In 2008, the national college entrance examination 1 mathematical composition probability questions.

In 2006, the national unified entrance examination for colleges and universities was held.

Liberal arts mathematics

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

Turn to page four. After the exam, return this paper together with the answer sheet.

volume one

Precautions:

1. Before answering questions, candidates must clearly fill in their names and admission ticket numbers on the answer sheet, and

Stick a barcode on it. Please carefully approve the admission ticket number, name and subject on the bar code.

2. After choosing the answer to each small question, black the answer label of the corresponding question on the answer sheet with 2B pencil. If you need to change it, clean it with an eraser, and then choose another answer label. The answer on the test paper is invalid.

3. There are *** 12 small questions in this volume, with 5 points for each small question and 60 points for * * *. Of the four options given in each question, only one meets the requirements of the topic.

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)？ Volume formula of P(B) sphere

If the probability of event A in the experiment is p, then

The probability of exactly k times in n independent repeated experiments, where r represents the radius of the ball.

I. Multiple choice questions

(1) It is known that vectors A and B satisfy | a |= 1, | b |=4, a? B=2, then the included angle between A and B is

(A) (B) (C) (D)

(2) Set a group, and then

(A) (B)

(3) Given that the image of the function and the image of the function are symmetrical about a straight line, then

(A) R) (B)？ ( )

(4) If the imaginary axis of hyperbola is twice as long as the real axis, then m=

(A) (B)-4 (C)4 (D)

(5) Let it be the sum of the first n items of arithmetic progression, if S7=35, then a4=

(A)8 (B)7 (C)6 (D)5

(6) The monotonic increasing interval of the function is

(A) Z (B) Z

Z (D) Z

(7) If two tangents are formed by points P (3 3,2) outside the circle, the cosine of the included angle between the two tangents is

(A) (B) (C) (D)0

(8) The opposite sides of the internal angles A, B and C of △ ABC are A, B and C respectively. If a, b and c form a geometric series, and

(A) (B) (C) (D)

(9) It is known that the height of a regular quadrangular prism with all vertices on the sphere is 4 and the volume is 16, then the surface area of the sphere is

16 20(C)24(D)32

(10) In the expansion, the coefficient of is

(A)- 120(B)- 120(C)- 15(D)

(1 1) What is the minimum distance from a point on a parabola to a straight line?

(A) (B) (C) (D)3

(12) A triangle is composed of five thin sticks with lengths of 2, 3, 4, 5 and 6 (unit: cm) (connection is allowed, but breaking is not allowed). The maximum area of the triangle can be obtained as follows.

(A) cm2 (B) cm2

20 square centimeters

In 2006, the national unified entrance examination for colleges and universities was held.

Liberal arts mathematics

Volume II

Precautions:

1. Before answering the questions, candidates should fill in their names and admission ticket numbers clearly with a black pen on the answer sheet, and then stick a bar code on it. Please carefully approve the admission ticket number, name and subject on the bar code.

2. On page ***2 of Volume 2, please use a black pen to answer each question in the answer area on the answer sheet. The answer on the test paper is invalid.

3. This volume *** 10, ***90 points.

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

(13) If the known function is odd function, then a=.

(14) Given that the volume of a regular pyramid is 12 and the length of the diagonal of the bottom is, the dihedral angle formed by the side and the bottom is equal to.

(15), where the variables x and y satisfy the following conditions.

The maximum value of z is.

(1 6) 7 staff members will be on duty from May1day to May 7, and each staff member will be on duty for one day. May 1 and 2/are not arranged by both parties. There are different arrangements. (Answer with numbers)

3. Solution: This big question is ***6 small questions, ***74 points. The solution should be written in proof process or calculus steps.

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

A general formula called geometric series solution.

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

The three internal angles of △ABC are A, B and C. Find the maximum value when A is, and find this maximum value.

(19) (The full mark of this small question is 12)

A and B are two drugs for treating the same disease. Several experimental groups were used for comparative tests. Each experimental group consists of four mice, two of which take A and the other two take B, and then observe the curative effect. If the number of mice taking A is much more than that taking B in an experimental group, this experimental group is called Group A. Let the probability that each mouse takes A is effective and the probability that each mouse takes B is effective.

(i) Find out the probability that an experimental group is Group A;

(2) Observe the three experimental groups and find the probability that there is at least one group A in these three experimental groups.

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

As shown in the figure, are mutually perpendicular straight lines in different planes, and MN is their common vertical line segment. Point a and point b are on it, and point c is on it.

AM = MB = MN。

(i) Evidence;

(ii) If yes, find the cosine of the angle between NB and plane ABC.

(2 1) (the full score of this small question is 14)

Let p be the endpoint of the short axis of the ellipse and q be the moving point on the ellipse, and find the maximum value of |PQ|.

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

Let A be a real number, and the sum of functions be increasing function.

The range of a.

In 2006, the national unified entrance examination for colleges and universities was held.

Reference answers to liberal arts mathematics test questions (compulsory+elective ⅰ)

I. Multiple choice questions

( 1)C (2)B (3)D (4)A (5)D (6)C

(7)B(8)B(9)C( 10)C( 1 1)A( 12)B

Step 2 fill in the blanks

( 13) ( 14) ( 15) 1 1 ( 16)2400

Three. solve problems

(17) solution:

Let the common ratio of geometric progression be q, then q≠0,

therefore

solve

while

therefore

while

therefore

(18) solution:

pass by

So there is

while

(19) solution:

(i) Let A 1 represent the event that "in an experimental group, I mice take A effectively", I = 0, 1, 2,

B 1 means the event "In an experimental group, I mice took B effectively", I = 0, 1, 2,

According to the meaning of the question

The probability of seeking is

P = P(B0？ A 1)+ P(B0？ A2)+ P(B 1？ A2)

(2) the probability of finding is

(20) Solution:

(I) from the known l2⊥l 1 l2⊥mn Mn l 1 = m,

The l2⊥ plane ABN can be obtained.

It is known that MN⊥l 1, AM = MB = MN,

We know that AN = NB, AN⊥NB ANd an are.

The projection of AC on ABN plane,

∴ AC⊥NB

(ⅱ)∵Rt△CAN = Rt△CNB，

∴ AC = BC, also known as ∠ ACB = 60,

So △ABC is a regular triangle.

∫Rt△ANB = Rt△CNB .

∴ NC = NA = NB, so the projection H of n on the plane ABC is the center of the regular triangle ABC, connecting BH, and ∞∠NBh is the angle formed by nb and the plane ABC.

At Rt △NHB,

Solution 2:

As shown in the figure, the spatial rectangular coordinate system M-XYZ is established.

Let MN = 1,

Then a (- 1, 0,0), b (1, 0,0), n (0, 1, 0).

(I) ∵ Mn is the common vertical line of l 1 and l2, l2⊥l 1,

∴l2⊥ Aircraft Company,

∴l2 is parallel to the z axis,

So you can set c (0, 1, m).

therefore

∴AC⊥NB.

(Ⅱ)