Mathematics examination questions (liberal arts)
Volume one multiple-choice questions (***50 points)
1. Multiple choice questions: Of the four options given in each question, only one meets the requirements of the topic (this big question * * 10, 5 points for each question, * * 50 points).
1. If the set is known, then =( A)
A.B.
C.D.
2. If the complex number (unit bit of imaginary number) is pure imaginary number, then the value of real number is ().
a . 6 B- 2 c . 4d-6
3. If it is known, ""is ""(b)
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
4. It is known that the point P(x, y) moves on the plane region represented by the inequality group.
The range of z = x-y is ()
A.[-2，- 1] B.[- 1，2] C.[-2， 1] D.[ 1，2]
5. The eccentricity of hyperbola is 2, and one focus coincides with parabola, so the value of mn is ().
A.B. C. D。
Grade one, grade two, grade three
Girls 373
Boys 377
6. There are 2000 students in a school, and the number of boys and girls in each grade is shown in the table. It is known that 1 student is randomly selected from the whole school, and the probability of drawing girls in the second grade is 0. 19. At present, 64 students are selected by stratified sampling from the whole school, but they should be selected in the third grade.
The number of students is ()
12
7. Plane vector = ()
A. 1 B.2 C.3 D
8. In arithmetic progression, the value of then is known as ().
A.-30 B. 15 C.-60 D.- 15
9. Let sum be two different planes, L and M are two different straight lines, and L and M have the following two propositions: ① If ‖, then L ‖ m; ② If it is l⊥m, then it is ⊥. Then ()
A.① is a true proposition, ② is a false proposition B. ① is a false proposition, ② is a true proposition.
C.① ② Both are true propositions D. ① ② Both are false propositions.
10. Given three views of a geometric figure as shown in the figure, then the volume of this geometric figure is ().
A.6 B.5.5
C.5 D.4.5
Volume 2 Non-multiple choice questions (* *100)
Fill-in-the-blank question: There are 7 small questions in this big question. Candidates answer 5 small questions, with 5 points for each small question, out of 25 points.
(1) Required questions (1 1 ~ 14)
1 1. Given the angle of the second quadrant,
Then _ _ _ _ _ _ _.
12. Execute the program block diagram on the right. If = 12, input.
out of =；
13. If this function
The value of is:;
14. The difference between the maximum distance and the minimum distance from a point on a circle to a straight line is: _ _ _ _ _ _ _.
(2) choose to do the problem (15 ~ 17, candidates can only choose to do one of them)
15. (4-4 coordinate system and parametric equation are optional) The positional relationship between curves is: (fill in "intersection", "tangency" or "separation");
16.(4-5 inequality elective course) The solution set of inequality is:
17.(4- 1 elective course of geometric proof) The tangent of the known circle is,. Is the diameter of a circle, and if it intersects the circle at a point, it is the radius of the circle.
Third, the solution: the solution should be written to explain the process or calculation steps (this answer is ***6 small questions, ***75 points)
18. (This little question is 12)
Given a directional quantity, suppose.
The value of (1);
(2) If, find the range of the function.
19. (This little question is 12)
Known function.
(1) If you select any element from the set, you select any element from the set.
Find the probability that the equation has two unequal real roots;
(2) If we take any number from the interval and any number from the interval, we can find the probability that the equation has no real root.
20. (This little question is 12)
In the plane rectangular coordinate system xoy, four points A (2 2,0), B (-2,0), C (0 0,2) and D (-2,2) are known, and the plane of the coordinate system is folded into a straight dihedral angle along the Y axis.
(1) Verification: BC ⊥ AD;
(2) Find the volume of the triangular pyramid C-AOD.
2 1. (This small problem is 12 points)
It is known that the sum of the first n terms of a sequence is 0 and satisfies,
( 1);
(2) Verification: the sequence is a geometric series;
(3) If, find the sum of the first n items in the series.
22. (This small problem 13 points)
It is known that the tangent equation of a function at a point is.
( 1);
(2) Find the monotone interval of the function;
(3) Find the range of the function.
23. (This subproblem is 14) It is known that the two focal points of an ellipse are F 1 and F2, respectively, where p is the point of the ellipse on the arc of the first quadrant and satisfies = 1. Two lines PA and PB with complementary inclinations passing through P intersect the ellipse at points A and B respectively.
(1) Find the coordinates of point P;
(2) Find the slope of straight line AB;
(3) Find the maximum value of delta △PAB area.
Reference answers and grading standards of liberal arts mathematics
First, multiple-choice questions:
Answer to multiple-choice questions in volume a
The title is 1 23455 6789 10.
Answer A D A B D C B A D C
Answer to multiple-choice questions in volume b
The title is 1 23455 6789 10.
answer
Second, fill in the blanks:
(1) Questions that must be done
1 1.； 12.4.； 13. 1 or; 14.。
(2) Select the question.
15. Intersection; 16.； 17.。
Third, answer questions:
18. Answer: =
=
=...........................(4 points)
( 1)
=.......................(8 points)
(2) When,
∴ ..................... (12 points)
19. Solution: (1)a takes any element in the set {0, 1, 2,3}, and B takes any element in the set {0, 1, 2}.
∴ The values of A and B are (0,0), (0,1) (0,2) (1,0) (1,1) (2).
(2, 1), (2, 2), (3, 0)(3, 1)(3, 2) where the first number represents the value of a, the second number represents the value of b, and the total number of basic events is 12.
Let "the equation has two unequal real roots" be event a,
When the equation has two unequal real roots, the necessary and sufficient conditions are as follows
When, the value of is (1, 0) (2,0) (2,1) (3,0) (3,2).
In other words, the number of basic events contained in A is 6.
Probability of an equation with two unequal real roots
................................... (6 points)
(2)∫a takes any number in the interval [0,2], and B takes any number in the interval [0,3].
Then all the test results constitute the area.
This is a rectangular area whose area
Let "the equation has no real root" be event B.
The area formed by event b is
That is, the trapezoid of the shaded part in the figure, its area
The probability that the equation has no real root can be obtained from the probability calculation formula of geometric probability.
................................ (12)
20. the solution 1: (1)∵BOCD is a square,
∴BC⊥OD, ∠AOB is the plane angle of dihedral angle B-Co-A.
∴AO⊥BO ∵AO⊥CO and bo ∩ co = o
∴AO⊥ plane BCO and
∴AO⊥BC and DO∩AO=O ∴BC⊥ aircraft ADO
∴ BC ⊥ AD .............. (6 points)
(2) ...................... (12)
2 1. Solution: (1) Because the solution is ... 1 min.
Order separately again and get 3 points.
(2) because,
So,
Subtract two algebraic expressions to get
So,
Because it, together with the first term 2 and the common ratio 2, constitutes a geometric series ... 7 points.
(3) Because it constitutes a geometric series with the first term of 2 and the common ratio of 2.
So, so ... eight points.
Because, so ... ...
therefore
manufacture
So 1 1 min.
So 12 points.
22. Solution: (1)
The tangent equation of a point is.
∴ …………………………(5)
(2) From (1),
x
2
+ 0 — 0 +
huge
minimum
The monotone increasing interval of ∴ is: sum.
The monotone decreasing interval of is (9)
(3) According to (2), when x=-1, the minimum value is taken.
When x= 2, the maximum value is taken.
And when,; When x< is at 0 o'clock,
Therefore, the range of values is .......................... (13).
23. Solution: (1), 0,0
Then,
Say it again, this is what you want ... (5 points)
(2) Establishment: simultaneous establishment
Get:
∵ ,∴ ,
rule
Similarly, ∴...( 10)
(3) Settings: Synchronization.
, get:, ∴
∴|AB|=
but
∴S=
If and only if m = 2, the equal sign holds. ........................... (14)