Mathematics and Science (3)

This volume is divided into two parts: the first volume (multiple-choice questions) and the second volume (non-multiple-choice questions). The full score is 150, and the examination time is 120 minutes.

The first volume (multiple choice questions, ***60 points)

First, multiple-choice questions: this big question * * 12 small questions. 5 points for each small question, 60 points for * * *. Of the four options given in each small question, only

One meets the requirements of the topic.

1. Then set a group.

A.( 1，2] B.[0，+)

C.[0，2]

Let it be a real number and a pure imaginary number, and then

A. 3rd century BC

3. If, then

A.B. C. D。

4. If, and then,

ABC or d

5. In arithmetic progression, if yes, the sum of the first 13 terms of this series is

A.24 B.39 C.52 D. 104

6. Let the tangent of the curve at this point be perpendicular to the straight line, then

BC 1 year

7. If the line and the function intersect at two adjacent points respectively, then

The maximum value of is

A.B. C. D。

8. Let the even function be a decreasing function, and then set the inequality solution.

for

A.B.

C.D.

9. If the image of the function is symmetrical with the image of the function, then

A.B.

C.D.

10. If a straight line passes through this point, then

A.B.

C.D.

1 1. As we all know, the bottom of a quadrangular prism is square, and its sides are equal to the sides of the bottom.

If the projection inside is the center of the square, then the sine of the angle with the bottom is equal to

A.B. C. D。

12. Color all faces of the cube. Any two adjacent edges are different colors. There are five different colors now.

Color, and painted the colors of the three faces of the vertex, then the painting scheme of the remaining three faces is * * *.

A. 15 B. 14 C. 13 D. 12.

Volume 2 (multiple choice questions, ***90 points)

Fill-in-the-blank question: This big question has four small questions, each with 5 points and ***20 points. Fill in the answers on the lines in the questions.

13. If the coefficients of the first three items in the expansion are arithmetic progression, the coefficients of each item in the expansion are

Answer with numbers.

14. If, and when, there is always, then take, as the coordinate point.

The plane area formed is.

15. Let the right directrix of hyperbola with focus on the axis intersect with two asymptotes at two points, and the right focus.

Eccentricity of hyperbola.

16. perpendicular to the plane, when

When the area is the largest, the distance from the point to the straight line is.

Third, the solution: this big question is ***6 small questions, 70 points. The solution should be written in words, proof process or calculus steps.

17. (Full score for this small question 10)

If the function.

(1) Find the monotone decreasing interval of the function;

(2) Given three sides, the corresponding angle is, and the area of the triangle is, if, the value range of.

18. (The full score of this small question is 12)

Due to the financial crisis, the export volume of a company declined. Therefore, relevant experts put forward two schemes to promote exports, each of which needs to be implemented within two years. If the first scheme is implemented, the probability that the export volume in the first year can be restored to pre-crisis 1.0 times, 0.9 times and 0.8 times is 0.3, 0.3 and 0.4 respectively. The probability that the export volume in the second year can be increased to 1.25 times and 1.0 times that in the first year is 0.5 and 0.5 respectively. If the second scheme is implemented, the probability that the export volume in the first year can be restored to 1.2 times, L.0 times and 0.8 times before the crisis is 0.2, 0.3 and 0.5 respectively. The probability that the export volume in the second year is 1.2 times and 1.0 times that in the first year is 0.4 and 0.6 respectively. The first year and the second year of each plan are independent of each other. Order(= 1, 2) indicates that after two years of implementation of the scheme, the export volume has reached multiple before the crisis.

(1) Write the allocation table;

(2) Which scheme is implemented, and the probability that the export volume will exceed the pre-crisis export volume after two years is greater?

(3) No matter which scheme, if the export volume fails to reach, it will just reach or exceed the export volume before the crisis in two years, and the estimated profits are 654.38+10,000 yuan,/kloc-0.50 million yuan and 200,000 yuan respectively. Which scheme has a larger average profit?

19. (The full score of this small question is 12)

In the pyramid, the bottom is a rectangle and the side is a regular triangle, which is the midpoint of the pyramid.

(1) proof: plane;

(2) Find the size of dihedral angle.

20. (The full score of this short question is 12)

It is known that, is a positive integer, the first term of arithmetic progression is, the tolerance is, and the first term of geometric progression is, the common ratio, and there is a sum in the sum of series, which makes, again.

(1) sum value;

(2) Find the smallest item in the series and explain the reason.

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

The center of the ellipse is the coordinate origin, the focus is on the axis, the distance and eccentricity from the focus to the corresponding directrix are, the straight line intersects the axis at one point, and the ellipse intersects at two different points, and.

(1) Find the elliptic equation;

(2) If yes, the value range of.

22. (The full score of this short question is 12)

Known function, where is a constant.

(1) The range of values to find if and when it holds;

(2) The monotone interval of the solution.

Reference answer

1.B 2。 D 3。 A 4。 C 5。 C 6。 D 7。 D 8。 B 9。 C 10。 B

1 1.A 12。 C

1., so choose B.

2. so choose D.

3. So choose.

4. or, so choose C.

5. so choose C.

6. Tangent slope

, so choose D.

7. Observe the image. So choose D.

8. become or, so choose B.

9. there is also symmetry, so choose C.

10. straight lines and ellipses have something in common, so choose B.

1 1. As shown in the figure, if, then,

,

Therefore, the sine value of the angle with the bottom surface is equal to. So choose a.

12. Classification coloring ① Only three colors are used, the reverse side is the same color, and there are 1 painting methods; (2) Four colors are used together, with painting method; ③ There are five colors and one painting method. * * * There are 13 painting methods. So I chose C.

Second,

13.7.By or (by),

The rest of the project is.

14. According to the topic, the plane area formed by a point is a square with a side length of 1 and its area is 1.

15., by, by

.

16.。

As shown, it is possible to set, also,

.

When the area is the largest, the distance from the point to the straight line is.

Third,

17.( 1)

Youde,

The monotone decreasing interval of is.

(2)

.

All values of 18. (1) are 0.8, 0.9, 1.0, 1. 125, 1.25 respectively, and their distribution lists are as follows.

0.8 0.9 1.0 1. 125 1.25

0.2 0. 15 0.35 0. 15 0. 15

All values of are 0.8, 0.96, 1.0, 1, 2, 1.44, and its distribution list is

0.8 0.96 1.0 1.2 1.44

0.3 0.2 0. 18 0.24 0.08

(2) Assuming that the probability of exceeding the pre-crisis export volume after two years of implementation of Scheme I and Scheme II is 0, then

The probability of exceeding the pre-crisis export volume after two or three years of implementation of the plan is even greater.

(3) The estimated profits of Scheme I and Scheme II are as follows

10 15 20

0.35 0.35 0.3

10 15 20

0.5 0. 18 0.32

The average profit of the first scheme is greater.

19.( 1) meets and intersects at this point.

So, that is to say, again, and

This plane is a regular triangle,

, therefore, the plane.

(2) Plane, plane and plane, plane and plane

Set to the midpoint of, connect, and then,

Plane, point, connect, and then.

Is the plane angle of dihedral angle.

Yes,.

Here we go again.

20.( 1) by, get, and then

It is also a positive integer,

, therefore.

(2)

When or, the minimum value is obtained.

2 1.( 1) from

∴ Elliptic circle equation is:

(2) from,

and

Let the equation of the straight line be:

allow

From this. ①

So, let it intersect with the ellipse.

allow

, organized

, organized

, the above types are not valid, ②

Derived from formulas ① and ②.

or

The value range of ∴ is.

22.( 1) Youde

Let's order then

When the world is monotonous.

The value range of is.

(2)

rule

(1) when, is a decreasing function.

At that time, it was to increase functions.

(2) when it is an incremental function.

To sum up; At that time, increase the interval to and decrease the interval to;

When is, increase the interval to.

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