1. Multiple choice questions (7 points for each question, ***56 points) Only one of the following four conclusions is correct. Please fill in the English letters of the correct answer in the brackets after the question.
1.a, b and c are positive integers, a >；; B, and it is equal to ()
A.b. or C. 1
Answer: Because A, B and C are all positive integers, they are all integers, and because of a>b. However, within the positive integer range, there is only one decomposition method. So still. So choose d
2. In the four digits consisting of numbers 2, 4, 5 and 7, each number only appears once. Arrange all four digits from small to large, and the four digits of ranking 13 are ().
AD 4527-5247
A: In the four digits consisting of numbers 2, 4, 5 and 7, each number only appears once, and there are 24 permutations of * *. When 2, 4, 5 and 7 are the highest bits, there are 6 arrangements respectively. From small to large, There are 12 kinds of * * * after row 2 and row 4, so the highest digit of 13 is 5, but the requirement is the lowest, so it is 5247. So choose B.
3. 1989 China's GDP (gross national product) was only equivalent to 53.5% of that of Britain at that time, and now it is equivalent to 8 1% of that of Britain. If Britain's current GDP is m times that of 1989, then China's current GDP is about 1989 ().
A. 1.5 times B. 1.5m times C.27.5 times D.m times.
Answer: Let the GDP of Britain be X in 1989. See the table below for other information.
My country, England.
1989 53.5% x
Currently it is 8 1%mx mx.
From the table:, so choose B.
4. If x is an integer, the value of the fraction is an integer, and the value of x is ().
A.3 B.4 C.6 D.8
Answer: The topic is a typical integer separation problem. . If the whole score is an integer, it must be a divisor of 6, and the divisor of 6 is
***8, but if you want X to be an integer, you have to get an odd number with four odd divisors. So choose B.
5. It is known that a is an integer and the root of the equation about x is a prime number, and it is satisfied, then a is equal to ().
A.2 B.2 or 5 c
Answer: from: x is a prime number and must be a divisor of 20, then,,. And because of substitution verification, I chose D.
6. As shown in the figure, Rt△ABC, ∠ C = 90 and ∠ A = 30 are known. Take a point P on the straight line BC or AC to make △PAB an isosceles triangle, then the qualified point P is ().
A.2 B.4 C.6 D.8
Answer: Many students will choose the wrong questions. Because we did a lot of such questions when we were studying at school, there are generally eight qualified P points. Because of inertial thinking, we chose D directly. But there is a special case of this problem, that is, when the angle between straight line AB and BC is 30, the obtained △PAB has an equilateral triangle. Some points overlap. So the solution to this problem should be C.
The practice of this kind of problem is:
(1) Make a circle with A as the center and AB as the radius, and the intersection with the straight line where AC and BC are located is the solution.
(2) Make a circle with B as the center and AB length as the radius, and the intersection with the straight line where AC and BC are located is the demand.
(3) Who is AB in the middle vertical line?
7. Three cubes with side lengths of 3, 5 and 8 are bonded together. Among these solids bonded together in various ways, the surface area of the one with the smallest surface area is ().
From 570 to 502 A.D.
A: In order to minimize the surface area, these surfaces must overlap to the greatest extent.
At this time, the surface area is the smallest three-dimensional figure. The surface area is
So choose B.
8. In quadrilateral ABCD, diagonal AC bisects ∠BAD, AB > AD, and the following conclusion is correct ().
A.AB-AD>CB-CD B.AB-AD=CB-CD
The relationship between C.a b-AD < CB-CD d. a b-AD and c b-CD is uncertain.
Answer: Axisymmetric transformation. The bisector of an angle is the symmetry axis of the angle. For geometric inequalities,
The comparison of line segment sizes usually has the following situations:
(1) Compare the sizes of two line segments, with a large angle for the big side and a small angle for the small side.
(2) Compare the sizes of the three line segments, make the three line segments into triangles as far as possible, and compare them with the trilateral relationship.
(3) Compare the sizes of four line segments, merge two of them into one, and form a triangle with the other two. Then compare it with the trilateral relationship.
This problem can be intercepted at AB AM=AD. Connect CM. △AMC and△△ ADC are identical, so AB-AD=MB.
CD=CM. You can get Ab-ad > CB-CD by studying in △BCM. So choose one
Fill in the blanks (7 points for each small question, 84 points for * * *)
9. The minimum value of a polynomial is _ _ _ _.
A: The formula is commonly used to find the maximum value of polynomial.
, the minimum available value is
10. If known, the value of is equal to _ _ _ _ _.
Answer: (1) Because, so.
Method (2), just substitute.
1 1. The picture shows a computer motherboard with right angles at each corner. The data is shown in the figure, and the unit is mm, so the perimeter of the motherboard is _ _ _ _ mm 。
Answer:
Can be cut and repaired. Form a rectangle with a length of 24 and a width of 20, but there are two more 4mm line segments. So the circumference is mm.
12. A school built a rectangular water tank without a cover. Now the inner wall and bottom of the tank are ground with a circular grinding wheel with radius r, so the area that the grinding wheel can't grind is _ _.
Answer: the four corners of the bottom rectangle cannot be polished, and the area of each corner is,
However, it should be noted that among the four surfaces, the upper corner can be polished off because it is not covered.
So the correct answer should be:
13. There are two acute angles and one obtuse angle in α, β and γ, and their values are given. When calculating the numerical value, three students worked out three different results, 23, 24 and 25 respectively, and one of them was indeed the correct answer, so α+β+γ = _ _ _ _.
Answer: there are two acute angles and an obtuse angle in α, β and γ, so,
So 23 should be the correct calculation result. therefore
14. Let a be a constant, and the remainder obtained by dividing by polynomial is, then A = _ _ _.
Answer: the calculation of the remainder theorem is relatively simple: let the quotient be
Judging from the meaning of the question:
Order. solve
15. In △ABC, the straight line where Gao BD and CE are located intersects at point O. If △ABC is not a right triangle, and
∠ A = 60, then∠ BOC = _ _ _ degrees.
Answer: 120 or 60.
The topic is double solution. Be sure to think that it may be an obtuse triangle:
16. Xiao Wang's school held a grade exam, chose several courses, and then tried another course. Wang Kao got 98 points, and Wang's average score was higher than the original 1 point. Later, he took an extra course, and Wang Kao got 70 points. At this time, Wang's average score dropped by 1 minute compared with the initial average score, so Wang * * * took _ _ courses (including two extra exams), and the final average score was _ _.
A: If there are m gates and the initial average is divided into n, the equation can be listed.
So the answers are 10 and 88.
17. As we all know, the range of is _ _ _ _ _ _.
A: Yes, because.
So ... you can tell from the title
Solution:
18. There is a reciprocal key on the calculator, which can find the reciprocal of the input non-zero number (note: sometimes it is necessary to press the or key first to realize this function, which will not be explained below). For example, if you enter 2 and press the key, you will get 0.5. Now enter a number on the calculator, and then press the key in the following order: The result on the display screen is -0.75, so the original number entered is _ _ _ _ _.
Answer: Note that the title is input continuously, and the columns are composed as follows:
, solution:
19. There are three different types of batteries, A, B and C, and the prices are different. If you have a sum of money, you can buy 4 A, 18 B,16 C; Or type a 2, type b 15, type c 24; Or A 6 type, B 12 type, C 20 type. If all this money is used to buy C-type batteries, you can buy _ _ _ _.
Solution:
In this equation, if both A and B are represented by C, you can get the sum. Divide by c and you get 48.
20. As shown in the figure, in the known pentagonal ABCDE, ∠ ABC = ∠ AED = 90, AB = CD = AE = BC+DE = 2, then the area of pentagonal ABCDE is _ _.
Answer: When the sum of two angles is fixed or the sum of two line segments is fixed, you can choose to rotate.
Connect AC and AD, rotate △ADE counterclockwise around A, and let AE after rotation.
Edge and AB edge overlap. This will result in congruent triangles. You can also calculate
The final area is 4.