1. Evaluate the product when = for 2 ( − ) 2. Simplify: ( − ) + ( − ) + ( − ) 3. Evaluate using identities: (i) 2 25 + 2 16 − 10 when = 1 and = 2 (ii) (0.5 + )(0.5 − ) when = 2 and = 5 (iii) 81 2 + 16 2 − 72 when = 4 and = 1 4. Find ( − ) 2 and ( + ) 2 if 2 + 2 = 49 and = 16 5. Show that (9 − 5) 2 + 180 = (9 + 5) 2 6. Divide 2 + 14 + 49 by + 7 7. Divide 39 2 + 17 + 3 − 57 by − 1 8. The difference between the squares of two consecutive numbers is 31. Find the numbers. 9. Is 784 a perfect square? 10.Express 64 as the sum of first n odd natural numbers and find the value of n. 11.Without actually finding the squares of the numbers, find the value of (153)2 -(152)2 12.Find a Pythagorean triplet whose one member is 20. 13.Find the squares of the following without actual multiplication. a) 98 b)103 14.Find the least number to be multiplied with 250 to make it as a perfect square. 15.Find the least number by which 20172 should be divided to get a perfect square.
Answers
Answer:
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Question 1:
Evaluate:
(i) 15 + (−8)
(ii) (−16) + 9
(iii) (−7) + (−23)
(iv) (−32) + 47
(v) 53 + (−26)
(vi) (−48) + (−36)
ANSWER:
(i) 15 + (−8) = 7
(ii) (−16) + 9 = −7
(iii) (−7) + (−23) = −30
(iv) (−32) + 47 = 15
(v) 53 + (−26) = 27
(vi) (−48) + (−36) = −84
Page No 4:
Question 2:
Find the sum of:
(i) 153 and − 302
(ii) 1005 and − 277
(iii) − 2035 and 297
(iv) − 489 and − 324
(v) − 1000 and 438
(vi) − 238 and 500
ANSWER:
(i) 153 + (−302) = −149
(ii) 1005 + (−277) = 728
(iii) (−2035) + 297 = −1738
(iv) (−489) + (−324) = −813
(v) (−1000) + 438 = −562
(vi) (−238) + 500 = 262
Page No 4:
Question 3:
Find the additive inverse of:
(i) − 83
(ii) 256
(iii) 0
(iv) − 2001
ANSWER:
(i) Additive inverse of −83 = −(−83) = 83
(ii) Additive inverse of 256 = −(256) = −256
(iii) Additive inverse of 0 = −(0) = 0
(iv) Additive inverse of 2001 = −(−2001) = 2001
Page No 5:
Question 4:
Subtract:
(i) 28 from − 42
(ii) − 36 from 42
(iii) − 37 from − 53
(iv) − 66 from − 34
(v) 318 from 0
(vi) − 153 from − 240
(vii) − 64 from 0
(viii) − 56 from 144
ANSWER:
(i) −42 − 28 = (−42) + (−28) = −70
(ii) 42 −(−36) = 42 + 36 = 78
(iii) -53 - (-37) = (-53) - (-37) = -16
(iv) -34 - (-66) = -34 + 66 = 32
(v) 0 - 318 = -318
(vi) (-240) - (-153) = -87
(vii) 0 - (-64) = 0 + 64 = 64
(viii) 144 - (-56) = 144 + 56 = 200
Page No 5:
Question 5:
Subtract the sum of − 1032 and 878 from − 34.
ANSWER:
Sum of −1032 and 878 = −1032 + 878
= -154
Subtracting the sum from −34, we get
−34 − (−154)
= (−34)+ 154
= 120
Page No 5:
Question 6:
Subtract − 134 from the sum of 38 and − 87.
ANSWER:
First, we will calculate the sum of 38 and −87.
38 + (−87) = −49
Now, subtracting −134 from the sum, we get:
−49 − (−134)
=(−49) + 134
= 85
Page No 5:
Question 7:
Fill in the blanks:
(i) {(−13) + 27} + (−41) = (−13) + {27 + (......)}
(ii) (−26) + {(−49) + (−83)} = {(−26) + (−49)} + (......)
(iii) 53 + (−37) = (−37) + (......)
(iv) (−68) + (−76) = (......) + (−68)
(v) (−72) + (......) = −72
(vi) − (−83) = ......
(vii) (−60) − (......) = − 59
(viii) (−31) + (......) = − 40
ANSWER:
(i) −41 (∵ Associative property)
(ii) −83 (∵ Associative property)
(iii) 53 (∵ Commutative property)
(iv) −76 (∵ Commutative property)
(v) 0 (∵ Additive identity)
(vi) 83 (∵ Additive inverse)
(vii) (−60) − (−59) = −1
(viii) (−40) − (−31) = −9
Page No 5:
Question 8:
Simplify:
{−13−(−27)} + {−25−(−40)}.
ANSWER:
{−13 − (−27)} + {−25 − (−40)}
= {−13 + 27} + {−25 + 40}
=14 + 15
= 29
Page No 5:
Question 9:
Find 36 − (−64) and (−64) − 36. Are they equal?
ANSWER:
36 − (−64) = 36 + 64 = 100
Now, (−64) − 36 = (−64) + (−36) = −100
Here, 100 ≠ −100
Thus, they are not equal.
Page No 5:
Question 10:
If a = − 8, b = − 7, c = 6, verify that (a+b) + c = a + (b+c).
ANSWER:
(a + b) + c = (−8 + (−7)) + 6 = −15 + 6 = −9
a + (b + c) = −8 + (−7 + 6) = −8 + (−1) = −9
Hence, (a + b) + c = a + (b + c) [i.e., Property of Associativity]
Page No 5:
Question 11:
If a = − 9 and b = − 6, show that (a−b) ≠ (b−a).
ANSWER:
Here, (a − b) = −9 − (−6) = −3
Similarly, (b − a) = −6 − (−9) = 3
∴ (a−b) ≠ (b−a)
Page No 5:
Question 12:
The sum of two integers is − 16. If one of them is 53, find the other.
ANSWER:
Let the other integer be a. Then, we have:
53 + a = −16
⇒ a = −16 − 53 = −69
∴ The other integer is −69.
Page No 5:
Question 13:
Ths sum of two integers is 65. If one of them is − 31, find the other.
ANSWER:
Let the other integer be a.
Then, −31 + a = 65
⇒ a = 65 − (−31) = 96
∴ The other integer is 96.
Page No 5:
Question 14:
The difference of an integer a and (−6) is 4. Find the value of a.
ANSWER:
We have:
a − (−6) = 4
⇒ a = 4 + (−6) = −2
∴ a = −2
Page No 5:
Question 15:
Write a pair of integers whose sum gives
(i) zero;
(ii) a negative integer;
(iii) an integer smaller than both the integers;
(iv) an integer greater than both the integers;
(v) an integer smaller than only one of the integers.
ANSWER:
(i) Consider the integers 8 and −8. Then, we have:
8 + (−8) = 0
(ii) Consider the integers 2 and (−9). Then, we have:
2 + (−9)= −7, which is a negative integer.
(iii) Consider the integers −4 and −5. Then, we have:
(−4) + (−5) = −9, which is smaller than −4 and −5.
(iv) Consider the integers 2 and 6. Then, we have:
2 + 6 = 8, which is greater than both 2 and 6.
(v) Consider the integers 7 and −4. Then, we have:
7 + (−4) = 3, which is smaller than 7 only.
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