Senior one preliminary examination questions.
First, multiple-choice questions (out of 36 points)
1. The quadratic function satisfying the condition f(x2)=[f(x)]2 is
A.f(x)=x2 B. f(x)=ax2+5
C.f(x)=x2+x D. -x2+2004
2. Functions y=sinx and y=sin2004 defined on R, where the number of even functions is
A.0 B. 1 C. 2 D. 3
3. There are exactly three real number solutions, so a is equal to.
A.0 B. 0.5 C. 1 D。
4. Real numbers A, B and C satisfy A+B >; 0、b+ c & gt； 0、c+a & gt； 0, f(x) is the odd function on R, and it is a strictly decreasing function, that is, if X 1
A.2f(a)+f(b)+f(c)= 0b . f(a)+f(b)+f(c)& lt； 0
C.f(a)+f(b)+f(c)>0 D. f(a)+2f(b)+f(c)=2004
5. It is known that among the four positive integers A, B, C and D, A divided by 9 is 1, B divided by 9 is 3, C divided by 9 is 5, and D divided by 9 is 7, so two numbers that are not completely square are
A. means
6. In the positive real sequence a 1, a2, a3, a4 and a5, a 1, a2 and a3 become arithmetic progression, a2, a3 and a4 become geometric progression, and the common ratio is not equal to 1, and the reciprocal of a3, a4 and a5 becomes geometric progression, then
A.a 1, a3, a5 do geometric series.
B.a 1, a3 and a5 become arithmetic progression.
The reciprocal of C.a 1, a3 and a5 becomes arithmetic progression.
The reciprocal of d.6a 1, 3a3, 2a5 becomes a geometric series.
Two. Fill in the blanks (out of 64)
1. Known, try to determine the value.
2. Given a= 1+2+3+4+…+2003+2004, find the remainder of a divided by 17.
3. It is known that if ab2≠ 1 exists, try to determine the value.
4. As shown in the figure, the right vertex C of isosceles right triangle ABC is on the hypotenuse DF of isosceles right triangle DEF, and e is on the hypotenuse AB of △ABC. If the area of convex quadrilateral ADCE is equal to 5 square centimeters, what is the area of convex quadrilateral ABFD?
5. If A, b∈R, a2+b2= 10, try to determine the value range of A-B.
6.a and B are two roots of the equation x4+m=9x2 about X, which satisfies a+b=4. Try to determine the value of m.
7. Find the value of cos20 cos40 cos60 cos80.
8. express 2004 as the sum of n unequal positive integers, and find the maximum value of n. ..
Preliminary answer sheet
Multiple choice questions: ADCBBA；; Fill in the blanks: 1, -0.52, 1.
3、- 1 4、 10 5、[ , ]
6、49/4 7、 1/ 16 8、62
200 1 Beijing middle school students' mathematics competition for the first year of high school.
1. Multiple-choice questions (full score is 36, each question has only one correct answer, please fill in the English letter code of the correct answer in the specified position of 1, the correct answer will get 6 points, and the wrong answer or no answer will get 0 points).
The number of subsets of 1. set {0, 1, 2,2001} is
16 (B) 15 (C)8 (D)7
2. in the cube ABCD-a1b1c 1d 1, m is a point ABove the side c1d1,n is a point above the side ab, ∠ mab = \.
(A)AM and CC 1 are straight lines in different planes. (B)AM and NB 1 are straight lines in different planes.
(C)AN and MB 1 are straight lines in different planes. (D)AN and MC 1 are straight lines in different planes.
3. The inverse function of the function y =-√ (1-x) (x ≤ 1) is (a) y = x2- 1 (- 1 ≤ x ≤ 0). (b) y = 1-。
4. A straight line intersects with the unequal sides AB and AC of ABC at D and E respectively. If the straight line DE bisects the perimeter and the area of Δ ABC, then the straight line DE must intersect Δ ABC.
(a) center of gravity (b) outer center (c) inner center (d) vertical center
5. Given f(x6)=log2x, then f(8) is equal to
(A)4/3(B)8(C) 18(D) 1/2
6. The picture on the right shows the plane expansion of the cube. In this cube,
①BM is parallel to ED;
②CN and BE are straight lines in different planes;
③CN and BM form an angle of 60;
④DM is perpendicular to BN.
The serial number of the correct proposition in the above four propositions is
(A)①②③ (B)②③④ (C)③④ (D)②④
2. Fill in the blanks (out of 64 points, 8 points for each small question, please fill in the answer at the designated position on 1 page)
1. In the regular tetrahedron ABCD, m is the midpoint of the side BD, n is the midpoint of the side AD, the angle between the out-of-plane straight line MN and CD is α, and the angle between AC and MN is β. Find the degree of α+β.
2. If the real numbers x, y and z satisfy √ x+√ (y-1)+√ (z-2) =1/2 (x+y+z), find the value of logz (x+y).
3. Let any real number X have f (x)=x2+lg(x+√(x2+ 1)) and f (a)=m, and find f (-a), which is represented by a and m. ..
4. Let f (x) be an even function defined on r, and f (x+2)=- 1/f (x). When 2≤X≤3, f (x)=x, and the value of f (5.5) is determined.
5. In the tetrahedron ABCD, the edge CD is perpendicular to the plane ABC, AB = BC = CA = 6, BD = 3 √ 7, let the dihedral angle D-AC-B be α, D-AB-C be β, B-DC-A be R, and find the value of sin α+TG β+Cosr.
6.max{x 1, x2, x3, …, xn}, min{x 1, x2, x3, …, xn} represent the maximum and minimum values of x 1, x2, x3, …, xn respectively.
If a+b+c= 1, (a, b, c∈R), determine the value of min{max{a+b, b+c, c+a}}.
7. Let 3x=0.03y= 10-2 and find the value of (1/x-1/y) 2001.
8. If the equation sin2x+sinx+a=0 about X has a real number solution, find the sum of the maximum and minimum values of the number A. ..
1. Fill in the blanks (out of 40 points, 8 points for each small question)
1.f (x+y)=f (x) Is it known? F (y) holds for any nonnegative real numbers X and Y, and f (1)=3, then f (1)/f (0)+f (2)/f (1)+f (3)/f (2)+f.
2. In the right figure, if AD = AB, ∠ ABC = ∠ Bad = 90, the area of quadrilateral ABCD is 22, and the area of square CDEF is 25, then the line segment AE = ().
3. Let a = √ (1+12+1/22)+√ (1+1/32).
4. Two different quadratic trinomials f (x) and g (x), their first coefficients are 1, and they satisfy f (1)+f (10)+f (100) = g (/kloc).
5. In tetrahedral ABCD, if dihedral angle B-AC-D is a straight dihedral angle, AB = BC = CD, BD = AC, and dihedral angle B-AD-C is denoted as α, then cosα= ().
2. (Full score 15) Does the integer coefficient polynomial f (x) satisfy f (1999)? F (2000)=200 1, please prove that f (x)=0 has no integer root.
(Full score 15) It is known that the quadratic function f (x) satisfies f (- 1)=0, and for all real numbers x, there is always x≤f (x)≤ 1/2(x2+ 1). Try to determine the expression of f (x).
(Full score 15) In tetrahedral ABCD, it is known that AB = 3, BC = 4, CD = 5, ∠ ABC = 45, ∠ BCD = 90, and the angle formed by straight line AB and CD is equal to 60, so find the side length AD.
5. (Full mark is 15) From the set m = {2, 3, 4, 5, 6, 7, 8, 9, 10/1,…, 958, 959, 960}