Integers (Exercise 1.1) RD Sharma Solutions | Mathematics (Maths) Class 7 PDF Download

Q.1. Determine each of the following products:

(i) 12 ☓ 7

(ii) (−15) ☓ 8

(iii) (−25) ☓ (−9)

(iv) 125 ☓ (−8)

Ans: 

(i) 12 × 7 = 84

(ii) (−15) × 8 =  −120

(iii) (−25) × (−9) =  225

(iv) 125 × (−8) =  −1000

Q.2. Find each of the following products:

(i) 3 ☓ (−8) ☓ 5

(ii) 9 ☓ (−3) ☓ (−6)

(iii) (−2) ☓ 36 ☓ (−5)

(iv) (−2) ☓ (−4) ☓ (−6) ☓ (−8)

Ans: 

(i) 3 × (−8) × 5 = −3×(8×5) = −120

(ii) 9 × (−3) × (−6) = 9× (3×6) = 162

(iii) (−2) × 36 × (−5) = 36 × (2×5) = 360

(iv) (−2) × (−4) × (−6) × (−8) =  (2×4×6×8) = 384

Q.3. Find the value of:

(i) 1487 × 327 + (−487) × 327

(ii) 28945 × 99 − (−28945)

Ans: 

(i) 1487 × 327 + (−487) × 327 = 327 (1487 − 487) = 327 ×1000 =327000

(ii) 28945 × 99 − (−28945) = 28945 (99 −(−1)) = 28945 (99 + 1) = 2894500

Q.4. Complete the following multiplication table:

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

Ans: 

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

Q.5. Determine the integer whose product with '−1' is

(i) 58

(ii) 0

(iii) −225

Ans: 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) = − (0×1) = 0

(iii) (−225) × (−1) =  225

Q.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?

Ans: 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 = positive × positive = positive integer

(ii) (negative) 21 times  × (positive) 3 times = negative × positive  = negative integer

(iii) (negative) 199 times × (positive) 10 times = negative × positive = negative integer

Q.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)

Ans: 

(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

Q.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?

Ans: 

(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

Q.9. Verify the following:

(i) 19 × {7 + (−3)} = 19 × 7 + 19 × (−3)

(ii) (−23) {(−5) + (+19)} = (−23) × (−5) + (−23) × (+19)

Ans: 

(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.

Q.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.

Ans: 

(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, a×b ≠ b×a ≠b