Ex-1.1 Integers, Class 7, Math RD Sharma Solutions | RD Sharma Solutions for Class 7 Mathematics PDF Download

Q 1. Determine each of the following products:
(i) 12 ☓ 7
(ii) (−15) ☓ 8
(iii) (−25) ☓ (−9)
(iv) 125 ☓ (−8)

(i) 12 × 7 = 84
(ii) (−15) × 8 =  −120
(iii) (−25) × (−9) =  225
(iv) 125 × (−8) =  −1000

(i) 3 × (−8) × 5 = −3 × (8 × 5) - 3 × (8 × 5) = −-120
(ii) 9 × (−3) × (−6) = 9 × (3 × 6) 9 × (3 × 6) = 162
(iii) (−2) × 36 × (−5) = 36 × (2 × 5)36 × (2 × 5) = 360
(iv) (−2) × (−4) × (−6) × (−8) =  (2 × 4 × 6 × 8)(2 × 4 × 6 × 8) = 384

Q 3. Find the value of:
(i) 1487 × 327 + (−487) × 327
(ii) 28945 × 99 − (−28945)

(i) 1487 × 327 + (−487) × 327 = 327 (1487 − 487) = 327 × 1000 = 327000327 (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? 

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

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 × 1) = 0
(iii) (−225) × (−1) =  225

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 × positive = positive integer
(ii) (negative) 21 times  × (positive) 3 times = negative ×positive negative ×positive  = negative integer
(iii) (negative) 199 times × (positive) 10 times = negative × positive negative × positive = negative integer 

(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

(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

(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.

(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