Mathematics examination for science students

This volume is divided into two parts, Volume I and Volume II, with a total of 4 pages, with a full score of 150 and an examination time of 120 minutes. After the exam, return this paper together with the answer sheet.

Precautions:

1. Before answering questions, candidates must fill in their name, seat number, admission ticket number, county and department in the position specified in the answer sheet and test paper with a 0.5mm black signature pen, and paste the bar code of the admission ticket number in the position specified in the answer sheet.

2. After selecting the answers to each small question in Book 1, use 2B pencil to black the answer labels of the corresponding questions on the answer sheet. If you need to change it, clean it with an eraser, and then choose another answer label. The answer can't be answered on the paper.

3. Volume 2 must be answered in the answer area of each question on the answer sheet with a 0.5mm black signature pen; Can't write on the test paper; If you need to change, take out the original answer first, and then write a new answer; Correction fluid, adhesive tape and correction tape cannot be used, and the answer that is not answered according to the above requirements is invalid.

Please fill in the answers directly. The answer should be written in words to prove the process or calculus steps.

Reference formula:

The volume formula of a cylinder is V=Sh, where s is the bottom area of the cylinder and h is the height of the cone.

The volume formula of a cone is V=, where s is the bottom area of the cone and h is the height of the cone.

If events A and B are mutually exclusive, then p (a+b) = p (a)+p (b); R if events a and b are independent, then P(AB)=P(A)P(B).

The probability that event A happens in one trial is, so the probability that event A happens exactly twice in an independent repeated trial is:

The first volume (***60 points)

First, multiple-choice questions: This big question is a * *12 small question, with 5 points for each small question and 50 points for * * *. Of the four options given in each question, only one meets the requirements of the topic.

1. Set if the value of is ().

A.0 B. 1 C.2 D.4

Analysis: ∵, ∴ ∴, so choose D.

Answer: d

Proposition: this question examines the union operation of sets, and obtains the corresponding elements through observation, thus obtaining the answer. This question is simple.

2. Complex number equals ().

A.B. C. D。

2. Analysis: C.W.W.W.K.S.5.U.C.O.M

Answer: c

Proposition: This question examines the division of complex numbers. Both the numerator and denominator need to be multiplied by the * * * yoke complex number of the denominator, so that the denominator can be turned into a real number and the division can be turned into a multiplication for operation.

3. Translate the image of the function by units to the left and then by 1 unit, and the resolution function of the obtained image is ().

A.B. C. D。

3. Analysis: Shift the image of the function to the left by one unit to get the image of the function, and then shift it up by 1 unit, and the resolution function of the obtained image is 0, so choose B. 。

Answer: b

Proposition conception: This topic examines the basic knowledge and skills of using inductive formula and double-angle formula to translate the image of trigonometric function and simplify analytical formula, and learns the deformation of formula. 5.u.c.o.m

4. The three views of a space geometry are shown in the figure, so the volume of the geometry is ().

A.B. C. D。

Analysis: Space geometry consists of a cylinder and a pyramid.

The radius of the cylinder bottom is 1, the height is 2, and the volume is, the bottom of the pyramid.

The side length is and the height is, so the volume is.

So the volume of this geometry is.

Answer: c

Proposition conception: This question examines the spatial imagination ability in solid geometry.

From these three views, we can accurately imagine the three-dimensional picture of space.

Calculate the volume of the geometry.

5. It is known that α and β represent two different planes, and m is in plane α.

A straight line, then ""is ""()

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

Analysis: From the judgment theorem that the plane is perpendicular to the plane, we know that if m is in the plane α,

Then, the straight line is not necessarily the other way around. So ""is the necessary and sufficient condition of ""w.w.w.k.s.5.u.c.o.m

Answer: B.

Proposition: This topic mainly examines the judgment of vertical relationship and the concept of necessary and sufficient conditions in solid geometry.

6. The image of the function is roughly ().

Analysis: If the function is meaningful, it needs to be defined as excluding C and D, and because the function is a subtraction function, A. w.w.w.k.s.5.u.c.o.m is chosen.

Answer: a.

Proposition conception: this question examines the image of function and the nature of function, such as definition domain, value domain, monotonicity and so on. The difficulty of this problem lies in the complexity of a given function, which needs to be deformed first, and then other properties are investigated in the definition domain.

7. Let p be a point on the △ABC plane, then ()

A.B. C. D。

Analysis: Because point P is the midpoint of line segment AC, B should be chosen.

Answer: B.

Proposition: This topic examines the addition operation of vectors and the parallelogram rule.

You can use diagrams.

8. A factory sampled a batch of products. The picture on the right is based on the sampling of w.w.w.k.s.5.u.c.o.m

The frequency distribution histogram drawn by the net weight (unit: gram) data of the product, in which the product

The range of net weight is an increasing function, if the equation f (x) = m (m >; 0) There are four different roots in the interval, then w.w.w.k.s.5.u.c.o.m

Analysis: Because it satisfies the odd function defined on R, it is odd function, so the function image is symmetrical and known about the line, so the function is a periodic function with a period of 8, and because it is increasing function in the interval, it is also increasing function in the interval. As shown in the figure, the equation f (x) = m (m >; 0) The interval has four different roots, so we might as well assume that it is symmetrically known.

Answer: -8

Proposition: This topic comprehensively examines the parity and monotonicity of functions.

Keywords symmetry, periodicity, solving equation problem with function image,

Answer the questions with the idea of combining numbers with shapes and the idea of functional equations.

Third, the answer: this big question ***6 points, ***74 points.

17. (The full mark of this small question is 12) Let the function f (x) = cos (2x+)+sin x. 。

(1) Find the maximum and minimum positive period of the function f(x).

(2) Let A, B and C be the three internal angles of A, B and C. If cosB= and C are acute angles, find Sina.

Solution: (1)f(x)=cos(2x+ )+sin x.=

So the maximum value of the function f(x) is the minimum positive period. 5.u.c.o.m

(2) = =-, so because c is an acute angle,

And because in ABC, cosB=, so, so W W W K S 5. U C O M.

.

Proposition: This topic mainly examines the properties of chord function formula, double angle formula, trigonometric function and trigonometric relationship in trigonometric function.

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

As shown in the figure, in the regular quadrangular prism A BCD-A B C D, the bottom ABCD is an isosceles trapezoid, AB//CD, AB = 4, BC = CD = 2, AA = 2, E, E and F are the midpoints of sides AD, AA and AB respectively.

(1) proof: straight line EE // plane FCC

(2) Find the cosine of dihedral angle b-fc-c ... 5.u.c.o.m

Solution 1: (1) Take the midpoint F 1 of A 1B 1 in ABCD-A B C D.

Connect A 1D, C 1F 1 and CF 1 because AB=4, CD=2, and AB//CD,

So CD=//A 1F 1, A 1F 1CD is a parallelogram, so cf1/a1d,

And because e and e are the midpoint of edges AD and AA respectively, ee1/a1d,

So cf1/ee1,and because of planar FCC, planar FCC,

So the straight line EE // plane FCC.

(2) Because AB=4, BC=CD=2, F is the midpoint of the side AB, BF = BC = CF, and △ BCF is a regular triangle. If we take the middle point O of CF, it is OB⊥CF, and because in the straight quadrangular prism ABCD-A B C D, CC 1⊥ plane ABCD, if O intersects, it is OP⊥C 1F in the plane CC 1F, and the vertical foot is P.

At Rt△OPF,,, so the cosine of dihedral angle B-FC -C is.

Solution 2: (1) Because AB = 4, BC = CD = 2, and F is the midpoint of the side AB,

So BF=BC=CF, △BCF is a regular triangle, because ABCD is

Isosceles trapezoid, so ∠ BAC = ∠ ABC = 60, take the midpoint m of AF,

Connect DM, then DM⊥AB, so DM⊥CD,

A rectangular coordinate system with DM as X axis, DC as Y axis and DD 1 as Z axis is established.

Then d (0 0,0,0), a (,-1, 0), f (,1, 0), c (0 0,2,0),

C 1 (0,2,2), E (0 0,0), E 1 (- 1,1), so let the normal vector of the plane CC 1F be, so, so the straight line ee.

(2) Let the normal vector of the plane BFC 1 be, so, take, then,

，w . w . w . w . k . s . 5 . u . c . o . m

So it can be seen from the figure that dihedral angle B-FC -C is acute, so the cosine of dihedral angle B-FC -C is w.w.w.k.s.5.u.c.o.m

Proposition conception: this topic mainly examines the concept of regular prism, the determination of the relationship between line and surface, the calculation of dihedral angle, the ability of spatial imagination, reasoning and operation, and the ability of applying vector knowledge to solve problems.

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

In a basketball fixed-point shooting training organized by a school, it is stipulated that each person can shoot at most 3 times; 3 points for each pitch in A and 2 points for each pitch in B; If the sum of the previous two scores exceeds 3 points, stop shooting; otherwise, for the third shooting, a classmate's hit rate at point A is 0.25, and the hit rate at point B is Q. The classmate chooses to shoot a ball at point A first, and then shoot at point B, indicating the total score of the classmate after shooting training. Its distribution list is

0 2 3 4 5

World Intellectual Property Organization (WIPO)

0.03 p 1 P2 P3 P4

(1) Find the value of q; 5.u.c.o.m

(2) Find the mathematical expectation e of random variables;

(3) Try to compare the probability that students choose to shoot at point B and score more than 3 points, and the probability that they choose to shoot more than 3 points.

Solution: (1) Let the students vote for event A in A and event B in B, then events A and B are independent of each other, p (a) = 0.25, p (b) = q,.

According to the allocation table: =0 =0.03, so q =0.8.

(2) when =2, p1= w.w.w.k.s.5.u.c.o.m.

=0.75 q ( )×2= 1.5 q ( )=0.24

When =3, P2 = =0.0 1,

When =4, P3= =0.48,

When =5, P4=

=0.24

So the distribution list of random variables is

0 2 3 4 5

p 0.03 0.24 0.0 1 0.48 0.24

Mathematical expectation of random variables

(3) The probability that students choose to shoot at point B and score more than 3 points is

;

The probability that the student chooses (1) more than 3 points is 0.48+0.24=0.72.

From this point of view, students choose to shoot at point B, and the probability of scoring more than 3 points is very high.

Proposition: This topic mainly examines the probability of mutually exclusive events, the probability and mathematical expectation of mutually independent events, and the ability to solve problems by using probability knowledge.

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

The sum of the first n terms of the geometric series {} is known to be a function and a constant for any point).

(1) Find the value of r;

(1 1) When b=2, remember

Prove: for anyone, inequality exists.

Solution: Because any point is on an image with a constant function, if, if, and because {} is a geometric series, the common ratio is,

When b=2,

So, so.

Let's prove this inequality by mathematical induction.

(1) if, left =, right =, because, inequality.

(2) Suppose that when the inequality holds, it holds. So when, left =

So when, inequality also holds.

The inequality obtained by ① and ② holds.

Proposition: this topic mainly examines the definition, general formula and known basic problems of geometric series, proves the proposition related to natural numbers by mathematical induction and proves inequalities by scale method.

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

The distance between two counties A and B is 20 kilometers. Now it is planned to choose a point C on the semi-circular arc with the diameter of AB outside the two counties to build a garbage treatment plant. Its impact on the city is related to the distance from the selected location to the city. The total impact on cities A and B is the sum of the impacts of cities A and B. Remember that the distance from point C to city A is X kilometers, while the total impact of building a garbage disposal plant in place C on cities A and B is Y. The statistical survey shows that the garbage disposal plant has a greater impact on city B, and its impact on city B is inversely proportional to the square of the distance from the location to city B, with a proportional coefficient of K. When the garbage disposal plant is built at the midpoint of, the total impact on cities A and B is 0.065.

(1) indicates that y is a function of x;

(1 1) discuss the monotonicity of the function in (1), and judge whether there is a point on the arc that makes the total impact of the garbage treatment plant built here on cities A and B minimal? If it exists, find the distance from this point to city A; If it does not exist, explain why.

Scheme 1: (1) as shown in the figure, AC⊥BC.

Where when, y=0.065, so k=9.

So y as a function of x is

(2), so, that is, when, that is, so the function is monotonically decreasing, and when, that is, so the function is monotonically increasing. So when the distance from point C to city A is 0, the function has a minimum value.

Option 2: (1) Same as above.

(2) Settings,

Then,, so

Take "=" if and only if.

It is proved that this function is a decreasing function at (0, 160) and a increasing function at (160,400).

Set 0

,

Because 0; 4×240×240

9m 1 m2 & lt； 9× 160× 160 So,

So the function is a decreasing function at (0, 160).

Similarly, the function is an increasing function at (160,400), let160.

Because 1600

So,

So the function is increasing function at (160,400).

Therefore, when m= 160, the function y has a minimum value.

So there is a point on the arc that can minimize the total impact of the garbage treatment plant built here on city A and city B.

Proposition: This topic mainly examines the application of functions in practical problems, the ability to solve the resolution function by using the undetermined coefficient method, and the monotonicity of functions by using method of substitution and basic inequalities.

(22) (The full score of this small question is 14)

Let an ellipse e: (a, b >；; 0) passes through M(2,) and n (,1), where o is the coordinate origin,

(i) Find the equation of ellipse E;

(II) Is there a circle whose center is at the origin, so that any tangent of the circle and ellipse E always has two intersections A and B? If it exists, write the equation of the circle and find the range of |AB |. If it does not exist, explain why.

Solution: (1) because ellipse e: (a, b >；; 0) After M(2,) and n (,1),

So the equation of ellipse e is solved as follows

(2) Suppose that there is a circle with the center at the origin, so that any tangent of the circle and ellipse E always has two intersections A and B, and suppose that the tangent equation of the circle is obtained by solving the equations, that is,

Delta =, that is

In order to do it, we have to do it, that is, this way, this way, this way, that is, or, because the straight line is the tangent of the circle whose center is at the origin, and the radius of the circle is, to find the circle is, at this time, all the tangents of the circle satisfy or, when the slope of the tangent does not exist, the two intersections of the tangent and the ellipse satisfy or. To sum up, there is a circle whose center is at the origin, so that any tangent of the circle intersects with or.

Because,

So,

,

(1) When

Because, therefore,

So,

So take "=" if and only if.

(2) when,

③ When the slope of AB does not exist, the two intersections are or, so at this time,

To sum up, the value range of |AB | is:

Proposition: this topic belongs to exploring whether there is a problem. This paper mainly examines the determination of elliptic standard equation, the positional relationship between straight line and ellipse, the positional relationship between straight line and circle, and the method of solving equation by undetermined coefficient method. It can use the method of solving the equation to study the related parameter problems and the relationship between the roots and coefficients of the equation.