Q.1 Use a suitable identity to get each of the following products.
(i) (ii)
(iii)
(iv) (v)
(vi)
(vii) (viii)
(ix)
(x)
Sol. (i)
=
= (Using identity
)
=
(ii)
=
= (Using identity
)
=
(iii)
=
= (Using identity
)
=
(iv)
=
= (Using identity
)
=
(v)
=
= (Using identity
)
(vi)
= (Using identity
)
=
(vii)
=
(Using identity
)
=
(viii)
=
=
= (Using identity
)
=
(ix)
=
= (Using identity
)
=
(x)
=
= (Using identity
)
=
Q.2 Use the identity (x + a) (x + b) = x2 + (a + b)x + ab to find the following products.
(i) (ii)
(iii) (iv)
(v) (vi)
(vii)
Sol. (i)
= (Using identity
)
=
(ii)
= (Using identity
)
=
(iii)
= (Using identity
)
=
(iv)
= (Using identity
)
=
(v)
= (Using identity
)
=
(vi)
= (Using identity
)
=
(vii)
= (Using identity
)
=
Q.3 Find the following squares by using the identities.
(i) (ii)
(iii)
(iv) (v)
(vi)
Sol. (i)
= (Using identity
)
=
(ii)
= (Using identity
)
=
(iii)
= (Using identity
)
=
(iv)
= (Using identity
)
=
(v)
= (Using identity
)
=
(vi)
= (Using identity
)
=
Q.4 Simplify
(i) (ii)
(iii) (iv)
(v) (vi)
(vii)
Sol. (i)
= (Using identity
)
=
(ii)
= (Using identity
and
)
=
=
=
(iii)
= (Using identity
and
)
=
=
=
(iv)
= (Using identity
)
=
=
(v)
= (Using identity
)
=
=
(vi)
= (Using identity
)
=
=
(vii)
= (Using identity
)
=
=
Q.5 Show that.
(i) (ii)
(iii) (iv)
(v)
Sol. (i)
LHS =
= (Using identity
)
=
=
RHS =
= (Using identity
)
=
We can see that, LHS = RHS
Hence,
(ii)
LHS =
= (Using identity
)
=
=
RHS =
= (Using identity
)
=
We can see that, LHS = RHS
Hence,
(iii)
LHS =
=
=
=
= RHS
Hence,
(iv)
LHS =
= (Using identity
and
)
=
=
=
= RHS
Hence,
(v)
LHS =
=
= 0
= RHS
Hence,
Q.6 Using identities, evaluate.
(i) 712 (ii)992 (iii)1022 (iv)9982
(v)5.22 (vi)297×303 (vii) 78×82 (viii)8.92
(ix) 1.05 × 9.5
Sol. (i) 712
712 = (70 + 1)2 = (70)2 + 2 × 70 × 1 + (1)2 (Using identity)
= 4900 + 140 + 1 = 5041
(ii) 992
992 = (100 - 1)2 = (100)2 - 2 × 100 × 1 + (1)2 (Using identity)
= 10000 - 200 + 1 = 9801
(iii) 1022
1022 = (100 + 2)2 = (100)2 + 2 × 100 × 2 + (2)2 (Using identity)
= 10000 + 400 + 4 = 10404
(iv) 9982
9982 = (1000 - 2)2 = (1000)2 - 2 × 1000 × 2 + (2)2 (Using identity)
= 100000 - 4000 + 4 = 996004
(v)5.22
5.22 = (5 + 0.2)2 = (5)2 + 2 × 5 × 0.2 + (0.2)2 (Using identity)
= 25 + 2 + 0.04 = 27.04
(vi)297×303
297×303 = (300 - 3) × (300 + 3)
= (300)2 – (3)2 (Using identity)
= 90000 – 9 = 89991
(vii) 78×82
78×82 = (80 - 2) × (80 + 2)
= (80)2 – (2)2 (Using identity)
= 6400 – 4 = 6396
(viii)8.92
8.92 = (8 + 0.9)2 = (8)2 + 2 × 8 × 0.9 + (0.9)2 (Using identity)
= 64 + 14.4 + 0.81 = 79.21
(ix) 1.05 × 9.5
1.05 × 9.5= (1 + 0.05) × (1 – 0.05) × 10
= [(1)2 - (0.05)2 ] × 10 (Using identity)
= 0.9975 × 10 = 9.975
Q.7 Using a2 – b2 = (a + b) (a – b), find
(i) (ii)
(iii)
(iv)
Sol. (i)
= (Using identity
)
=
= 200
(ii)
= (Using identity
)
=
= 0.08
(iii)
= (Using identity
)
=
= 1800
(iv)
= (Using identity
)
=
= 84
Q.8 Using (x + a) (x + b) = x2 + (a + b) x + ab, find
(i) 103 × 104 (ii) 5.1 × 5.2 (iii) 103 × 98 (iv) 9.7 × 9.8
Sol. (i) 103 × 104
= (100 + 3) × (100 + 4)
= (100)2 + (3 + 4) ×100 + 3 × 4 (Using identity (x + a) (x + b) = x2 + (a + b) x + ab)
= 10000 + 7 × 100 + 12
= 10000 + 700 + 12
= 10712
(ii) 5.1 × 5.2
= (5 + 0.1) × (5 + 0.2)
= (5)2 + (0.1 + 0.2) ×5 + 0.1 × 0.2 (Using identity (x + a) (x + b) = x2 + (a + b) x + ab)
= 25 + 0.3 × 5 + 0.02
= 25 + 1.5 + 0.02
= 26.52
(iii) 103 × 98
= (100 + 3) × (100 - 2)
= (100)2 + (3 - 2) ×100 + 3 × (-2) (Using identity (x + a) (x + b) = x2 + (a + b) x + ab)
= 10000 + 100 - 6
= 10094
(iv) 9.7 × 9.8
= (10 - 0.3) × (10 - 0.2)
= (10)2 + ((-0.3) + (-0.2)) ×10 + (-0.3) × (-0.2) (Using identity (x + a) (x + b) = x2 + (a + b) x + ab)
= 100 - 5 + 0.06
= 95.06