

(C) (D)

(2) Complex number

(A)0 (B)2 (C)-2i (D)2

(3) Arguing with the observation data of variables X and Y (,) (i= 1, 2, …, 10), and getting the scatter diagram of 1; Variables u and v have observed data (,) (i= 1, 2, …, 10), and scatter plots are obtained. 2. Judging from these two scatter charts.

(a) the variable x is positively correlated with y, and u is positively correlated with v (b) the variable x is positively correlated with y, and u is negatively correlated with v.

(c) The variable X is negatively correlated with Y, and the variable U is positively correlated with V (d) The variable X is negatively correlated with Y, and the variable U is negatively correlated with V..

(4) The distance from the focus of hyperbola -= 1 to the asymptote is

(A) (B)2 (C) (D) 1

(5) There are four propositions about trigonometric functions:

:x R，+ = : x、y R，sin(x-y)=sinx-siny

: x, = sinx:sinx = comfortable x+y=

One of the false propositions is

(1), (2), (3) and (4),

(6) Let X and Y satisfy

(a) The minimum value is 2 and the maximum value is 3; (b) There is a minimum value of 2 and no maximum value.

(c) There is a maximum of 3, and there is no minimum. (d) There is neither a minimum nor a maximum.

(7) The sum of the first n terms of the geometric series is, 4, 2 becomes arithmetic progression. If = 1, then =

7 (B)8 (3) 15 (4) 16

(8) As shown in the figure, the length of the edge line of the cube is 1, and there are two moving points E and F on the line segment. The following conclusion is wrong.

(1)

(2)

(c) The volume of the triangular pyramid is unchanged.

(d) The angle formed by straight lines in different planes is constant.

(9) Given that O, N and P are in the plane, the points O, N and P are in turn.

(a) Out of the center of gravity; (b) Out of the center of gravity.

(c) centre of gravity other than the centre of gravity; (d) Outside the center of gravity and inside the center of gravity.

(Note: The three high lines of a triangle intersect at a point, which is the vertical center of the triangle. )

(10) If the program block diagram on the right is executed and input, the sum of the output numbers is equal to.

(A)3 (B) 3.5 (C) 4 (D)4.5

(1 1) The three views of a pyramid are shown in the figure, so the total area of the pyramid (unit: c) is

(A)48+ 12(B)48+24(C)36+ 12(D)36+24

(12) min{a, b, c} is used to indicate the minimum value of a, b, c.

Let f(x)=min{, x+2,10-x} (x 0), then the maximum value of f(x) is

(A)4 (B)5 (C)6 (D)7

Volume II

Second, fill in the blanks; This big question is ***4 small questions, each with 5 points.

(13) Let the vertex of the known parabola C be at the coordinate origin, the focus be f (1 0), and the straight line L and parabola C intersect at point A and point B. If the midpoint of AB is (2,2), the equation of the straight line is _ _ _ _ _ _ _.

(14) known function y = sin (x+) (>; 0，-& lt； ) as shown in the figure, then = _ _ _ _ _ _ _ _ _ _

(15) Six of the seven volunteers will participate in community public welfare activities on Saturday and Sunday. If three people are arranged every day, there are _ _ _ _ _ _ _ _ _ _ * different arrangements (answer with numbers).

(16) The sum of the first n items in arithmetic progression is. Given +-=0 and =38, then m = _ _ _ _ _ _

Third, solve the problem: the solution should write descriptive words, prove the process or calculus steps.

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

In order to measure the distance between M and N at the top of two mountains, the plane is measured at two points A and B along the horizontal direction, and A, B, M and N are on the same vertical plane (as shown in the schematic diagram). The data that can be measured on the plane include the depression angle and the distance between A and B. Please design a scheme, which includes: ① pointing out the data to be measured (expressed in letters and marked in the drawing); ② Write out the steps of calculating the distance between m and n with words and formulas.

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

A factory has 1000 workers, of which 250 workers have participated in short-term training (called Class A workers) and 750 workers have participated in long-term training (called Class B workers). At present, stratified sampling method (divided into two layers according to Class A and Class B) is used to randomly select 100 workers from the factory to investigate their production capacity (output here).

(i) Find out the probability that both workers A and B are extracted, where A is a class worker and B is a class worker;

(2) The sampling results of Class A workers and Class B workers are shown in Table 1 and Table 2 respectively.

(1) Determine X and Y first, and then complete the following frequency distribution histogram on the answer sheet. As far as production capacity is concerned, which is the smaller individual difference between Class A workers and Class B workers? (You can answer the conclusion directly by observing the histogram without calculation. )

(2) Estimate the average production capacity of workers in Class A and Class B respectively, and estimate the average production capacity of workers in our factory. The data in the same group are expressed by the midpoint value of the group interval)

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

As shown in the figure, the bottom of the quadrangular pyramid S-ABCD is square, the length of each side is twice the length of the ground, and p is the point on the side SD.

(i) Verification: communication ⊥ self-destruction;

(ii) If SD⊥ plane PAC, find out the size of dihedral angle P-AC-D.

(iii) Under the condition of (ii), is there a little E on the side SC?

Make it a flat package. If it exists, find the value of se: EC;

If it does not exist, please explain why.

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

It is known that the center of ellipse C is the origin of rectangular coordinate system xOy, and the focus is on the S axis. The distances from a vertex to two focal points are 7 and 1 respectively.

(1) Find the equation of ellipse c;

(2) If P is the moving point on the ellipse C, and M is the point on the straight line passing through P and perpendicular to the X axis, = λ, find the trajectory equation of point M, and explain what curve the trajectory is.

(2 1) (the full score of this small question is 12)

known function

(i) For example, the monotone interval of the solution;

(II) If it monotonically increases and decreases, prove that