# Class 9 RD SHARMA Solutions Maths Chapter 4 - Algebraic Identities

## Algebraic Identities Exercise Ex. 4.1

### Solution 1(i)

### Solution 1(ii)

### Solution 1(iii)

### Solution 1(iv)

### Solution 1(v)

### Solution 2(i)

### Solution 2(ii)

### Solution 2(iii)

### Solution 2(iv)

### Solution 3(i)

### Solution 3(ii)

### Solution 3(iii)

### Solution 3(iv)

### Solution 4

### Solution 5

### Solution 6

### Solution 7

### Solution 8

### Solution 9

### Solution 10(i)

### Solution 10(ii)

### Solution 11

### Solution 12

### Solution 13(i)

### Solution 13(ii)

### Solution 13(iii)

### Solution 13(iv)

### Solution 14

## Algebraic Identities Exercise Ex. 4.2

### Solution 1(i)

### Solution 1(ii)

### Solution 1(iii)

### Solution 1(iv)

### Solution 1(v)

### Solution 1(vi)

### Solution 1(vii)

### Solution 1(viii)

### Solution 1(ix)

### Solution 1 (x)

### Solution 1 (xi)

### Solution 1 (xii)

### Solution 2

### Solution 3

### Solution 4

### Solution 5

### Solution 6(i)

### Solution 6(ii)

### Solution 6(iii)

### Solution 6(iv)

### Solution 6(v)

### Solution 7(i)

### Solution 7(ii)

### Solution 7(iii)

## Algebraic Identities Exercise Ex. 4.3

### Solution 1(i)

### Solution 1(ii)

### Solution 1(iii)

### Solution 1(iv)

### Solution 2

### Solution 3

### Solution 4

### Solution 5

### Solution 6

### Solution 7

### Solution 8

### Solution 9

### Solution 10

### Solution 11(i)

### Solution 11(ii)

### Solution 11(iii)

### Solution 11(iv)

### Solution 11(v)

### Solution 11(vi)

### Solution 12(i)

### Solution 12(ii)

### Solution 12(iii)

### Solution 12(iv)

### Solution 13

### Solution 14(i)

### Solution 14(ii)

### Solution 15

### Solution 16

### Solution 17(i)

### Solution 17(ii)

### Solution 17(iii)

### Solution 17(iv)

### Solution 18

### Solution 19

## Algebraic Identities Exercise Ex. 4.4

### Solution 1 (i)

### Solution 1 (ii)

### Solution 1 (iii)

### Solution 1 (iv)

### Solution 1 (v)

### Solution 1 (vi)

### Solution 1 (vii)

### Solution 1 (viii)

### Solution 1 (ix)

### Solution 1 (x)

### Solution 1 (xi)

### Solution 1 (xii)

### Solution 2 (i)

### Solution 2 (ii)

### Solution 2 (iii)

### Solution 2 (iv)

### Solution 2 (v)

### Solution 3

### Solution 4

### Solution 5

### Solution 6 (i)

### Solution 6 (ii)

### Solution 6 (iii)

## Algebraic Identities Exercise Ex. 4.5

### Solution 1 (i)

### Solution 1 (ii)

### Solution 1 (iii)

### Solution 1 (iv)

### Solution 2(i)

### Solution 2(ii)

### Solution 2(iii)

### Solution 2(iv)

### Solution 3

### Solution 4

### Solution 5

## Algebraic Identities Exercise 4.30

### Solution 1

### Solution 2

### Solution 3

### Solution 4

### Solution 5

### Solution 6

### Solution 7

### Solution 8

We know that (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2(ab + bc + ca)

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

Thus, we have

(9)^{2} = a^{2} + b^{2} + c^{2} + 2(23)

81 = a^{2} + b^{2} + c^{2} + 46

a^{2} + b^{2} + c^{2} = 81 - 46

a^{2} + b^{2} + c^{2} = 35

Hence, correct option is (a).

### Solution 9

### Solution 10

### Solution 11

a - b = -8

(a - b)^{2} = 64

a^{2} + b^{2} - 2ab = 64

a^{2} + b^{2} - 2ab + 3ab = 64 + 3ab

a^{2} + b^{2} + ab = 64 + 3(-12)

a^{2} + b^{2} + ab = 64 - 36

a^{2} + b^{2} + ab = 28

Now a^{3} - b^{3} = (a - b)(a^{2} + b^{2} + 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 = 3x^{2} - 27 (given)

abc = 3(x^{2} - 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 × a + 2 × a × b + b × b

= a^{2} + 2ab + b^{2}

= (a + b)^{2}

= (75 + 25)^{2}

= (100)^{2}

= 10000

Hence, correct option is (a).

### Solution 14

(x - y)(x + y) = x^{2} - y^{2} [by identity (a + b)(a - b) = a^{2} - b^{2}]

(x^{2} - y^{2})(x^{2} + y^{2}) = x^{4} - y^{4}

(x^{4} - y^{4})(x^{4} + y^{4}) = x^{8} - y^{8}

Now,

(x - y)(x + y)(x^{2} + y^{2})(x^{4} + y^{4})

= (x^{2} - y^{2})(x^{2} + y^{2})(x^{4} + y^{4})

= (x^{4} - y^{4})(x^{4} + y^{4})

= x^{8} - y^{8}

Hence, correct option is (b).

### Solution 15

### Solution 16

### Solution 17

### Solution 18

### Solution 19

a^{2} + b^{2} + c^{2} - ab - bc - ca = 0

Multiplying by 2 on both the sides, we have

2(a^{2} + b^{2} + c^{2} - ab - bc - ca) = 0

2a^{2 }+ 2b^{2} + 2c^{2} - 2ab - 2bc - 2ca = 0

a^{2} + a^{2} + b^{2} + b^{2 }+^{ }c^{2 }+ c^{2} - 2ab - 2bc - 2ca = 0

(a^{2} + b^{2} - 2ab) + (b^{2} + c^{2} - 2bc) + (a^{2} + c^{2} - 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 20

### Solution 21

### Solution 22

Given, a + b + c = 9

Hence, (a + b + c)^{2} = 81

So, a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca = 81

i.e. a^{2} + b^{2} + c^{2} + 2(ab + bc + ca) = 81

i.e. a^{2} + b^{2} + c^{2} + 2(23) = 81

i.e. a^{2} + b^{2} + c^{2} = 81 - 46 = 35

Now, a^{3} + b^{3} + c^{3} - 3abc

= (a + b + c)(a^{2} + b^{2} + c^{2} - ab - bc - ca)

= (a + b + c)[(a^{2} + b^{2} + c^{2)} - (ab + bc + ca)]

= (9)[35 - 23]

= 9 × 12

= 108

Hence, correct option is (a).

### Solution 23

## Algebraic Identities Exercise 4.32

### Solution 24

### Solution 25

Given expression is (x^{2} - 1)(x^{4} + x^{2} + 1)

Let x^{2} = A and 1 = B

Then, we have

(A - B)(A^{2} + AB + B^{2})

= A^{3} - B^{3}

= (x^{2})^{3} - (1)^{3}

= x^{6} - 1

Hence, correct option is (c).

### Solution 26

### Solution 27