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**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) = x^{2 }+ (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) 71^{2} (ii)99^{2} (iii)102^{2} (iv)998^{2}

(v)5.2^{2} (vi)297Ã—303 (vii) 78Ã—82 (viii)8.9^{2}

(ix) 1.05 Ã— 9.5

** Sol.** (i) 71

71^{2} = (70 + 1)^{2} = (70)^{2 }+ 2 Ã— 70 Ã— 1 + (1)^{2 }(Using identity)

= 4900 + 140 + 1 = 5041

(ii) 99^{2}

99^{2} = (100 - 1)^{2} = (100)^{2 }- 2 Ã— 100 Ã— 1 + (1)^{2} (Using identity)

= 10000 - 200 + 1 = 9801

(iii) 102^{2}

102^{2} = (100 + 2)^{2} = (100)^{2 }+ 2 Ã— 100 Ã— 2 + (2)^{2} (Using identity)

= 10000 + 400 + 4 = 10404

(iv) 998^{2}

998^{2} = (1000 - 2)^{2} = (1000)^{2 }- 2 Ã— 1000 Ã— 2 + (2)^{2 }(Using identity)

= 100000 - 4000 + 4 = 996004

(v)5.2^{2}

5.2^{2} = (5 + 0.2)^{2} = (5)^{2 }+ 2 Ã— 5 Ã— 0.2 + (0.2)^{2} (Using identity)

= 25 + 2 + 0.04 = 27.04

(vi)297Ã—303

297Ã—303 = (300 - 3) Ã— (300 + 3)

= (300)^{2} â€“ (3)^{2} (Using identity)

= 90000 â€“ 9 = 89991

(vii) 78Ã—82

78Ã—82 = (80 - 2) Ã— (80 + 2)

= (80)^{2} â€“ (2)^{2} (Using identity)

= 6400 â€“ 4 = 6396

(viii)8.9^{2}

8.9^{2} = (8 + 0.9)^{2} = (8)^{2 }+ 2 Ã— 8 Ã— 0.9 + (0.9)^{2} (Using identity)

= 64 + 14.4 + 0.81 = 79.21

(ix) 1.05 Ã— 9.5

1.05 Ã— 9.5= (1 + 0.05) Ã— (1 â€“ 0.05) Ã— 10

= [(1)^{2 }- (0.05)^{2} ] Ã— 10 (Using identity)

= 0.9975 Ã— 10 = 9.975

**Q.7 Using a^{2} â€“ b^{2} = (a + b) (a â€“ 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) = x^{2} + (a + b) x + ab, find **

(i) 103 Ã— 104 (ii) 5.1 Ã— 5.2 (iii) 103 Ã— 98 (iv) 9.7 Ã— 9.8

** Sol.** (i) 103 Ã— 104

= (100 + 3) Ã— (100 + 4)

= (100)^{2} + (3 + 4) Ã—100 + 3 Ã— 4 (Using identity *(x *+* a) (x *+* b) *=* x ^{2} *+

= 10000 + 7 Ã— 100 + 12

= 10000 + 700 + 12

= 10712

(ii) 5.1 Ã— 5.2

= (5 + 0.1) Ã— (5 + 0.2)

= (5)^{2} + (0.1 + 0.2) Ã—5 + 0.1 Ã— 0.2 (Using identity *(x *+* a) (x *+* b) *=* x ^{2} *+

= 25 + 0.3 Ã— 5 + 0.02

= 25 + 1.5 + 0.02

= 26.52

(iii) 103 Ã— 98

= (100 + 3) Ã— (100 - 2)

= (100)^{2} + (3 - 2) Ã—100 + 3 Ã— (-2) (Using identity *(x *+* a) (x *+* b) *=* x ^{2} *+

= 10000 + 100 - 6

= 10094

(iv) 9.7 Ã— 9.8

= (10 - 0.3) Ã— (10 - 0.2)

= (10)^{2} + ((-0.3) + (-0.2)) Ã—10 + (-0.3) Ã— (-0.2) (Using identity *(x *+* a) (x *+* b) *=* x ^{2} *+

= 100 - 5 + 0.06

= 95.06