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Class 9 RD SHARMA Solutions Maths Chapter 4 - Algebraic Identities

Algebraic Identities Exercise 4.30

Solution 8

We know that (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

Here, a + b + c = 9, ab + bc + ca = 23

Thus, we have

(9)2 = a2 + b2 + c2 + 2(23)

81 = a2 + b2 + c2 + 46

a2 + b2 + c2 = 81 - 46

a2 + b2 + c2 = 35

Hence, correct option is (a).

Solution 11

a - b = -8

(a - b)2 = 64

a2 + b2 - 2ab = 64

a2 + b2 - 2ab + 3ab = 64 + 3ab

a2 + b2 + ab = 64 + 3(-12)

a2 + b2 + ab = 64 - 36

a2 + b2 + ab = 28

Now a3 - b3 = (a - b)(a2 + b2 + ab)

= (-8)(28)

= -224

Hence, correct option is (c).

Algebraic Identities Exercise 4.31

Solution 12

Volume of a cuboid of side a, b and c = abc

Now, Volume = 3x2 - 27    (given)

abc = 3(x2 - 9)

abc = 3(x - 3)(x + 3)

So, possible dimensions are 3, x - 3 and x + 3

Hence, correct option is (b).

Solution 13

Given expression is 75 × 75 + 2 × 75 × 25 + 25 × 25

Let 75 = a and 25 = b

Then, we have

× a + 2 × a × b + b × b

a2 + 2ab + b2

= (a + b)2

= (75 + 25)2

= (100)2

= 10000

Hence, correct option is (a).

Solution 14

(x - y)(x + y) = x2 - y2  [by identity (a + b)(a - b) = a2 - b2]

(x2 - y2)(x2 + y2) = x4 - y4

(x4 - y4)(x4 + y4) = x8 - y8

Now,

(x - y)(x + y)(x2 + y2)(x4 + y4)

= (x2 - y2)(x2 + y2)(x4 + y4)

= (x4 - y4)(x4 + y4)

= x8 - y8

Hence, correct option is (b).

Solution 19

a2 + b2 + c2 - ab - bc - ca = 0

Multiplying by 2 on both the sides, we have

2(a2 + b2 + c2 - ab - bc - ca) = 0

2a+ 2b2 + 2c2 - 2ab - 2bc - 2ca = 0

a2 + a2 + b2 + b2 + c+ c2 - 2ab - 2bc - 2ca = 0

(a2 + b2 - 2ab) + (b2 + c2 - 2bc) + (a2 + c2 - 2ac) = 0

(a - b)2 + (b - c)2 + (a - c)2 = 0

(a - b)2 = 0(b - c)2 = 0, (a - c)2 = 0

(a - b) = 0(b - c) = 0, (a - c) = 0

a = b, b = c, a = c

or we can say a = b = c

Hence, correct option is (d).

Solution 22

Given, a + b + c = 9

Hence, (a + b + c)2 = 81

So, a2 + b2 + c2 + 2ab + 2bc + 2ca = 81

i.e. a2 + b2 + c2 + 2(ab + bc + ca) = 81

i.e. a2 + b2 + c2 + 2(23) = 81

i.e. a2 + b2 + c2 = 81 - 46 = 35

Now, a3 + b3 + c3 - 3abc

= (a + b + c)(a2 + b2 + c2 - ab - bc - ca)

= (a + b + c)[(a2 + b2 + c2) - (ab + bc + ca)]

= (9)[35 - 23]

= 9 × 12

= 108

Hence, correct option is (a).

Algebraic Identities Exercise 4.32

Solution 25

Given expression is (x2 - 1)(x4 + x2 + 1)

Let x2 = A and 1 = B

Then, we have

(A - B)(A2 + AB + B2)

= A3 - B3

= (x2)3 - (1)3

= x6 - 1

Hence, correct option is (c).