Class number, name and grade

(This exam is divided into two parts: Volume 1 (multiple-choice questions) and Volume 2 (non-multiple-choice questions), with a full score of 150. The examination time is 120 minutes. )

The first volume (multiple choice questions ***50 points)

A, multiple-choice questions (this topic is entitled *** 10, each small item 5 points, * * * 50 points. Only one of the four options given in each small question meets the requirements)

1. If it is known, "Yes" (★★★★).

A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions

C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition

2. Let, be different straight lines, and, be different planes. There are the following four propositions:

① If then ② If,,, then

③ If, then ④ If, then

The serial number of the real question is (★★★★).

A.①④ B. ②③ C.②④ D.①③

There are 40 balls in the bag, including 16 red balls, 12 blue balls, 8 white balls and 4 yellow balls. If 10 balls are randomly selected as a sample, the probability of this sample obtained by stratified sampling method is (★★★★).

A.B. C. D。

4. In △ ABC, the area of is (★★★).

A.B. 1

5. The zero point of the function must be within the interval (★★★).

A.B. C. D。

6. It is known that the following inequality must hold (★★★).

A.B. C. D。

7. It is known that Sn represents the sum of the first n terms of arithmetic progression, which is equal to (★★★★).

A.B. C. D。

8. The following proposition is false (★★★★).

A.

B. The negative form of ""is ""

C. decreasing upward

D. They are not even functions.

9. The function defined on satisfies that the minimum value of when, when, is (★★★★).

A.B. C. D。

10. If the three views of a geometric figure are shown on the right, in which the front view is a regular triangle with a side length of and the top view is a regular hexagon, then the side view area of the geometric figure is (★★★★★★).

A.B. C. D。

Volume 2 (multiple choice questions *** 100)

Fill in the blanks (***5 small questions, 4 points for each small question, ***20 points. Fill in the answers on the lines in the questions)

1 1. Given that x and y are satisfied, the maximum value of is equal to ★★★★★★.

12. According to the program block diagram shown on the right, if input,

Then output ★★★★★★★★.

13. Then the constant term in binomial expansion is ★★★★★★ ★.

14. The inclination of the straight line passing through the point is twice that of the straight line, so the trajectory equation of the center of the circle tangent to the straight line and the X axis is ★★★★★★★★★ ★.

65438+

vectorial

Maximum value ★★★★★★★.

Third, the solution (this big question is ***6 small questions, ***80 points. The solution should be written in words, proof process or calculus steps; Where 16 to 19 score13,20 and 2 1 score 14).

16.△, the opposite sides of the angle are vectors respectively.

, and

(1) Find the size of the acute angle;

(2) If, find the maximum area of △.

17. As shown in the figure, in a straight triangular prism,

It's a little off-side.

(1) verification: plane;

(2) Find the size of dihedral angle;

(3) Find the distance from the point to the plane.

18. As we all know, a large number of red crucian carp and goldfish are raised in a pond. In order to estimate the number of these two kinds of fish in the pond, the breeders captured 1000 red crucian carp and goldfish from the reservoir, marked each fish as not affecting its survival, and then put it back into the pond. After a period of time, 1000 fish were randomly caught from the pond at a time.

(1) Calculate the average number of labeled red crucian carp and goldfish respectively according to the stem leaf diagram, and estimate the number of red crucian carp and goldfish in the pond;

(2) Assume that the number of red crucian carp in five fish is randomly selected from the pond and counted one by one, and the distribution list and mathematical expectation are obtained.

19. It is known that the minimum value of the quadratic function whose domain is r is 0 and exists, the chord length of the line cut by the image is 0, and the sequence satisfies.

(1) function;

(2) Assume that the series is geometric series;

(3) Find the general term formula of the sequence.

20. If there is a sum of real constants, so that the sum of functions is constant for any real number in its definition domain, the line is called an "isolated line" of the sum. Known (where is the base of natural logarithm).

Extreme value of (1);

(2) Is there an isolated straight line between function and? If it exists, solve the isolated straight line equation; If it does not exist, please explain why.

▲ Topic selection: Choose two of the following three small questions and only calculate the scores of the first two questions.

2 1.( 1) (4-2 matrix and matrix transformation are optional) If matrix A has eigenvalues, sum its corresponding eigenvectors respectively to find matrix A;

(2) (4-4 coordinate system and parametric equation are selected) Given the point and inclination of a straight line, let it intersect with a circle and two points, and find the product of the distance from one point to two points;

(3) (Inequality 4-5 Elective Course) Understanding and Verification:

Fuzhou No.3 Middle School (Science) in 2009, the beginning of the next semester, the quality inspection of mathematics examination paper reference answer.

First, multiple-choice questions: ADACA BBDCA

2. Fill in the blanks: (11); ( 12)4; ( 13)24;

( 14) ; ( 15) ;

16. Solution: (1)

That's two points.

This is also an acute angle of .................................... 4 points.

Six points.

(2)

................................., eight.

Substitute the above formula: (if and only if the equal sign holds) … 10 point.

(If and only if the equal sign holds)

.................................. 13.

17. Prove: (1) It is easy to know the curved surface in the straight triangular prism ABC-a1b1.

ACC 1A 1⊥ aircraft ABC,

Acb = 90, ∴BC⊥ surface ACC 1A 1, ... 2 points.

∴bc⊥am face ACC 1a 1

∫ Also,

∴ AM？ The plane ... four points.

Solution: (2) Let the intersection of AM and A 1C be O, and connect BO, from (1).

Me? OB and AM OC,

So ∠BOC is the plane angle of dihedral angle B-AM-C, ... 6 points.

In Rt△ACM and Rt△A 1AC, ∠ OAC+∠ ACO = 90, ∠ AA 1C = ∠ MAC.

∴Rt△ACM∽Rt△A 1AC,∴

∴

∴ in Rt△ACM,

∵ ,∴

∴, Tan in Rt△BCO

So the dihedral angle is 45 degrees ... 9 minutes.

(3) The distance from point C to plane ABM is h, which is easy to know.

You can get ... 10 points.

∵

∴ ,

∴

The distance from point C to ABM plane is ... 13 minutes.

Solution 2: (1) Same solution 1 4 points.

(2) As shown in the figure, with C as the origin, the straight lines of CA, CB and CC 1 are respectively

For x-axis, y-axis and z-axis, a spatial rectangular coordinate system is established, and then

, setting

∵ ,∴

So, that is to say,

So ... six points.

Let the vector be the normal vector of the plane AMB, then

, that is,

Let x= 1, then a normal vector of plane AMB is, obviously, a normal vector of plane AMC,

It is easy to know that the angle between and is equal to the dihedral angle B-AM-C, so the dihedral angle is 45.

..... 9 points

(3) The projection length of the vector on the normal vector is the sought distance,

∵

The distance from point C to ABM plane is ... 13 minutes.

18.( 1) 10 recorded an average number of 20 red crucian carp and China goldfish, so it can be considered that the number of red crucian carp in the pond is the same as that of China goldfish. If the total number of two kinds of fish in the pond is 0, then there is.

-Four points.

That is to say,

Therefore, it can be estimated that the number of red crucian carp and China goldfish in the reservoir is 25,000. -Seven points.

② Obviously, 9 points.

Its distribution list is

0 1 2 3 4 5

- 1 1.

Mathematical expectation. -13 points.

19. Solution: (1) Hypothesis,

Then the two intersections of the line and the image are (1, 0).

............................., 4 points.

(2)

; A sequence is a geometric series whose first term is 1, the common ratio.

9 points ... 9 points.

(3)∵

............................. 13.

20. Solution: (1).

When.

When,, function decreases;

When, when the function increases;

∴ When appropriate, take the minimum value, and the minimum value is 6 points for ............................

(2) Solution 1: It can be seen from (1) that the image of the sum of functions has a common point.

So, if there is an isolated sum straight line, this straight line passes through this common point.

Let the slope of an isolated straight line be, then the equation of the straight line is, that is.

By, a constant can be established.

By, by.

The following proves that Dang Shiheng was established. Orders,

Then,

When.

When, when the function increases;

When,, function decreases;

∴ When appropriate, take the maximum value, and its maximum value is.

Therefore, it is constant.

∴ Function sum has a unique isolated straight line ..................................... 14 score.

Solution 2: We can know the time from (1), (when and when to take the equal sign).

If there is an isolated sum line, there is a real constant sum, which makes the sum constant hold.

Order, and then and

That is, the following steps are the same as the solution 1.

2 1. solution: (1) hypothesis, derived from

Seven points ... Seven points.

(2) The parameter equation of the straight line is,

Substitute the parametric equation of a straight line into the equation.

Seven points ... Seven points.

(3)

According to the ranking inequality, sum of disorder > inverse sum.

Seven points ... Seven points.

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