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In 2008, the national unified entrance examination for colleges and universities was held.
Liberal Arts Mathematics (National Volume Ⅰ) (Compulsory 1+ Elective Ⅰ)
This paper consists of two parts: the first volume (multiple choice questions) and the second volume (non-multiple choice questions). Volume 1 1 to 2 pages. Volume II, pages 3-9. After the exam, return this paper together with the answer sheet.
Precautions for candidates:
1. Before answering questions, candidates must clearly fill in their name and admission ticket number on the answer sheet with a black ink pen with a diameter of 0.5 mm, and affix the bar code. 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=4∏R2
If events A and B are independent of each other, then R represents the radius of the ball.
P(A+B)=P(A)+P(B) S=4∏R2
P(A？ B)=P(A)？ Volume formula of P(B) sphere
If the probability of event A in an experiment is p, then V= ∏R3.
The probability that event A happens exactly k times in n independent repeated tests, where r represents the radius of the ball.
Pn(k)=CknPk( 1-p)n-k(k=0， 1，2，…，n)
First, multiple choice questions
The domain of (1) function y= is
(A){ x | x≤ 1 }(B){ x | x≥ 1 }
(c) {x | x ≥ 1 or x ≤ 0} (d) {x | 0 ≤ x ≤ 1}
(2) If the car stops after starting, accelerating, driving at a constant speed and slowing down, if the driving distance S of the car in this process is regarded as a function of time t, its image may be
(3) the coefficient x in the expansion (1+)
(A) 10 (B)5 (C) (D)
(4) The inclination of the tangent of curve Y = x-2x+4 at (1, 3) is
(A)30 (B)45 (C)60 (D) 12
(5) in △ABC, = c, = B. If point D satisfies =2, then =
(A) (B) (C) (D)
(6) y = (sinx-cosx)- 1 Yes
(a) An idol function with a minimum positive period of 2π; (b) odd function with a minimum positive period of 2π.
(c) Even function with minimum positive period π; (d) odd function with minimum positive period π.
(7) If the known geometric series {a} satisfies a+a +a =3 and a+a = 6, then a =
64(B)8 1(C) 128(D)243
(8) If the image of function y = f (x) and the image of function y= 1n are symmetrical about the straight line y=f(x), then f (x) =
(A) (B) (C) (D)
(9) In order to get the image of function y=cos(x+), we only need to change the image of function y=sinx.
(a) Translate the length unit to the left; (b) Translate the length unit to the right.
(c) Translate the length unit to the left (d) Translate the length unit to the right.
(10) If the straight line = 1 has a common point with the graph, then
(A) (B) (C) (D)
(1 1) It is known that the edge of the triangular prism ABC- is equal to the edge of the bottom surface, and the projection on the bottom surface ABC is the center of △ABC, then the sine value of the angle formed by A and the bottom surface ABC is equal to
(A) (B) (C) (D)
(12) Fill in the 3×3 grid with 1, 2,3, and there are no duplicate numbers in each row and column. The following is a filling method, and different filling methods use * * *.
(A)6 species (B) 12 species (c), 24 species (d) and D)48 species.
In 2008, the national unified entrance examination for colleges and universities was held.
Liberal arts mathematics (compulsory+elective 1)
Volume II
Precautions:
1. Before answering questions, candidates should clearly fill in their name and admission ticket number with a black ink pen with a diameter of 0.5 mm on the answer sheet, and then stick a bar code. Please carefully approve the admission ticket number, name and subject on the bar code.
2. Book 2 * * * Page 7, please use a black ink pen with a diameter of 0.5 mm to answer the questions 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 5 points and ***20 points. Fill in the answers on the lines in the questions.
(Note: the answer on the test paper is invalid)
(13) If x and y satisfy the constraints, the maximum value of z = 2x-y is.
(14) Given the focal point y of a parabola = ax2-1as the coordinate origin, the area of a triangle with three intersections and two coordinate axes as the vertices is.
(15) in △ABC, ∠ a = 90, and tanB= =. If an ellipse with focal points A and B passes through point C, the eccentricity of the ellipse is e =.
(16) In the known rhombic ABCD, AB = 2, ∠ A = 120, and the dihedral angle A-BD-C 120 is made by folding △ABD along the diagonal BD, then the distance from point A to the plane where △BCD is located is equal to.
Third, the solution: this big question is ***6 small questions, with ***70 points. The solution should be written in proof process or calculus steps.
(17) (the full score of this small question is 10)
(Note: the answer on the test paper is invalid)
Let the side lengths of the internal angles A, B and C of △ABC be A, B, C, ACOSB = 3 and BSINA = 4, respectively.
(i) Find the side length a;
(2) If the area of △ABC is S = 10, find the perimeter l of △ABC.
(18) (the full score of this small question is 12)
(Note: the answer on the test paper is invalid)
In pyramid A-BCDE, the bottom BCDE is rectangular, while the side ABC⊥ bottom BCDE, BC = 2, CD =, AB=AC.
(1) proof: ad ⊥ ce;
(2) Let the side ABC be an equilateral triangle and find the dihedral angle C-ad-e. 。
(19) (the full score of this small question is 12)
(Note: the answer on the test paper is invalid)
In sequence {0}, = 1, an+ 1=2an+2n.
(i) Let bn=. Prove that the sequence {BN} is arithmetic progression;
(ii) Find the first n terms of the sequence {an} and Sn.
(20) (The full score of this small question is 12)
(Note: the answer on the test paper is invalid)
It is known that 1 of the 5 animals has a certain disease, and blood test is needed to determine the diseased animal. If the blood test result is positive, the sick animal is found; if it is negative, it is not sick. The following are two test schemes:
Scheme A: Test one by one until the sick animals can be identified.
Scheme B: First, take three animals and mix their blood for testing. If the result is positive, it means that the diseased animals are 1 of these three animals, and then test them one by one until the diseased animals can be identified. If the result is negative, take the other 2 1 for testing.
Find out the probability that the number of tests required by scheme A is not less than that required by scheme B. 。
(2 1) (the full score of this small question is 12)
(Note: the answer on the test paper is invalid)
The function f(x)=x3+a x2+x+ 1, a R.
(i) Discuss the monotone interval of function f(x);
(2) Let the function f(x) be a decreasing function in the interval (-) and find the range of α.
(22) (The full score of this small question is 12)
(Note: the answer on the test paper is invalid)
The center of hyperbola is the origin O, and the focus is on the X axis. The two asymptotes are l 1 and l2. A straight line perpendicular to l 1 through the right focus F intersects L 1, and L2 is at points A and B. It is known that |||| | | forms a arithmetic progression with the same direction.
(1) Find the eccentricity of hyperbola;
(Ⅱ) Let the line segment length of hyperbola section AB be 4, and find the equation of hyperbola.