Mathematics (Literature) of College Entrance Examination in Liaoning Province in 2008

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In 2008, the national unified examination for enrollment of ordinary colleges and universities (Liaoning volume)

Mathematics (for liberal arts candidates)

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

This paper is divided into two parts: the first volume (multiple choice questions) and the second volume (non-multiple choice questions). Volume I 1 to 2 pages, and volume II, 3 to 4 pages.

After the exam, return this paper together with the answer sheet.

Reference formula:

If events A and B are mutually exclusive, then the surface area formula of the ball

P(A+B)=P(A)+P(B) S=4πR2

If events A and B are independent of each other, then R represents the radius of the ball.

P (a b) = p (a) p (b) Volume formula of sphere.

If the probability of event A in an experiment is p, then V=πR3.

The probability that event A happens exactly k times in n independent repeated tests, where r represents the radius of the ball.

Pn(k)=CknPk( 1-p)n-k(k=0， 1，2，…，n)

Where represents the radius of the ball.

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 set is known, then ()

A.B. C. D。

2. If the function is even, then a= ()

A.B. C. D。

3. The necessary and sufficient condition for a circle and a straight line to have no common point is ().

A.B.

C.D.

4. If,,, is known, then ()

A.B. C. D。

5. Given three vertices of a quadrilateral,, and, the coordinates of the vertices are ().

A.B. C. D。

6. Let p be the point on the curve C: and the tangent inclination range of the curve C at the point P is, then the abscissa range of the point P is ().

A.B. C. D。

7.4 The numbers 1, 2, 3 and 4 are written on the card respectively. If two cards are randomly drawn from these four cards, the probability that the sum of the numbers on the two cards is odd is ().

A.B. C. D。

8. Get the image of the function according to the image of the vector translation function, and then ()

A.B. C. D。

9. The maximum value of a known variable that satisfies the constraint condition is ().

A.B. C. D。

10. The first production process has four processes, and each process needs one person to participate. At present, Party A, Party B and Party C each arrange four workers to participate in one process. In the first process, only Party A and Party B can arrange 1 worker, and in the fourth process, only Party A and Party C can arrange 1 worker, so there will be different arrangements.

A.24 species B. 36 species C. 48 species D. 72 species

1 1. Given that the distance between a vertex of hyperbola and an asymptote of hyperbola is, then ().

A. 1

12. In a cube, the midpoint of an edge is a straight line () that intersects all three straight lines in space.

A. There is no B, only two C's, only three D's and countless D's.

Volume 2 (non-multiple choice questions ***90 points)

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

13. The inverse of this function is.

14. If there are three points A, B and C on the surface of a ball with volume, AB= 1, BC=, and the spherical distance between the points A and C is, then the distance from the center of the ball to the plane ABC is _ _ _ _ _ _ _ _.

15. The constant term in the expansion is.

16. If is, the minimum value of the function is.

Third, the solution: this big question is ***6 small questions, and the score is ***74. The solution should be written in proof process or calculus steps.

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

The side length of the opposite side of the middle angle and the inner angle is known.

(i) If the area of is equal to, find;

(ii) If yes, the area to be sought.

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

A wholesale market makes statistics on the weekly sales volume (unit: tons) of a commodity, and the statistical results of the latest 100 week are shown in the following table:

Weekly sales volume 2 3 4

Frequency 20 50 30

(1) According to the above statistical results, find out the frequency of selling 2 tons, 3 tons and 4 tons per week respectively;

(2) If the above-mentioned frequency is taken as the probability, and the weekly sales volume is independent of each other, then,

(i) The probability that the sales volume of the commodity will be 4 tons in at least one of 4 weeks;

(2) The probability that the total sales around the commodity is at least 15 tons.

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

As shown in the figure, in a cube with a side length of 1, AP = bq = b (0

(i) Prove that plane PQEF and plane PQGH are perpendicular to each other;

(ii) It is proved that the sum of the areas of part PQEF and part PQGH is a constant value,

And find this value;

(iii) If yes, find the sine value of the angle with the plane PQEF.

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

In a series, it is a geometric series whose terms are all positive numbers.

(1) Is the sequence a geometric series? Prove your conclusion;

(ii) Set the sum of the previous paragraphs of a series as. If, find the sum of the previous paragraph of the series.

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

In the plane rectangular coordinate system, the sum of the distances from point P to two points is equal to 4, and the trajectory of point P is.

(i) Write the equation of c;

(Ⅱ) Let a straight line intersect C at points A and B. What is the value of k? What is the value at this time?

22. (The full score of this short question is 14)

Let the function get the extreme value at and.

(i) If, the value of, the monotone interval of;

(ii) If yes, the range of values to be found.

In 2008, the national unified examination for enrollment of ordinary colleges and universities (Liaoning volume)

Mathematics (for liberal arts candidates)

Reference answers and scoring references

First, multiple-choice questions: This question examines basic knowledge and basic operations. 5 points for each small question, 60 points for * * *.

1.D 2。 C 3。 B 4。 C 5。 A six. A

7.C 8。 A nine. B 10。 B 1 1。 D 12。 D

Fill in the blanks: This question examines the basic knowledge and basic operation. 4 points for each small question, full score 16.

13. 14. 15. 16.

Third, answer questions.

17. This small question mainly examines the basic knowledge and comprehensive calculation ability of the relationship between the angles of a triangle. The perfect score is 12.

Solution: (1) Derived from cosine theorem,

Because the area is equal to, so, get 0.4 points.

0.6 points for solving simultaneous equations.

(ii) According to the sine theorem, the known condition is 8 points.

Solutions of simultaneous equations.

So./kloc-the area of point 0/2.

18. This small topic mainly examines basic knowledge such as frequency and probability, and examines the ability to use probability knowledge to solve practical problems. The perfect score is 12.

Solution: (1) The frequency of selling 2 tons, 3 tons and 4 tons per week is 0.2, 0.5 and 0.3.4 minutes respectively.

(2) According to the meaning of the question, the weekly sales volume is 2 tons, and the frequencies of 3 tons and 4 tons are 0.2, 0.5 and 0.3 respectively. The probability of finding is

(1) .8 points

(2). 12 points

19. This small topic mainly examines the basic knowledge of line-plane relationship, plane-plane relationship and triangle solution in space, and examines the spatial imagination and logical thinking ability. The perfect score is 12.

Solution 1:

(i) Prove that in a cube, it can be obtained from what is known.

,,,

So,,,

So the plane.

So the plane and the plane are perpendicular to each other 4 points

(2) Evidence: According to (1)

The PQEF section and PQGH section are both rectangular, and PQ= 1, so the sum of the areas of PQEF section and PQGH section is

, is a fixed value. Eight points.

(3) Solution: Set intersection points and connect them.

Because of the plane,

So it's at an angle to the plane.

Because they are the midpoint of,, respectively.

It can be seen.

So. 12 point

Solution 2:

Taking d as the origin and rays DA, DC and DD' as the positive half axes of the X, Y and Z axes respectively, a spatial rectangular coordinate system D-XYZ as shown in the figure is established.

,,,,

,,,

,,.

(1) Proof: In the established coordinate system, we can get

Because it is the normal vector of the plane PQEF.

Because it is the normal vector of the plane PQGH.

Because, therefore,

So the plane PQEF and the plane PQGH are perpendicular to each other. Four points.

(ii) Proof: Because, therefore, PQEF is a rectangle, and PQGH is also a rectangle.

In the established coordinate system, we can get,

So, once again,

Therefore, the sum of the areas of PQEF and PQGH is a constant value of 0. 8 points.

(3) Solution: According to (1), it is the normal vector of the plane.

From the midpoint, we can know that they are the midpoint of,, respectively.

So, the sine of the angle with the plane is equal to

. 12 point

20. This small question mainly examines arithmetic progression, geometric progression, Logarithm and other basic knowledge, and examines the ability of comprehensively applying mathematical knowledge to solve problems. The perfect score is 12.

Solution: (1) Geometric series. Two points.

Proof: If the common ratio of is set and the common ratio is set, then

, so it is a geometric series. Five points.

(Ⅱ) The sum of sequence is arithmetic progression of allowable sum respectively.

Conditional, i.e.

.7 points

So by the way,,,

therefore

Substitute,,. 10 point.

So there it is.

Therefore, the sum of the previous paragraphs in this series is

. 12 point

2 1. This topic mainly examines the basic knowledge such as plane vector, the definition of ellipse, standard equation, the positional relationship between straight line and ellipse, and the ability to solve problems by using analytical geometry knowledge comprehensively. The perfect score is 12.

Solution:

(i) Let P(x, y). According to the definition of ellipse, the locus c of point P is an ellipse with the long axis 2 as the focus. Its short axis,

So the equation of curve C is .4 points.

(ii) Set whose coordinates satisfy

Eliminate y and arrange it,

So ... 6 points

, that is.

Besides,

So ...

So when, so. Eight points.

At that time,

but

So. 12 point

22. This small question mainly examines the basic knowledge of derivative, monotonicity, extreme value and maximum value of a function, and examines the ability to study the related properties of a function by comprehensively using derivatives. The perfect score is 14.

Solution:. ① 2 points

(1) At that time,

;

From the meaning of the problem, it is called the two roots of the equation, so

To get 0.4 points.

Therefore,.

At that time,; At that time,

Therefore, it is monotonically decreasing, while it is monotonically increasing. Six points.

(ii) From the meaning of the formula 1 and the question, we can see that it is two equations.

So ...

Therefore,

From the above formulas and questions. 8 points