Eighth grade mathematics volume II first monthly examination paper Beijing Normal University Edition
Class ———————————— Name _ _ _ _ Seat number _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Teacher's message: if you do it seriously, you can do it! Believe in yourself!
First, multiple-choice questions (***20 points)
1. It is correct to say "x2 is nonnegative" with inequality ().
A.x2 0③x = 4④x+y⑤x≠5⑥x+2 & gt； y+3
A. 1
3. Given x>y, the following inequality must be correct ().
a . x—6 & lt； y—6b . 3x & lt； 3y c .—2x & lt； —2y d . 2x+ 1 & gt； 2y+ 1
4. On the number axis, the solution set of inequality group can be expressed as
A B C D
5. The deformation from left to right of the following equation is decomposed into sum ()
A.6a2b=3a2？ 2b B.mx+nxy-xy=mx+xy(n- 1)
c . am-a = a(m- 1)d .(x+ 1)(x- 1)= x2- 1
6. () is the solution of the inequality x-4 ≥ 0.
A. 1
7. Three line segments with lengths of 3, 7 and x can form a triangle, so x can be ().
A.3 B.4 C.5 D. 10
8. The inequality x+3≥0 has () negative integer solutions.
A. 1
9. 19992+ 1999 is divisible by ().
A.b . 1995 c . 2000d . 200 1
10. given AB = 7, A+B = 6, the value of polynomial a2b+ab2 is ().
A. 13 b . 1 c . 42d . 14
2. Fill in the blanks.
1 1. The inequality means "x+ 1 is a negative number": _ _ _ _ _ _.
12. Known
13. The solution set of the inequality is represented on the number axis as shown in the figure, so the inequality may be _ _ _ _ _ _ _ _ _ _ _ _.
14. Transform the inequality x+3- 1 into "x >；; An or X.
15. The solution set of inequality 2x-3 ≤ 0 is _ _ _ _ _ _ _ _.
16. The positive integer of inequality 4(x+ 1)≤64 is _ _ _ _ _ _ _.
17. When y 1 =-x+3 and y2 = 3x-4 are known, when x _ _ _ _ _ y 1 >; y2。
18. The common factor of the terms in the polynomial 2x2+x3-x is _ _ _ _ _ _ _ _ _.
19. Decomposition factor: x (a+b)+y (a+b) = _ _ _ _ _ _.
20. If there is a two-digit number, its ten digits are more than one digit 1, and the two-digit number is more than 30 and less than 42, then this two-digit number is _ _ _ _ _ _ _ _ _.
3. Answer the questions.
2 1. Solve the following inequalities (groups).
( 1)3—x & lt； 2x+6 (2)
Two. Fill in the blanks (***32 points)
9. inequality (m-2) x >; The solution set of 2-m is x <-1, so the value range of m is _ _ _ _ _ _ _ _ _ _.
10. The common factor of polynomial AX2-4A and polynomial X2-4x+4 is _ _.
1 1. If x-y = 2, then x2-2xy+y2 =.
12.
13. The solution set of inequality is _ _ _ _ _ _ _ _
14. Given that the length of a cuboid is 2a+3 b, the width is a+2b, and the height is 2a-3b, the surface area of the cuboid is _ _.
15. If the polynomial 4a2+M can be decomposed by the square difference formula, then the monomial m = _ _ _ _ (just write one).
16. Passwords are needed in daily life, such as withdrawing money and surfing the Internet. There is a password generated by factorization method, which is easy to remember. The principle is: for the polynomial x4-y4, the result of factorization is (x-y) (x+y) (x2+y2). If x = 9, y = 9. (x2+y2) = 162, so "0 18 162" can be used as a six-digit password. For the polynomial 4x3-xy2, when x = 10 and y = 10, the password generated by the above method is:
Iii. Answering questions (***39 points)
17. Factorize the following polynomials: (***2 1 point)
( 1)a3- 16a。
(2)4ab+ 1-a2-4b2。
(3)9(a-b)2+ 12(a2-B2)+4(a+b)2。
(4)x2-2xy+y2+2x-2y+ 1。
(5)(x2-2x)2+2x2-4x + 1。
(6)49(x-y)2-25(x+y)2
.
(7)8 1x5y5- 16xy。
(8)(x2-5x)2-36。
18, please write a decomposable quadratic quartic formula and decompose it. (5 points)
19, please select two of the following formulas to make difference and factorize the obtained formula. 4a2，(x+y) 2， 1，9b2。 (5 points)
20. A park plans to build a fountain, as shown in Figure ①. Later, someone suggested changing it to the shape shown in Figure ②, and the diameter of the outer circle remained unchanged, but I was worried that the materials originally prepared were not enough. Please compare the two schemes, which one needs more materials? (8 points)
Four, broaden the topic (***47 points)
2 1, please follow the following equation before filling in the blanks: (10)
32- 12=8× 1,52-32=8×2.
( 1)72-52=8× ;
(2)92-( )2=8×4;
(3)( )2-92=8×5;
(4) 132-( )2=8× .
(5) Through observation and induction, write an equation containing the natural number n to express this law and verify it.
22. Solve the inequality group and express its solution set on the number axis: (10 point)
23. When x is a value, the value of the formula is not greater than the value of the formula. (10)
24.( 1) Calculation:1× 2× 3× 4+1= _ _ .2× 3× 4× 5+1= _ _.
3×4×5×6+ 1=__.4×5×6×7+ 1=__.
(2) Observing the results of the above calculation, it is pointed out that they have the same characteristics.
(3) Do the above characteristics still exist for any given sum of the products of four consecutive positive integers and 1? Explain your guess and verify the conclusion of your guess. (10)
25. It is known that A and B are positive integers, and A2-B2 = 45. Find the values of a and b. (5 points)
26. Tintin and Dongdong made a cuboid and a cylinder out of plasticine, respectively, and put them together at the same height. Tintin and Dongdong want to know which is bigger, but there is no ruler around, so they have to find a short rope. They measured that the length of the bottom of the cuboid is exactly three times that of the rope, the width is twice that of the rope, and the circumference of the bottom of the cylinder is 10 times that of the rope. Can you know which is bigger? How much bigger? (Hint: You can set the length of rope as cm and the height of cuboid and cylinder as h cm) (5 points)
Reference answer:
I.1.c2.c3.d4.c5.b6.c7.d8.a.
Second, 3, m < 2；; 10，(x-2)； 1 1,4; 12. 13. 14; 14， 16 a2+ 16ab- 18 B2； It is suggested that the surface area of a cuboid is 2 (2a+3b) (2a-3b)+2 (2a+3b) (a+2b)+2 (2a-3b) =16a2+16ab-18b2; 15, the answer is not unique. For example, when m =- 1, 4a2+m = 4a2-1= (2x+1); Or when m =-B2, 4a2+m = 4a2-B2 = (2x+b) (2x-b) and so on. 16, 1030 10, or 30 10 10, or10/030.
Three. 17,25.( 1)A(A+4)(A-4)； (2)( 1+a+2b)( 1-a-2b)； (3) ; (4)(x-y+ 1)2； (5)(x- 1)4； (6)4(6x-y)(x-6y)； (7)xy(9x2y 2+4)(3xy+2)(3xy-2)； (8)(x-2)(x-3)(x-6)(x+ 1)； 18, according to the meaning of the question, the polynomial of "a decomposable quadratic quartic formula" and "decompose it" is compiled, so the answer is not unique. For example, A4-B4 = (A2+B2) (A+B) (A-B), A4-2A2+B4 = (A2-B2) 2. 19, the answer to this question is not unique. * * There are 12 different difference results, namely 4a2- 1, 9b2- 1, 4a2-9b2, 1-4a2, 1-9b2. 1-(x+y) 2，4a2-(x+y) 2，9b2-(x+y) 2。 The decomposition factors are as follows: 4a2-9b2 = (2a+3b) (2a-3b); 1-(x+y)2 =[ 1+(x+y)][ 1-(x+y)]=( 1+x+y)( 1-x-y)。 20. Let the diameter of the great circle be d and the circumference be π d; Let the diameters of three small circles be d 1, d2 and d 3 respectively, then the sum of the perimeters of the three small circles is π d1+π d2+π d3 = π (d1+d2+d3). Because D = D 1+D2+D3, π D = π D
4.2 1, ( 1) 3; (2)7; (3) 1 1; (4) 1 1,6; (4)(2n+ 1)2-(2n- 1)2 = 8n。 The correctness of this conclusion can be verified by decomposing the left factor; 22. solution: solve inequality ≥x to get x≤3, and solve inequality to get x >-2. So the solution set of the original inequality group is -2 < x ≤ 3. On the number axis, it is expressed as
23. solution: from the meaning of the question: +2x ≥
24.( 1) After calculation, the easy results are 25, 12 1, 36 1 and 841respectively; (2) 25, 12 1, 36 1 and 84 1 are all complete squares; (3) The sum of the products of any four consecutive positive integers and 1 is a complete square number for the following reasons: Let the smallest positive integer be n, then the sum of the products of four consecutive positive integers and 1 is expressed as n (n+1) (n+2) (n+3)+1. That is n (n+ 1) (n+2) (n+3)+65438+.
25. Because A2-B2 = (a+b) (a-b) = 45 =1× 3× 3× 5, and both A and B are positive integers, it is still or therefore still.
26. The volume of a cuboid is: 3a? 2a？ H = 6a2h (cm3), and the volume of the cylinder is = a2h (cm3). A2h-6a2h = (-6) A2h, and -6 > 0, so A2h-6a2h > 0, A2h > 6a2h. Answer: The volume of a cylinder is larger, which is (-6) A2h cm.