RD Sharma solution class 7 chapter 1 Integers

EXERCISE 1.1

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Question 1:

Determine each of the following products:
(i) 12 ☓ 7
(ii) (−15) ☓ 8
(iii) (−25) ☓ (−9)
(iv) 125 ☓ (−8)

Answer 1:

(i) 12 × 7 = 84

(ii) (−15) × 8 =  −120

(iii) (−25) × (−9) =  225

(iv) 125 × (−8) =  −1000

Question 2:

Find each of the following products:
(i) 3 ☓ (−8) ☓ 5
(ii) 9 ☓ (−3) ☓ (−6)
(iii) (−2) ☓ 36 ☓ (−5)
(iv) (−2) ☓ (−4) ☓ (−6) ☓ (−8)

Answer 2:

(i) 3 × (−8) × 5 = $-3\times (8\times 5)$ = $-$120

(ii) 9 × (−3) × (−6) = $9\times (3\times 6)$= 162

(iii) (−2) × 36 × (−5) = $36 \times (2\times 5)$ = 360

(iv) (−2) × (−4) × (−6) × (−8) =  $(2\times 4\times 6\times 8)$ = 384

Question 3:

Find the value of:
(i) 1487 × 327 + (−487) × 327
(ii) 28945 × 99 − (−28945)

Answer 3:

(i) 1487 × 327 + (−487) × 327 = $327 (1487-487) = 327 \times 1000 =327000$

(ii) 28945 × 99 − (−28945) = $28945 (99 -(-1\left)\right) = 28945 (99+1) = 2894500$
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Question 4:

Complete the following multiplication table:

Is the multiplication table symmetrical about the diagonal joining the upper left corner to the lower right corner?

Answer 4:

×	−4	−3	−2	−1	0	1	2	3	4
−4	16	12	8	4	0	−4	−8	−12	−16
−3	12	9	6	3	0	−3	−6	−9	−12
−2	8	6	4	2	0	−2	−4	−6	−8
−1	4	3	2	1	0	−1	−2	−3	−4
0	0	0	0	0	0	0	0	0	0
1	−4	−3	−2	−1	0	1	2	3	4
2	−8	−6	−4	−2	0	2	4	6	8
3	−12	−9	−6	−3	0	3	6	9	12
4	−16	−12	−8	−4	0	4	8	12	16

Yes, the table is symmetrical along the diagonal joining the upper left corner to the lower right corner.

Question 5:

Determine the integer whose product with '−1' is
(i) 58
(ii) 0
(iii) −225

Answer 5:

The integer, whose product with −1 is the given number, can be found by multiplying the given number by −1.

Thus, we have:

(i) 58 × (−1) =  −58

(ii) 0 ​× (−1) = $- (0\times 1)$= 0

(iii) (−225) ​× (−1) =  225

Question 6:

What will be the sign of the product if we multiply together
(i) 8 negative integers and 1 positive integer?
(ii) 21 negative integers and 3 positive integers?
(iii) 199 negative integers and 10 positive integers?

Answer 6:

Negative numbers, when multiplied even number of times, give a positive number. However, when multiplied odd number of times, they give a negative number. Therefore, we have:

(i) (negative) 8 times × (positive)  1 time = $\mathrm{positive} \times \mathrm{positive}$ = positive integer

(ii) (negative) 21 times  × (positive) 3 times = $\mathrm{negative} \times \mathrm{positive}$ = negative integer

(iii) (negative) 199 times × (positive) 10 times = $\mathrm{negative} \times \mathrm{positive}$= negative integer

Question 7:

State which is greater:
(i) (8 + 9) × 10 and 8 + 9  × 10
(ii) (8 − 9) × 10 and 8 − 9 × 10
(iii) {(−2) − 5} × (−6) and (−2) −5 × (−6)

Answer 7:

(i) ( 8 + 9) × 10 = 170   >   8 + 90 = 98

(ii) (8 − 9) ​× 10 = −10  >  8 − 90 = − 82

(iii) {(−2) − 5 } ​× (−6) = −7 ​× (−6) = 42  >   (−2) − 5 × (−6)  = ( −2 ) −  (−30)  = −2 + 30 = 28

Question 8:

(i) If a × (−1) = −30, is the integer a positive or negative?
(ii) If a × (−1) = 30, is the integer a positive or negative?

Answer 8:

(i) a × (−1) = −30

When multiplied by a negative integer, a gives a negative integer. Hence, a should be a positive integer.

a = 30

(ii) a × (−1) = 30  

​When multiplied by a negative integer, a gives a positive integer. Hence, a should be a negative integer​.

a = −30

Question 9:

Verify the following:
(i) 19 × {7 + (−3)} = 19 × 7 + 19 × (−3)
(ii) (−23) {(−5) + (+19)} = (−23) × (−5) + (−23) × (+19)

Answer 9:

(i)
LHS = 19 ​× { 7 + (−3) } = 19 × {4} =  76
 
RHS =  19 × 7 + 19 × (−3) = 133 + (−57) = 76

Because LHS is equal to RHS, the equation is verified.

(ii)
LHS = (−23) {(−5) + 19} = (−23) { 14} = −322

RHS = (−23) × (−5) + (−23) × 19 = 115 + (−437) = −322

Because LHS is equal to RHS, the equation is verified.

Question 10:

Which of the following statements are true?
(i) The product of a positive and a negative integer is negative.
(ii) The product of three negative integers is a negative integer.
(iii) Of the two integers, if one is negative, then their product must be positive.
(iv) For all non-zero integers a and b, a × b is always greater than either a or b.
(v) The product of a negative and a positive integer may be zero.
(vi) There does not exist an integer b such that for a> 1, a × b = b × a = b.

Answer 10:

(i) True. Product of two integers with opposite signs give a negative integer.

(ii) True. Negative integers, when multiplied odd number of times, give a negative integer.

(iii) False. Product of two integers, one of them being a negative integer, is not necessarily positive. For example, (−1) × 2 = −2

(iv) False. For two non-zero integers a and b, their product is not necessarily greater than either a or b. For example, if a = 2 and  b = −2, then, a × b = −4, which is less than both 2 and −2.

(v) False. Product of a negative integer and a positive integer can never be zero.

(vi) True. If a > 1, then, $\mathrm{a}\times \mathrm{b} \ne \mathrm{b}\times \mathrm{a} \ne \mathrm{b}$