Mathematics examination paper (science)

A, fill in the blanks (this big question out of 44 points)

1. known, and then.

2. If, is a real number and an imaginary number unit, then.

3. Arithmetic series, if, then.

4. In the polar coordinate system, the distance from a point to a straight line is.

5. Known vector, if, then real number.

6. At △, if, then.

7. If any two elements are selected from the set (), the probability that the curve corresponding to the equation represents a hyperbola with the focus on the axis is.

8. Let, if it exists, make, then the range of real numbers is.

9. If the fifth item in the binomial expansion of is, then.

10. If a function is set, it is defined on an interval, and it is not a monotone function if and only if.

1 1. In the rectangular coordinate plane, it is not difficult to draw the conclusion that "for any point on the hyperbola (), if the projection of the point on the axis and the axis are respectively, it must be constant". Similarly, for any point (,) on the hyperbola in the rectangular coordinate plane, if, then.

Second, multiple-choice questions (full mark for this big question 16)

12. ""is "straight line and vertical line" ()

A. sufficient non-essential conditions; B. necessary and insufficient conditions; C. necessary and sufficient conditions; It is neither a sufficient condition nor a necessary condition.

13. Let the sum be a non-zero vector. If the image sum of a function is a straight line, there must be ().

A.； b； c； d。

14. If a section of a cube just cuts the cube into two parts with equal volume, then this section ()

A. it must pass through the center of the cube; B it must pass through the center of one face of the cube;

C. it must pass through a vertex of the cube; It must form a regular polygon.

15. The range of set, set and real number is ().

A.； b； c； d。

Third, answer questions (this big question is 90 points)

16. (The full mark of this question is 12.) As shown in the figure, in a triangular pyramid with a volume of,,, points m and n are the midpoint respectively. Find the angle formed by the straight line on the different plane (the result is expressed by the value of the inverse trigonometric function).

17. (The full mark of this question is 12)

Known theorem: "If two non-zero vectors are not parallel, then the necessary and sufficient condition of () is". Try the above theorem to solve the problem:

Let the relationship between non-zero vector and nonparallel vector, known vector and sum vector; And when, the range of values.

18. (The full score of this question is 14, where 1 gives 5 points and 2 items give 9 points. )

Known function,.

(1) If the equation about has a solution, find the range of numbers;

(2) If so, find the minimum value.

19. (The full score of this question is 16, of which 1 gives 6 points for sub-item and 10 for sub-item 2. )

In economics, there is a model to measure the production capacity of enterprises, which is called "capacity boundary". It represents various possible combinations of the output of several products that an enterprise can produce in a certain period of time under the condition of maximum capacity. For example, if an enterprise can produce product A and product B in a certain period of time under the condition of maximum capacity, the function formed between them is the "capacity boundary function" of the enterprise.

(1) Try to analyze the capacity boundary of this enterprise, and fill in the following table with a serial number of ①, ② and ③:

Output combinations corresponding to points

Practical significance

(1) This is the combination of underutilized production capacity;

(2) This is a production combination in which the production target is divorced from the actual production capacity;

(3) This is the yield combination that maximizes the production capacity.

(2) Assume that the profit of each product A is RMB, and the profit of each product B is twice that of each product A (,). Under the boundary condition of enterprise productivity, try to make a decision for the enterprise. How many products should A and B produce to make the enterprise get the maximum profit?

20. (The full score of this question is 16, of which 1 item is 4 points, the second item is 5 points, and the third item is 7 points).

In the known infinite series, it is the arithmetic progression with the first term as the tolerance; Geometric series, who takes the first term as the norm; It applies to all positive integers.

(1) If yes, please write the first 12 items of the series in turn;

(2) If yes, try to evaluate it;

(3) Let the sum of the first few terms of the series be, and ask whether there is a value to make it true. If it exists, it is the calculated value; If it does not exist, please explain why.

2 1. (The full score of this question is 20, of which 1 item is 5, 2 items are 7 and 3 items are 8. )

It is known that the coordinates of a point are 0, and the line intersects at point P, and the product of its slopes is 0.

(1) Verify that the trajectory of point P is on an ellipse, and write the equation of ellipse;

(2) Let the ellipse in the problem of straight line passing through the origin (1) be at a point and the coordinates of the fixed point be, and try to find the maximum area and slope of the straight line at this time;

(3) Reflect on the solution of the problem (2), and discuss the relationship between the slope of the straight line and the slope of the straight line in the conclusion of the problem (2), so as to generalize the general situation of the point position or the general situation of the ellipse (making the conclusion of the problem (2) a special case after generalization), and try to put forward a guess or design a problem and try to study and solve it.

Note: This question will be graded according to your guess or the quality of the question.