High School Mathematics 2007-2009 Hubei Science Mathematics College Entrance Examination Paper and Analysis

In 2007, the national unified examination for enrollment of ordinary colleges and universities (Hubei volume)

Mathematics (science, engineering, agriculture and medicine)

This volume is ***4 pages, full mark 150, examination time 120 minutes.

Precautions:

1. Before answering questions, candidates must fill in their own names and admission ticket numbers on the test papers and answer sheets, and stick the bar code of the admission ticket number on the designated position on the answer sheet.

2. After choosing the answer for each multiple-choice question, use 2B pencil to black the answer label of the corresponding question on the answer sheet. If you need to change it, clean it with an eraser, and then choose to apply other answer labels. The answer on the test paper is invalid.

3. Fill in the blanks directly in the answer area corresponding to each question on the answer sheet with 0.5 mm black ink pen or black ink pen, and the answers on the test paper are invalid.

Please hand in this paper together with the answer sheet after the exam.

I. Multiple-choice questions: This topic is entitled * *10, with 5 points for each question and 50 points for each question. Of the four answers given in each question, only one meets the requirements of the topic.

1. If the expansion of contains non-zero constant terms, the minimum value of positive integer n is

A.3

B.5

D. 10

2. according to the vector a= translation image, the analytical formula of translation image is

3. Let p and q be two sets, and define the set P-Q=, if p = {x | log2x.

A.{ x | 0 & ltx & lt 1}

B.{ x | 0 & ltx≤ 1}

C.{ x | 1≤x & lt； 2}

D.{ x | 2≤x & lt； 3}

There are two straight lines m and n outside the plane α. If the projections of m and n on plane α are M' and N' respectively, the following four propositions are given:

①m'⊥n'm⊥n

②m⊥n m'⊥n'

③M' and N' intersect, and M and N intersect or overlap.

④M' and N' are parallel, and M and N are parallel or coincident.

The number of incorrect propositions is

A. 1

B.2

C.3

Dingsi

5. It is known that P and Q are two unequal positive integers, and q≥2, then

A: 0

B. 1

6. If the series {an} satisfies N*), then {an} is called "equal square ratio series".

Answer: The sequence {an} is an equal square ratio sequence; B: The sequence {an} is a geometric series.

A.a is a sufficient condition for B, but not a necessary condition.

B.a is a necessary but not sufficient condition for B.

C.a is a necessary and sufficient condition for b.

D.A. is neither a sufficient condition nor a necessary condition for B.

7. The hyperbola C 1: (A > 0, b>0) is L, and the left focus and right focus are F 1 and F2, respectively; The directrix of parabola C2 is L and the focus is F2; The intersection of C 1 and C2 is m, which is equal to

A.- 1

B. 1

8. It is known that the sum of the first n terms of two arithmetic progression {An} And {Bn} is an and Bn, respectively, so the number of positive integers n that are integers is

A.2

B.3

D.5

9. If the number of points obtained by rolling the dice twice is m and n respectively, and the angle between vector a=(m, n) and vector b=( 1,-1) is θ, then the probability is

10. It is known that a straight line (a, b are nonzero constants) and a circle x2+y2= 100 have a common * * * point, and the abscissa and ordinate of the common * * * point are integers, so such a straight line has * * *.

A.60

b66

C.72

D. Article D.78

Fill in the blanks: This big question is ***5 small questions, with 5 points for each small question and 25 points for * * *.

1 1. Given that the inverse of the function y=2x-a is y=bx+3, then a =；; b= .

12. Complex number z=a+bi, a, b∈R, b≠0. If z2-4bz is a real number, the ordered real number pair (a, b) can be. Just write an ordered real number pair. )

13. Let the variables x and y satisfy the constraint conditions, then the minimum value of the objective function 2x+y is.

14. A basketball player's hit rate at the three-point line is that he throws 10 times, that is, the probability of just hitting three balls. (Answer with numerical value)

15. In order to prevent influenza, a school disinfected the classroom by fumigation. It is known that in the process of drug release, the drug content y (mg) per cubic meter of indoor air is directly proportional to the time t (hours); After drug release, the functional relationship between y and t is (A is constant), as shown in the figure. According to the information provided in the picture, answer the following questions:

(i) The functional relationship between the drug content y (mg) per cubic meter of air and the time t (hours) is as follows:

(2) According to the measurement, only when the drug content per cubic meter in the air drops below 0.25 mg can students enter the classroom. After the drug is released, students need at least one hour to return to the classroom.

Third, answer: This big question is ***5 small questions, ***75 points. The solution should be written in words, proving the process or calculation steps.

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

It is known that the area of △ABC is 3 and satisfies 0≤≤≤6, and the included angle of the sum is θ.

(i) Find the range of θ;

(2) Find the maximum and minimum values of the function f(θ)=2sin2.

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

group

frequency

four

Combination plan

100

In the production process, measure the size of fiber products (the quantity indicating the fiber thickness).

* * * There are 100 data. Group data as shown in the right table:

(i) Complete the frequency distribution table on the answer sheet and draw it in the given coordinate system.

Frequency distribution histogram;

(ii) Estimate the probability that the size falls in the middle and the probability that the size is less than 1.40.

What is the rate;

(3) In statistical methods, the midpoint value of the same set of data is often represented (for example, the midpoint value of an interval is 1.32). On this basis, the expected value of fineness is estimated.

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

As shown in the figure, in the triangular pyramid V-ABC, VC⊥ bottom ABC, AC⊥BC and D are the midpoint of AB, AC=BC=a, and ∠VDC=θ.

(i) Verification: VAB⊥ aircraft vcd；;

(2) When the angle θ changes, find the range of the angle formed by the straight line BC and the plane VAB.

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

In the plane rectangular coordinate system xOy, the straight line passing through the fixed point C(0, p) and the parabola x2 = 2py (p >; 0) intersects a and b.

(i) If point N is the symmetrical point of point C relative to the coordinate origin O, find the minimum value of △ANB area;

(2) Is there a straight line L perpendicular to the Y axis, so that the chord length of L is cut into a constant by a circle with a diameter of AC? If it exists, find the equation of L; If it does not exist, explain why. (This question doesn't need to be drawn on the answer sheet)

20. (The full score of this short question is 13)

It is known that the function f(x)=x2+2ax, g(x)=3a2lnx+b defined on the positive real number set, where a >；; 0。 Let two curves y=f(x) and y=g(x) have a common point with the same tangent.

(i) Use a to represent b and find the maximum value of b;

(ii) Verification: f (x) ≥ g (x) (x >; 0)。

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

It is known that m and n are positive integers.

(i) Prove by mathematical induction that when x >; At - 1，( 1+x)m≥ 1+MX；

(ii) For n≥6, it is known and verified that m= 1, 2…, n;

(iii) Find all positive integers n that satisfy the equation 3n+4m+…+(n+2) m = (n+3) n.

In 2007, the national unified examination for enrollment of ordinary colleges and universities (Hubei volume)

Mathematics (science, engineering, agriculture and medicine)

Reference answer

First, multiple-choice questions: This question examines basic knowledge and basic operations. 5 points for each small question, out of 50 points.

1.B2 a3 . B4 . D5 . C6 . B7 . A8 . d9 . c 10。 A

Fill in the blanks: This question examines the basic knowledge and basic operation. 5 points for each small question, out of 25 points.

1 1.6;

12.(2, 1) (or any set of non-zero real number pairs (a, b) satisfying a=2b)

13.—

14.

15.； 0.6

Third, answer: This big question is ***6 small questions, ***75 points.

16. This small topic mainly examines the calculation of the product of plane vectors, the basic knowledge of solving triangles, trigonometric formulas and trigonometric functions, and the ability of reasoning and operation.

Solution:

(i) Let the opposite sides of angles A, B and C in △a, B and C be A, B and C respectively,

Then by.

(Ⅱ)

Right away.

17. This topic mainly investigates the concepts of frequency distribution histogram, probability and expectation, and the statistical method of estimating the overall distribution with sample frequency, and investigates the ability of solving practical problems by using probability and statistics knowledge.

group

frequency

Frequency rate

four

0.04

0.25

0.30

0.29

0. 10

0.02

Combination plan

100

1.00

(2) The probability that the fineness falls in the middle is about 0.30+0.29+0. 10 = 0.69, and the probability that the fineness is less than 1.40 is about 0.04+0.25+× 0.30 = 0.44.

(3) The expected value of the overall data is about

1.32×0.04+ 1.36×0.25+ 1.40×0.30+ 1.44×0.29+ 1.48×0. 10+ 1.52×0.02= 1.4088.

18. This topic mainly examines the knowledge about the relationship between line and surface, the angle between line and surface, the ability of spatial imagination and reasoning operation, and the ability of applying vector knowledge to solve mathematical problems.

Solution 1:

(i) is an isosceles triangle, and d is the midpoint of AB.

and

(ii) If the intersection c makes H CH⊥VD in the plane VD, BH is connected by (i) knowing the plane VAB of CH⊥, so ∠CBH is the angle formed by the straight line BC and the plane VAB.

In Rt△CHD, let,

That is, the range of the angle formed by the straight line BC and the plane VAB is (0.degree.).

Solution 2:

(1) Take the straight lines of CA, CB and CV as the X-axis, Y-axis and Z-axis, respectively, and establish a spatial rectangular coordinate system as shown in the figure, and then c (0,0,0), a (a,0,0), b (0,a,0), d (),

therefore

In the same way; In a similar way

that is

and

(ii) Let the angle formed by the straight line BC and the plane VAB be φ and the normal vector of the plane VAB be n=(x, y, z),

And then by n.

19. This small question mainly examines the basic knowledge of plane analytic geometry such as straight lines, circles and parabolas, and examines the ability of reasoning and solving problems by comprehensively applying mathematical knowledge.

Solution 1:

(1) According to the meaning of the question, the coordinate of point N is N(0, -p), which can be set as A (X 1, Y 1) and B (X2, Y2). The equation of straight line AB is y=kx+p, and when combined with x2=2py, it is x2-2pkx-2p2 =.

X 1+x2 = 2pk, x 1x2 =-2p2 of Vieta's theorem.

therefore

(2) Assuming that a straight line L meets the conditions exists, its equation is y = a, the circle whose diameter is at the midpoint of AC intersects at points P and Q, and the midpoint of PQ is H, then

Make, get a fixed value, so there is a straight line l that satisfies the condition, and its equation is,

That is, the straight line where the trajectory of a parabola lies.

Solution 2:

(i) The previous solution is 1, which is obtained from the chord length formula.

It is also derived from the distance formula from a point to a straight line.

Therefore,

(ii) Assuming that a straight line T exists, its equation is y=a, and the equation of a circle with a diameter of AC is

Substitute into the linear equation y=a to get

Let the intersection of a straight line L and a circle with a diameter of AC be p (x2, y2) and q (x4, y4), then there are

Set to a constant value, then there is a straight line L that meets the conditions, and its equation is.

That is, the straight line where the parabolic trajectory lies.

20. This small question mainly examines the application of functions, inequalities and derivatives, and examines the ability to solve problems by comprehensively applying mathematical knowledge.

Solution:

(I) let y=f(x) and y = g (x) (x >; 0) The tangents at the common points (x0, y0) are the same,

that is

Have it at once

Zero years old

while

So it's a decreasing function,

So h(t) is

(ii) Establishment

rule

So F(x) is a decreasing function at (0, a) and a increasing function at (a,+).

So this function

Therefore, when x> is 0, there is

2 1. This small topic mainly examines the basic knowledge and basic operation skills such as mathematical induction, summation of series, inequality, and the ability to analyze problems and reason.

Solution 1:

(1) Proof: Prove by mathematical induction:

(i) When m= 1, the original inequality holds; When m=2, the left side = 1+2x+x2, and the right side = 1+2x, because x2≥0,

So left ≥ right, the original inequality holds;

(2) Assuming that the inequality holds when m=k, that is, (1+x)k≥ 1+kx, then when m=k+ 1,

Multiply both sides by1+x.

Therefore, the inequality also holds.

Based on (i) and (ii), the inequality holds for all positive integers m 。

(2) syndrome: when n ≥ 6 and m ≤ N, it is derived from (1).

therefore

(3) Solution: According to (2), when n≥6,

So we only need to discuss the case of n= 1, 2, 3, 4, 5;

When n= 1, 3≠4, the equation does not hold;

When n = 2,32+42 = 52, the equation holds;

When n = 3,33+43+53 = 63, the equation holds;

When n=4, 34+44+54+64 is an even number and 74 is an odd number, so 34+44+54+64≠74, the equation does not hold;

When n=5, it can be analyzed that the equation is not valid.

To sum up, the only n to be found is n = 2,3.

Solution 2:

(1) Prove that when x=0 or m= 1, the equal sign in the original inequality is obviously established, which is proved by mathematical induction:

When x>- 1 and x≠0, m≥2, (1+x) m > 1+mx。 1

(i) When m=2, the left side = 1+2x+x2, and the right side = 1+2x, because x≠0, x2>0, that is, left side > right side, inequality ① holds;

(2) suppose that when m=k(k≥2), inequality ① holds, that is, (1+x) k >; 1+kx, then when m=k+ 1, because x >；; -1, so1+x > 0. Because x ≠ 0 and k ≥ 2, kx2>0.

So in the inequality (1+x) k >; Both sides of 1+kx are multiplied by1+x.

( 1+x)k ( 1+x)>( 1+kx)( 1+x)= 1+(k+ 1)x+kx2 & gt； 1+(k+ 1)x，

So (1+x) k+1>; 1+(k+ 1)x, that is, when m = k+ 1, inequality ① also holds.

To sum up, the proved inequality holds.

(ii) Certificate: when

And through (i),

(3) Solution: Assuming there is a positive integer,

That is, () += 1 ②.

It can also be obtained from (II)

()+

+contradiction 2,

So when n≥6, there is no positive integer n satisfying this equation.

So we only need to discuss the case of n= 1, 2, 3, 4, 5;

When n= 1, 3≠4, the equation does not hold;

When n = 2,32+42 = 52, the equation holds;

When n = 3,33+43+53 = 63, the equation holds;

When n=4, 34+44+54+64 is an even number and 74 is an odd number, so 34+44+54+64≠74, the equation does not hold;

When n=5, it can be analyzed that the equation is not valid.

To sum up, the only n to be found is n = 2,3.

In 2007, the national unified examination for enrollment of ordinary colleges and universities (Hubei volume)

Mathematics (science, engineering, agriculture and medicine)

This volume is ***4 pages, full mark 150, examination time 120 minutes.

Precautions:

3. Fill in the blanks directly in the answer area corresponding to each question on the answer sheet with 0.5 mm black ink pen or black ink pen, and the answers on the test paper are invalid.

Please hand in this paper together with the answer sheet after the exam.

1. If the expansion of contains non-zero constant terms, the minimum value of positive integer n is

A.3

B.5

D. 10

2. according to the vector a= translation image, the analytical formula of translation image is

3. Let p and q be two sets, and define the set P-Q=, if p = {x | log2x.

A.{ x | 0 & ltx & lt 1}

B.{ x | 0 & ltx≤ 1}

C.{ x | 1≤x & lt； 2}

D.{ x | 2≤x & lt； 3}

There are two straight lines m and n outside the plane α. If the projections of m and n on plane α are M' and N' respectively, the following four propositions are given:

①m'⊥n'm⊥n

②m⊥n m'⊥n'

③M' and N' intersect, and M and N intersect or overlap.

④M' and N' are parallel, and M and N are parallel or coincident.

The number of incorrect propositions is

A. 1

B.2

C.3

Dingsi

5. It is known that P and Q are two unequal positive integers, and q≥2, then

A: 0

B. 1

6. If the series {an} satisfies N*), then {an} is called "equal square ratio series".

Answer: The sequence {an} is an equal square ratio sequence; B: The sequence {an} is a geometric series.

A.a is a sufficient condition for B, but not a necessary condition.

B.a is a necessary but not sufficient condition for B.

C.a is a necessary and sufficient condition for b.

D.A. is neither a sufficient condition nor a necessary condition for B.

A.- 1

B. 1

8. It is known that the sum of the first n terms of two arithmetic progression {An} And {Bn} is an and Bn, respectively, so the number of positive integers n that are integers is

A.2

B.3

D.5

9. If the number of points obtained by rolling the dice twice is m and n respectively, and the angle between vector a=(m, n) and vector b=( 1,-1) is θ, then the probability is

A.60

b66

C.72

D. Article D.78

Fill in the blanks: This big question is ***5 small questions, with 5 points for each small question and 25 points for * * *.

1 1. Given that the inverse of the function y=2x-a is y=bx+3, then a =；; b= .

12. Complex number z=a+bi, a, b∈R, b≠0. If z2-4bz is a real number, the ordered real number pair (a, b) can be. Just write an ordered real number pair. )

13. Let the variables x and y satisfy the constraint conditions, then the minimum value of the objective function 2x+y is.

14. A basketball player's hit rate at the three-point line is that he throws 10 times, that is, the probability of just hitting three balls. (Answer with numerical value)

(i) The functional relationship between the drug content y (mg) per cubic meter of air and the time t (hours) is as follows:

Third, answer: This big question is ***5 small questions, ***75 points. The solution should be written in words, proving the process or calculation steps.

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

It is known that the area of △ABC is 3 and satisfies 0≤≤≤6, and the included angle of the sum is θ.

(i) Find the range of θ;

(2) Find the maximum and minimum values of the function f(θ)=2sin2.

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

group

frequency

four

Combination plan

100

In the production process, measure the size of fiber products (the quantity indicating the fiber thickness).

* * * There are 100 data. Group data as shown in the right table:

(i) Complete the frequency distribution table on the answer sheet and draw it in the given coordinate system.

Frequency distribution histogram;

(ii) Estimate the probability that the size falls in the middle and the probability that the size is less than 1.40.

What is the rate;

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

As shown in the figure, in the triangular pyramid V-ABC, VC⊥ bottom ABC, AC⊥BC and D are the midpoint of AB, AC=BC=a, and ∠VDC=θ.

(i) Verification: VAB⊥ aircraft vcd；;

(2) When the angle θ changes, find the range of the angle formed by the straight line BC and the plane VAB.

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

In the plane rectangular coordinate system xOy, the straight line passing through the fixed point C(0, p) and the parabola x2 = 2py (p >; 0) intersects a and b.

(i) If point N is the symmetrical point of point C relative to the coordinate origin O, find the minimum value of △ANB area;

20. (The full score of this short question is 13)

It is known that the function f(x)=x2+2ax, g(x)=3a2lnx+b defined on the positive real number set, where a >；; 0。 Let two curves y=f(x) and y=g(x) have a common point with the same tangent.

(i) Use a to represent b and find the maximum value of b;

(ii) Verification: f (x) ≥ g (x) (x >; 0)。

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

It is known that m and n are positive integers.

(i) Prove by mathematical induction that when x >; At - 1，( 1+x)m≥ 1+MX；

(ii) For n≥6, it is known and verified that m= 1, 2…, n;

(iii) Find all positive integers n that satisfy the equation 3n+4m+…+(n+2) m = (n+3) n.

There are too many words to copy. I'll send it to you if you want.

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