1. if a < 0, b > 0 and | a | < | b|, then a+b= [].
A.| b |-| a | b .-| a |-| b | c . | a |-| b | d . | a |+| b |
Among these figures, the smallest is []
A.； b； c； D.3. 14 16。
3. The positions of A, B and C on the number axis are shown in Figure 6. -,-A, C-B and C+A, the largest is [].
A.-a； b . c-b； c . c+a； D ...
4. If, then N=[]
A. 199 1 B. 1993。 c 1995d 1997
5. The positions of A and B on the number axis are shown in Figure 7.
Then the number of negative numbers in a+b, b-2a, |a-b|, |b|-|a| is [].
A. 1
6. If the equation1992+1994+1996+1998 = 5000-□ holds, the number to be filled in □ is [].
A.5. B.-980 C.- 1990 D.-2980
7. It is reported that the largest prime number discovered by supercomputers at present is 2859433- 1, and the last number of this prime number is [].
A. 1
8. After replacing one of the non-zero numbers with the number 3 in -0. 1428, the number obtained is the largest, and the replaced number is [].
A. 1
9. When-1 < a < 0, there is [].
A.& gta； B. buy a3 and buy > a3; c .-a & gt； a2； D.a3 & lt-a2。
10. There are three conclusions:
A: If at least two of A, B and C are inverses, then A+B+C = 0.
B: If at least two of A, B and C are inverse numbers, (A+B) 2+(B+C) 2+(C-A) 2 = 0.
C: If at least two of A, B and C are antonyms, then (A+B) (B+C) (C+A) = 0.
The number of correct conclusions is []
A.0 .B. 1。 C.2. D.3
Fill in the blanks: (4 points for each question, ***40 points)
1. In Figure 8, there are _ _ _ _ lines with points A, B, C, D, E and O as endpoints.
2. Among the first n natural numbers of1,2, 3…, n, * * has p prime numbers, q composite numbers, m odd numbers and n even numbers, then (p-m)+(q-n) = _ _.
4. If three times of a six-digit number is equal to, then the six-digit number is _ _ _ _ _ _ _ _.
5. The ratio of time a seamstress spends making a shirt, a pair of trousers and a coat is 1: 2: 3. He can make two shirts, three trousers and four coats in ten working hours. So he needs _ _ _ _ to make 14 shirts, 10 pants and two coats.
6. If both P and Q are prime numbers, and the root of the equation px+5q=97 with X as unknown is 1, then P2-Q = _ _ _ _ _
7.N is a natural number, and we call the product of non-zero numbers of N "exponent". For example, the exponent of 1 is 1, the exponent of 27 is 14, and the exponent of 40 is 4, so the sum of the exponents of 99 natural numbers of 1 ~ 99 is _ _.
8. in the equation y=ax2+bx+c, when x= 1, y=-2, when x=- 1, y=20, ab+BC+9b2 = _ _ _.
9. We use
10. The electronic flea falls on a point k0 on the number axis. In the first step, it jumps from k0 to the left 1 unit to k 1, in the second step, it jumps from k 1 to k2 by 2 units, in the third step, it jumps from K to k3 by 3 units, and in the fourth step, it jumps from k3 to k4, … by 4 units, … according to the above rules.
Iii. Answer: (each question 10, out of 20)
1. In the rectangular ABCD, put six rectangles with the same shape and size, and mark the dimensions as shown in Figure 9.
Try to find the total area of the shaded part in the figure (write a concise process of step-by-step solution)
2.( 1) has a "template" of 19 (Figure 10). Please design a method to draw an angle of 1 on paper with this template and a pencil.
(2) There is a 17 "template" and a pencil. Can you draw an angle of 1 on the paper?
(3) Can you draw an angle of 1 on the paper with a "template" of 2 1 and a pencil?
For questions (2) and (3), if possible, please briefly describe the drawing steps; If not, please explain why.
The answer? point out
First, multiple choice questions
Tip:
1. According to the addition law of rational numbers, select (a).
3. As can be seen from Figure 6,-1 < a < 0, 0 < b < c < 1.
∴- 1 < C+A < 1。 And C-B < 1-0 = 1.
5. As can be seen from Figure 7, a < 0, b > 0, | a | > | b |.
∴ a+b < 0，b-2a > 0，| a-b | > 0，| b |-| a | < 0。 Select "b"
6. Let the number of □ be x, then1992+1994+1996+1998 = 5000-x, that is, 7980 = 5000-x.
∴ x = 5000-7980 =-2980。 Select (d).
The index of the last digit of 7.2n changes periodically with 4, and 2 1 = 2, 22 = 4, 23 = 8, and 24 is 6. Generally, the last digit of 24k+ 1 is 2, the last digit of 24k+2 is 4, the last digit of 24k+3 is 8, and the last digit of 24k+4 is 6.
859433=2 14858×4+ 1,2859433=24×2 14858+ 1
The last number is 2.
∴2859433- 1 The last digit is 1. Select (a).
8. Actually -0.3428(3 pairs 1), -0 1328(3 pairs 4), -0. 1438(3 pairs 2) and -0. 1423(3 pairs 8).
10. For example, A = 5, B =-5, c = 3,5, -5,3 are at least two opposite numbers, but 5+(-5)+3 = 3 ≠ 0. Know that (a) is not true. [5+(-5)] 2
Second, fill in the blanks
Tip:
1.* * There are 13 different line segments, AB, AC, BC, AE, EC, CD, BD, Bo, OE, Be, Ao, AD and OD.
2.p+q = n- 1, m+n = n. Then (p-m)+(q-n) = p-m+q-n = (p+q)-(m+n) = (n-1).
3. Because the unit is 23 3s, and the last digit of 23x3 = 69 is 9, so enter 6 into the decimal digits.
The decimal digit is the sum of 22 3s, 22×3=66, and the six digits add up to 72, so the decimal digit is 2, and the percentile is 7.
The hundredth bit is the sum of 213, 2 1×3=63, plus the tenth bit of 7, which is 70, so the hundredth bit is 0, the thousandth bit is 20×3=60, and the hundredth bit is 7, which is 67.
The four digits are 7029.
This six-digit number is 2857 13.
Suppose the sewing time of a shirt is x, then the time of a pair of trousers is 2x, and the time of making a coat is 3x.
Two shirts, three trousers, four tops, ten working hours, that is, 2x+3x (2x)+4x (3x) = 10 (working hours).
That is, 20x= 10 (man-hour), so to complete 2 tops, 10 pants and 14 shirts * * needs:
2× (3x)+10× (2x)+14x = 40x = 20 (working time)
6. Because 1 is the root of equation px+5q=97, one of P+5q = 97. P and 5q must be odd and the other must be even.
If P is odd, 5q is even, and only Q can be even prime number 2, then p=97-5×2=87=3×29, which does not meet the condition that P is prime number, so only P can be even prime number 2, 5q=95, and Q = 19.
∴p2-q=4- 19=- 15.
7. The sum of indicators of1~ 9 is
1+2+3+4+5+6+7+8+9=45
The sum of indicators from 10 to 19 is
1+ 1+2+3+4+5+6+7+8+9=46
The sum of the indicators from 20 to 29 is
2×( 1+ 1+2+3+4+5+6+7+8+9)=2×46
The sum of indicators from 30 to 39 is
3×( 1+ 1+2+3+4+5+6+7+8+9)=3×46
The sum of the indices from 40 to 49 is
4×( 1+ 1+2+3+4+5+6+7+8+9)=4×46
The sum of 50 ~ 59 indicators is
5×( 1+ 1+2+3+4+5+6+7+8+9)=5×46
The sum of the indices from 60 to 69 is
6×( 1+ 1+2+3+4+5+6+7+8+9)=6×46
The sum of the indices of 70 ~ 79 is
7×( 1+ 1+2+3+4+5+6+7+8+9)=7×46
The sum of 80 ~ 89 indicators is
8×( 1+ 1+2+3+4+5+6+7+8+9)=8×46
The sum of the indices from 90 to 99 is
9×( 1+ 1+2+3+4+5+6+7+8+9)=9×46
Therefore, the sum of the indicators of 1 ~ 99 is
45+( 1+2+3+4+5+6+7+8+9)×46=45×47=2 1 15
8. Replace y=a2+bx+c with X = 1 and Y =-2 to get a+b+c=-2 ①.
Substitute X =- 1 and Y = 20 into y=ax2+bx+c to get a-b+c=20 ②.
①-②, 2b=-22, so b =- 1 1. So a+c = 9. So ...
a b+ BC+9 B2 = b(a+c)+9 B2 =(- 1 1)×(9)+9× 1 12 = 990。
9. By definition,
∴<； & lt48 >； ×& lt； 6.7 & gt-& lt； 10. 1 & gt； & gt= & lt 15×3-4 & gt； = & lt4 1 & gt； = 13.
10. Let the number corresponding to k0 point be x, then
(x- 1)+2-3+4-5+6-, …, -99+ 100= 19.94, that is, x+50= 19.94.
∴x=-30.06.
Third, answer questions.
1. solution: let a small rectangle be x in length and y in width, as shown in figure 1 1.
x+3y= 14 ①
X+y-2y=6, that is, x-y=6 ②.
4y=8, from ① to ②y = 2, substitute ②x = 8.
Therefore, the width of the large rectangular ABCD is AD = 6+2Y = 6+2× 2 = 10.
The area of the rectangle ABCD =14×10 =140 (square centimeter).
Total shadow area = 140-6×2×8=44 (square centimeter).
2. Solution: (1) Take a point o on the plane and draw a straight line AOB through it. The vertex of the template coincides with O at 19, one side coincides with the OB ray, and the other side falls on the ray OB 1, still taking O as the vertex, and one side of the corner coincides with OB 1
∠BOB 19 is the angle of 1
(2) Using the template with the angle of 17, the key to drawing the angle of 1 is to find the integers m and n so that17× m-180× n =1.
Actually,17× 53-180× 5 = 901-900 =1. So the method is as follows:
Take any point O on the plane, draw a straight line AOB through the point O, with OB as the starting edge and O as the vertex, and draw 53 angles of 17 in turn counterclockwise. Let the last edge be OB53, and the last edge of 5× 180 be in the OA ray, then ∠AOB53 is 1.
(3) If the angle of 1 can be drawn with the template of 2 1, there are integers m, n, so
2 1 ×m- 180 ×n= 1
(Only one of the four conclusions of each question below is correct. )
1. When a =-0.0 1, in-(-a) 2, -|-a |, -a2, -(-a2), the positive value is [].
A.-(-a)2 B.-|-a|。 C.-a2 D.-(-a2)
2. If =0, then rational numbers A, b[]
A. all zeros B. reciprocal C. reciprocal D. incomplete zeros
3. The positions of the five rational numbers A, B, C, D and E on the number axis are shown in Figure 5: Then A+B-D × C÷÷e equals [].
A.-8.5 B- 4。 C.5 D.8.5
4. if a < 0 and ab < 0, | b-a+ 1 |-a-b-5 | is equal to [].
A.4. B.-4。 C.-2a+2b+6。 D. 1996
The distance between a and b is s kilometers. The speeds of a and b are a km/h and b km/h respectively (a > b).
Everyone goes to a meeting from A to B. If A leaves 1 hour earlier than B, then the number of hours when B arrives at B later than A is [].
A.； b； c； d。
6. If |x|=a, | x-a | = []
A.2x or 2a b.x-a.c.a-x d.zero
7. Let the equation A (X-A)+B (X+B) = 0 about X have infinite solutions, then []
a . a+b = 0； b . a-b = 0； c . ab = 0； D. =0。
8. After deleting two addends, the sum of the remaining four addends is exactly equal to 1, so the deleted two addends are [].
A., ; b .，； c .，； d。
9. If the solution of equation 3(x+4)=2a+5 about x is greater than the solution of equation about x, then [] a.a >; 2; B.a & lt2; C.a & lt； D.a & gt。
10. After adding a glass of water to a certain concentration, a new brine with a concentration of 20% is obtained. When adding pure salt with the same weight as the above glass of water to the new brine, the brine concentration becomes 33%, and the original brine concentration is [] A.23%; b . 25%； c . 30%； D.32%。
Second, fill in the blanks
1 1. If (x- 1996) 2+(7+y) 2 = 0, then x+y3 = _ _ _ _.
12. Natural numbers m and n are two different prime numbers, and the minimum value of m+n+mn is p, so = _ _ _ _.
13. There are two acute angles and an obtuse angle in the angle, and their values are given. When calculating the numerical value, the whole class got three different results, 23.50, 24.50 and 25.50, among which there were correct answers, so = _ _ _ _.
14. It is known that the sum A+B and the difference A-B of rational numbers A and B are shown in Figure 6 on the number axis, then | 2a+B |-2 | A |-B-7 | is simplified, and the obtained value is _ _ _ _.
15. In the rectangular ABCD, m is the midpoint of the CD edge, an arc is centered on A, an arc is centered on B, an = A, bn = B, then the area of the shaded part in Figure 7 is _ _ _ _ _.
16. The lengths of the fast train and the slow train are150m and 200m respectively, and they run on parallel tracks. If the time for people sitting on the local train to see the express train passing through the window is 6 seconds, then the time for people sitting on the local train to see the local train passing through the window is _ _ _ _ _ _.
17. If the base A of the triangle is increased by 3cm and the area remains unchanged after the height ha on the base decreases by 3cm, then HA-A = _ _ _ _ _ cm.
18. The full score of a math exam is 100, and the average score of 38 students in the class is 67. If the scores of A, B, C, D and E are removed, the average score of others is 62, so in this test, the score of C is _ _ _ _ _ _.
19. It takes _ _ _ _ _ minutes from 3: 00, 15, until the hour hand and the minute hand form an angle of 30 for the first time.
20. Party A and Party B start from point A of the 400-meter circular runway at the same time. Eight minutes later, they met for the third time. It is known that Party A travels 0. 1 m more than Party B every second, so the shortest distance from the place where they met for the third time to point A along the runway is _ _ _ _ _.
Third, answer questions.
2 1.( 1) Please write the sum of prime numbers of natural numbers not exceeding 30.
(2) Please answer, how many even natural numbers * * * are there in 1?
(3) The thousands of four even natural numbers are 1. When it is divided by four different prime numbers, the remainder is also 1. Try to find all the natural numbers that meet these conditions. What is the biggest one?
22.( 1) Use 1× 1, 2×2, 3×3 three types of square floor tiles to pave a 23×23 square floor. Please design an auxiliary scheme, and only make a floor tile of 1× 1
(2) Please prove that there are only 2×2 and 3×3 floor tiles, and 23×23 square floor cannot be paved without gaps.
Answers and tips
First, multiple choice questions
Tip:
1. When < 0, (-a) 2 > 0, |-a | > 0, A2 > 0.
So -(-a) 2 < 0, -|-a | < 0, -a2 < 0, so exclude A, B, C and choose D. 。
In fact, when a < 0, A2 > 0, -(-A2) > 0. Of course, this is especially true when a =-0.0 1.
3.a=-3，b=-6，c=- 1，d=2，e=4，
A+B-D× C ÷ E = (-3)+(-6)-2× (-1) ÷ 4 =-8.5, choose A. 。
4. b-a>0 can be known from a < 0 and ab < 0, so b-a > 0,
b-a+ 1>0，a-b