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 = 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 9
Solution 10
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 × 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 15
Solution 16
Solution 17
Solution 18
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
2a2 + 2b2 + 2c2 - 2ab - 2bc - 2ca = 0
a2 + a2 + b2 + b2 + c2 + 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 20
Solution 21
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).
Solution 23
Algebraic Identities Exercise 4.32
Solution 24
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).
Solution 26
Solution 27
