Mathematics examination paper for enrollment in other provinces and cities (the second time)
(The full mark of this question is 100, and it will be completed in 90 minutes. )
1. Multiple-choice question: (The full score of this big question is 30) This big question * * * has 10 small questions, and each small question gives four answers, code (a), (b), (c) and (d), of which one and only one answer is correct. Please write the code of the correct answer in brackets after the question. 3 points for each small question; If there is more than one representative letter that is not selected, wrong or wrong (whether written in brackets or not), score zero.
1, in Rt△ABC, the length of each side is expanded by 2 times, then the sine value of acute angle A (b)
(a) A threefold reduction; (b) it has not changed; (c) It has tripled; (d) It can be expanded or reduced.
2. The solution of the equation is (c)
(A)x=4, y=0(B)x=0, y=0(C) no solution (d) infinite group solution.
3. Given that the average value of a-2, b+ 1, c-5, d+8 and e-7 is m, then the average value of a, b, c, d and e is (a).
(A)m+ 1(B)m- 1(C)m+5(D)m-5
4. As shown on the right, the three median lines of a regular triangle form a small regular triangle. If a small regular triangle
The area (shaded part) is 25, so the perimeter of the large regular triangle is (a)
(A)60(B) 100(C)
5. As shown in the figure, at Rt△ABC, ∠ C = 90, DE⊥AB, AC=BE= 15, BC=20.
Then the area of the quadrilateral is (d)
54(B)75(C)90(D)96
6. If (3,4) is an inverse proportional function y= a point on the image, then this function must pass through point (a).
(A)(2，6)(B)(2，-6)(C)(4，-3)(D)(3，-4)
7. It is known that the angle between the diameter AB of ⊙O and the chord AC is 30, and the tangent PC passing through point C intersects the extension line of AB at p. ..
PC=5, then the radius of ⊙O is (b).
(A)(B)(C)5(D) 10
8. The inner angle of a convex N polygon has exactly four obtuse angles, so the maximum value of n is (c).
5(B)6(C)7(D)8
9. If a2+ma+ 18 can be decomposed into the product of two linear factors in the integer range, then the integer m can't be (c).
9(B) 1 1(C) 12(D) 19
10, Party A and Party B bought grain in the same grain store twice, and the unit price was different. Every time Party A buys 100 kg of grain, Party B buys 100 yuan.
If it is stipulated that whoever buys grain twice and whose average unit price is low, then the way to buy grain is cost-effective. Then these two grain purchases (b)
(a) A is cost-effective; (b) B is cost-effective; (c) A and B are the same; (d) It depends on the price situation twice.
Fill-in-the-blank question: (The full mark of this big question is 36) This big question * * * has 12 small questions, and each small question only needs to fill in the final and most accurate result above the horizontal line, and each question will get 3 points if it is filled correctly, otherwise it will get 0 point.
1 1, if the positions of the points corresponding to the three numbers A, B and C on the number axis are as shown in the figure,
Simplified: = _ _ 3 _ _ _.
12, the upper bottom of the isosceles trapezoid is equal to the waist length, and the lower bottom is equal to twice the waist length, so the smaller inner angle is _ _ _ _ 60 _ _ _ _ _.
13, the test scores of a class (10) are as follows: 10, 4 people, 9 people, 8 people, 14 people, 7 people, 18 people, 6 people,
Five points for two people. Then the median of this test is _ _ _ 7.5 _ _ _ _ _.
14, the symmetry center of function (c≠0) is known as (a, b), and the symmetry center of: is _ _ (3,4).
15, it is known that θ is an acute angle, and the equation x2+3x+2sinθ=0 is that the difference between the two roots of X is, then θ = _ _ 30 _ _.
16, let ∠ XOy = 30, a is a point on the ray Ox, OA=2, d is a point on the ray Oy, OD=3, and c is any point on the ray Ox.
B is any point on the ray Oy, then the minimum value of the length AB+BC+CD of the polyline ABCD is _ _ _ _ _ _ _.
17, as shown in the figure, AD∨BE∨CF, AB=AD= 1, BE=2, CF=4, then BC = _ _ _ _ _
18. In a unit circle with a radius of 5, the diameters AB and CD are perpendicular to each other, and the chords ch and AB intersect at k,
And CH=8, then │ AK-BK │ = _ _ _ 7.5 _ _ _ _ _ _
19, the known equation (x- 19)(x-90)=p has a real root r 1, r2, where p is a real number,
Then the minimum real root of equation (x-r 1)(x-r2)=-p is _ _19 _ _.
20. Master Wang surrounded a rectangular garden, with an area of 50m2 on one side against the wall. If it's not against the wall on three sides,
Enclose it with a bamboo fence. Then, the bamboo fence needs to be at least _ _ _ 20 _ _ meters long.
2 1, add a number 6 to the right of the number A, then this number will increase by 2004, then A = _ 222 _ _.
22. For any two positive integers x and y, define an operation "★" as x★y=2(x+2xy+y). If positive integers a and b satisfy a★b= 1 154,
Then the ordered positive integer pair (a, b)*** has a _14 _ _ pair.
3. Answer: (The full score of this big question is 34 points) This big question is entitled ***3 questions. You must write the necessary steps to answer the following questions.
23. (The full score of this question is 10) There are two small questions in this question. The full score of 1 small question is 6, and the full score of the second small question is 4.
The circle Q intersects the X axis at points A and B to the right of the origin, and is tangent to the Y axis at point C below the origin.
As shown in the figure. Known │AB│=3, │AC│=.
(1) Find the coordinates of points A, B and C;
(2) If the parabola passes through three points: A, B and C, find the analytical formula of this parabola.
Solution: (1) Let A(x, 0), B(x+3, 0) and C(0, y). According to secant theorem,
Solution: x= 1 or x= (give up). ∴y=-2。
∴ The coordinates of points A, B and C are A (1 0), B (4 4,0) and C (0 0,2) respectively.
(2) Let the parabola passing through two points be y=a(x- 1)(x-4),
It passes through point C(0, -2), ∴-2=a(0- 1)(0-4), and ∴a==. The equation of parabola is y=(x- 1)(x-4).
24. (The full score of this question is 12) There are three small questions in this question. The full score of 1 small question is 4, the full score of the second small question is 6 and the full score of the third small question is 2.
The last k(k≥2) bit of a complete square number n is the same non-zero number A. Q:
(1) which number is a?
(2) What is the maximum k?
(3) When k is the largest, use this property to write the smallest number (without proof).
Solution: (1) The last digit (not 0) of a square number can only be 1, 4, 5, 6, 9.
The last two digits of the number n must be 1 1, 44, 55, 66, 99,
N is a square number, ∴n≡0 or 1 (modulo 4).
The last two digits are 1 1, 55, 99, and the last two digits are 66, which is 3(mod 4).
∴a can only be 4, such as 144= 122.
(2) If there are at least four consecutive 4s, that is, n = m2 = t 104+4444. ∴ Let m=2m 1, m12 = 25t102+11≡ 3 (mod 4).
The same as (1) shows that 25t102+11cannot be a complete square number. ∴ 4 points for 3 consecutive times. (Yes, see (3))
(3) When k is maximum, the minimum number with this property is 1444=382.
25. (The full mark of this question is 12)
As shown in the figure, in a rhombic ABCD with a side length of a, ∠ A = 60, the extension line of C where AB and AD intersect with E and F respectively, and the connecting lines de and BF intersect with M. If the circumscribed circle radii of △BEM and △DFM are R 1 and R2 respectively, it is proved that R 1 R2 is a constant value.
Proof: △BEC∽△DCF
∴。
∴△BED∽△DBF，
∴∠BED=∠DBM。
∴∠bme=∠bdm+∠dbm=∠bdm+∠bed=∠abd=60 .
∴:2r 1 = from sine theorem, 2R2=,
∴R 1 R2= == .