Mathematics test questions (liberal arts) subject network
This paper is divided into two parts: the first volume (multiple choice questions) and the second volume (non-multiple choice questions). After the exam, the answer sheet and the answer sheet (Volume II) should be returned together. Theme network
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Volume I Theme Network
Note to candidates: subject network
1. Before answering questions, candidates must clearly fill in their names and quasi-evaluation numbers with 0.5mm black ink pen on the answer sheet. Theme network
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. Theme network
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. Theme network
Reference formula: topic network
If events A and B are mutually exclusive, then the lateral area formula of the topic network of regular pyramid and cone.
P(A+B)=P(A)+P(B) subject network.
If events A and B are independent of each other, the topic network
P(A？ B)=P(A)？ P(B) where c represents the perimeter of the bottom surface, and l represents the subject network of inclined height or bus length.
If the probability of event A in an experiment is the volume formula of the ball, the topic network
P, then the K-subject network occurs in n independent repeated experiments.
Probability of the number of times, where r represents the radius of the ball.
First, multiple-choice network
1. If it is collected, then () Theme Network
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2. If yes, the following conclusion is incorrect () subject network.
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3. The known function value is () topic network.
A.-1B.1c.2d.4 Subject Network
4. If there are two different intersections between a straight line and a circle, the positional relationship between points P(a, b) and C is () topic network.
A. the point is on the circle B. the point is inside the circle C. the point is outside the circle D. the topic network cannot be determined
5. It is known that the maximum values of non-negative real numbers x and y that meet the conditions are () topic net.
A. 50b. 40c. 38d. 18 Subject Network
6. Let A and B be two straight lines and two upper planes, then a sufficient condition is () main body network.
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7. According to the vector translation image, the analytical formula of the translation image is () topic network.
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8. Given a function, the image of its inverse function is roughly () topic network.
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9. Known proposition P: inequality; Proposition Q: In triangle ABC, if necessary but not sufficient, then () topic network.
A.p true q false B.P and q true C.P and q false D.P false q true topic network
10. Let vectors I and J be unit vectors in the positive direction of X axis and Y axis in rectangular coordinate system. If the vector discipline network
The trajectory equation of the point P(x, y) satisfying the above conditions is () main network.
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1 1. Geometric Series () Subject Network
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12. It is known that A, B and C are three points on the plane, O is any point on the plane ABC, and the moving point P satisfies the equation, then the trajectory of point P must pass through the () discipline network of △ABC.
A. inner heart B. loving heart C. outer heart D. center of gravity discipline network
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Volume II Theme Network
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Note: topic network
1. Before answering questions, candidates should use a black ink pen with a diameter of 0.5 mm to fill in their names and admission ticket numbers clearly on the answer sheet (Volume II) ...
2. Please answer the questions in the answer sheet (Volume 2) with a black ink pen with a diameter of 0.5 mm. The answer on the test paper is invalid. Theme network
3. This volume *** 10, ***90 points. Theme network
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2. Fill in the blanks: This big question has four small questions, each with 5 points and * * 20 points. Theme network
(Note: The answer on the test paper is invalid. ) theme network
13. A school has 200 teachers, 1200 boys and 1000 girls. At present, the stratified sampling method is adopted to extract a sample with a capacity of n from all teachers and students. Given that the number of female students is 80, the value of n is.
In the extension of 14. The constant term is equal to the topic network.
15. If the midpoint of the radius passing through the ball is taken as the cross section of the ball perpendicular to the radius, the ratio of the cross section area to the surface area of the ball. Theme network
16. For functions, the following propositions are given: topic network.
(1) When a= 1, it is a monotone increasing function in the domain; Theme network
② The image is symmetrical about the point (1, a); Theme network
(3) For any non-odd function; Theme network
④ When a =- 1, it is an even function; Theme network
⑤ When a=2, for all subject networks that meet the conditions,
The serial number of the correct proposition is. Theme network.
Third, answer: This big question is ***6 small questions, ***70 points. The solution should be written in words, proving the process or calculation steps. Theme network
17. (The full mark of this small question is 10) (Note: the answer on the test paper is invalid. ) theme network
In △ABC, it is known that the area of △ABC is equal to 6. Theme network
(1) find c; Theme network
(2) Find the lengths of the three sides of △ABC. Theme network
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18. (The full mark of this small question is 12) (Note: it is invalid to answer the questions on the test paper. ) theme network
Decorate the garden with red, yellow, blue, white and orange flowers as shown in the picture. It is required to use flowers of the same color in the same area and flowers of different colors in adjacent areas. Theme network
(1) If only four different colors of flowers are used for layout, how many different layout schemes are there? Theme network
(2) Find the probability that exactly two areas use safflower. Theme network
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19. (The full mark of this small question is 12) (Note: it is invalid to answer the questions on the test paper. ) theme network
As shown in the figure, in the regular triangular prism ABC-a1b1c1,AB=AA 1, and E is the midpoint of AC. Theme network
(1) Find the cosine of the angle formed by the straight line AB 1 and BC 1; Theme network
(2) Find the sine e-bc 1-c of dihedral angle.
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20. (The full mark of this small question is 12) (Note: the answer on the test paper is invalid. )
The sum of the first n terms of a known sequence is
(1) Find the general term formula of series;
(2) For a series, find the sum of the first n items of {}.
2 1. (The full mark of this small question is 12) (Note: it is invalid to answer the questions on the test paper. )
Find the extreme value of a known function.
(1) Find the analytical formula of the function;
(2) If the three tangents of the intersection intersect, the range of the number m should be realistic.
22. (The full mark of this small question is 12) (Note: the answer on the test paper is invalid. )
It is known that the center symmetry point of an ellipse with eccentricity about a straight line is on the right alignment of ellipse C.
(1) Find the equation of ellipse c;
(2) Let it be two points on the X-axis, the intersection point of a straight line with a slope other than 0 intersects with ellipse C at two points M and N, and the intersection point of straight line BN and ellipse C at another point E ... It is proved that △BME is an isosceles triangle.
Reference answer
First, multiple choice questions
The title is123455678911112.
Answer d d d d b c b c b c c d
Second, fill in the blanks
13. 192 14. 15 15. 16.②③⑤
Third, answer questions.
17. Solution: (1) Let the three sides corresponding to the three internal angles A, B and C of a triangle be A, B and C respectively.
∵ ,
∴, from sine theorem, 3 points.
According to cosine theorem,
That is,
So it's Rt, and ................... scored 6 points.
(ii) let a = 4k and b = 3k(k >；; 0) ............................ scored 8 points.
Then,
∴: The lengths of the three sides are a=4, b=3 and C = 5 respectively. ..............................................................................................................................................
18.(I) As shown, first, from five different colors.
Choose four kinds of flowers,
The arrangement of four-color flowers can be divided into two situations:
Areas A and D have the same color, and areas B and E have the same color.
There are all kinds of seeds, 3 points.
Therefore, there are * * * different placement schemes, only four different colors of flowers are used. .........................................................................................................................................................
(ii) Let m stand for the event "exactly two areas use safflower",
As shown in the figure, when area A and area D are of the same color, * * * has species;
When the colors of area A and area D are different, there is one kind of * * *;
Therefore, the total number of all basic events is:180+240 = 420 ................................. 8 points.
They are equally possible. And because when A and D are red, there is one kind of * * *;
When B and E are red, * * * has seeds;
Therefore, the basic event contained in event M is: 36+36=72. ......................................................................................................................................................
Therefore, the probability of using safflower only in two areas is = ................................................................................................................................................................
19. (i) If it extends to m, make, connect, connect, or the remaining angles are angles formed by straight lines on different planes (set to), then
Let AA 1=AB= 1, then in,
therefore
Therefore, the cosine of the angle formed by straight lines on different planes is .............................................................................................................................................................
(ii) is a flat regular triangular prism,
Planes,
Planes, planes,
Do something excessive,
Then the plane,
Overreaction derived from the three vertical theorems,
Therefore, ∠ is the plane angle of dihedral angle .......................... 9 points.
Let's say AA 1=AB=2,
Then, in Delta,
.
The sine value of dihedral angle is 12 points.
20. Solution: (1) From the knowledge, when …… when …… when … when … when … when … when … when … when … when … when … when … when … when … when … when … when … when … when ….
.
It is also determined by inspection that ................. scored 4 points.
By, by, ∴ p =
∴ ............................. 6 points.
(2) Get (1), 7 points.
2 ; ①
(2) .................................... scored 9 points.
②-① Yes,
= = ............... 12 point.
2 1. Solution: (i) f ′ (x) = 3ax2+2bx-3,
According to the meaning of the question, f ′ (1) = f ′ (-1) = 0, ...
that is
The solution is a= 1 and b = 0. ∴ f (x) = x3-3x ...................................... 4 points.
(ⅱ)f′(x)= 3 x2-3 = 3(x+ 1)(x- 1)，
∫ The curve equation is y = x3-3x,
Point A( 1, m) is not on the curve.
Let the tangent point be M(x0, y0), then the coordinates of point m satisfy.
Because,
So the slope of the tangent is,
Tidy up ..............................................................................................................................................................................
∵ Intersection point A( 1, m) can be used as three tangents of the curve.
The equation =0 about x0 has three real roots.
Let g(x? 0)=, then g'(x0)=6,
From g'(x0)=0, x0=0 or x0? =1.......................... 9 points.
∴g(x0) monotonically increases at (-∞, 0), (1,+∞), and monotonically decreases at (0, 1).
The extreme point of the function g(x0)= is x0=0, x0= 1.
∴x0 equation =0 has three real roots if and only if
Solution -3
Therefore, the range of real number A is -3.
22. Solution: (Ⅰ) ∵,
Let the abscissa of O about the symmetrical point of a straight line be
, .............................. 2 points.
You are straight.
Get the midpoint coordinates of the line segment (1, -3).
∴ ,
The elliptic equation is 5 points.
(ii) Set a point, when the slope of the straight line L exists,
Then the equation of the straight line L is, ... 6 points.
Replaced:
, ……①
Similarly, ① can become:
, ……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
By the known, there are
∵ ............... 10 point
In the same way; In a similar way
Solve,
∴ ..................11min.
So the straight line ME is perpendicular to the X axis. From the symmetry of the ellipse, it can be seen that points M and E are symmetrical about X axis, while point B is on the X axis.
∴|BM|=|BE|, that is, △BME is an isosceles triangle.
When the slope of the straight line L does not exist, the conclusion clearly holds. ..............................................................................................................................................................