IDENTITIES FORMULA AND FACTORISATION

SPECIAL PRODUCTS AS IDENTITIES FORMULA AND FACTORISATION. Find the product of : (i) (n + ) nd ((n + 5) ( + 0.) nd ( + 0.5) (iii) (y + 0.7) nd (y + 0.) (iv) + 3 nd + 3 (v) y + 5 nd 3 y + (iv) 5 + 7 nd + 7 Ans. (i) (n + ) (n + 5) n + ( + 5) n + 5 n + 7n + 80 ( + 0.) ( + 0.5) + ( 0. + 0.5) + 0. 0.5 + 0.9 + 0. (iii) (y + 0.7) (y + 0.) y + (0.7 + 0.) y + 0.7 0. y + 0.8y + 0.07 (iv) (v) + 3 nd + 3 + + + 3 3 3 3 3 + + + + 3 9 9 y + 5 nd y + 5 y + + y + y 5 5 5 5 + 6 y + 8 5 5 3 (vi) + 7 nd + 7 3 3 + + + + 5 + 6 7 7 7 7 7 9. Find the product of : (i) (n + 5) nd (n ) ( + 0.8) nd ( 0.5) 3 (iii) (y + 0.7) nd (y 0.) (iv) + nd Ans. (i) (n + 5) (n ) n + (5 ) n + 5 ( ) n + 3n 80 ICSE Mth Clss VII Question Bnk

( + 0.8) ( 0.5) + (0.8 0.5) + 0.8 ( 0.5) + 0.3 0. (iii) (y + 0.7) (y 0.) y + (0.7 0.) y + 0.7 ( 0.) y + 0.6y 0.07 (iv) 3. Find : 3 + + + 3 3 + 3 3 + 6 6 (i) ( 8) ( + 3) (.6) ( + 0.9) 7 (y 0.8) (y + 0.) (iv) + 5 5 Ans. (i) ( 8) ( + 3) + ( 8 + 3) + ( 8) (3) 5 5 (.6) ( + 0.9) + (.6 + 0.9) + (.6) (0.9) 0.7. (iii) (y 0.8) (y + 0.) y + ( 0.8 + 0.)y + ( 0.8) (0.) y 0.7y 0.08 (iv) 7 + 5 5 + 7 7 + + 5 5 5 5 5 5 5 3 5. Use the formul to find : (i) ( 5b) ( + 5b) (y 3pq) (y + 3pq) (iii) (0. + 0.5y) (0. 0.5y) (iv) + b b 3 3 Ans. (i) ( 5b) ( + 5b) () (5b) 5b (y 3pq) (y + 3pq) (y) (3pq) 6 y 9p q (iii) (0. + 0.5y) (0. 0.5y) (0.) (0.5y) 0.0 0.5y (iv) + b b 3 3 b 3 b 9 5. Using the identity ( + ) ( + b) + ( + b) + b evlute the following : (i) 36 09 06 9 (iii) 98 97 Ans. (i) 36 09 (00 + 36) (00 + 9) (00) + (36 + 9) 00 + 36 9 0000 + 5 00 + 3 0000 + 500 + 3 8 06 9 (00 + 6) (00 6) (00) + [(6) + ( 6)] 00 + 6( 6) 0000 + 0 00 36 0000 + 0 36 996 Mth Clss VII Question Bnk

(iii) 98 97 (00 ) (00 3) (00) + [( ) + ( 3)] 00 + ( ) ( 3) 0000 5 00 + 6 0000 500 + 6 9506 6. Epnd : (i) Ans. (i) (iii) y + y + 3 p + q 8. (i) Evlute (87) by using the epnsion for ( b) Ans. (87) (90 3) (90) + (3) (90) (3) 800 + 9 50 809 50 7569 Evlute (96) by using the epnsion for ( b). Ans. (96) (00 ) (00) + () (00) () 0000 + 6 800 006 800 96 9. Evlute ech of the following using ( + b) ( b) ( b ) : (i) 8 96 57 63 (iii) 8. 7.8 (iv) 6.5 7.5 Ans. (i) 8 96 (90 6) (90 + 6) (90) (6) 800 36 806 57 63 (60 3) (60 + 3) (60) (3) ICSE Mth Clss VII 3600 9 359 3 Question Bnk (iii) + ( y) + ( y) 3 3 3 ( p + q) ( p) + p q + q p + 6pq + 9 q 3 y 3 + 9 + 3 y y + 9 9 9 + y + y 6 3 8 3 y + 9 7. (i) Evlute (56) by using the epnsion for ( + b) Ans. (56) (50 + 6) (50) + (6) + (50) (6) 500 + 36 + 600 336 Evlute (73) by using the epnsion for ( + b). Ans. (73) (70 + 3) (70) + (3) + (70) (3) 900 + 9 + 0 539 6 + y + y

(iii) 8. 7.8 (8 + 0.) (8 0.) (8) (0.) 6 0.0 63.96 (iv) 6.5 7.5 (7 0.5) (7 + 0.5) (7) (0.5) 9 0.5 8.75 0. Epress ech of the following epressions s perfect squre : (i) 5c + 60cd + 36d + + (3m) 3m n + n 5 5 6 9m mn + n. 5 5 Mth Clss VII Question Bnk (iii) + + Ans. (i) 5c + 60cd + 36d (5c) + 5c 6d + (6d) (5c + 6d) + + () + + (iii) + + 3. Evlute (i) p + q 5 3 Ans. (i) ( ) + + + + + y 3 3 p + q 5 3 Since [( + b) + b + b ] + + + + 3 3 p + p q + q 5 5 5 3 9 p + pq + q 5 5 9 + y 3 [since ( + b) + b + b ] 3 3 + y + y 3 9 y y + + 3. Evlute (i) p q 3 Ans. (i) 3m n 5 3 p q 3 [since ( b) b + b ] 3 3 p p q + q 3 3 3 m n 5 [since ( b) b + b ] 9 p pq + q 9 6

7 3. Evlute (i) 3 Ans. (i) 7 3 3 p q 7 7 (3 ) 3 + 9 + 9 3 p q ICSE Mth Clss VII 5 Question Bnk [Since ( b) b + b ] [Since ( b) b + b ] 3 3 9 ( p) p q + q p 3p q + 6 q. Simplify the following, using formul, b ( + b) ( b) (i) 89 79 39 (iii) 35 (iv) 8 8 Ans. (i) 89 79 (89 + 79) (89 79) 68 0 680 39 ( + 39) ( 39) 83 5 5 (iii) 35 ( + 35) ( 35) 76 6 56 (iv) 8 8 (8 + 8) (8 8) 56 00 5600 5. Simplify the following using formul, b ( + b) ( b) (i) (iii) m (v) 6 9 5 Ans. (i) y y (iv) m n 5 b (vi) 0.6 ( ) + ( y) + y y (iii) m () m + m m (iv) m n 5 b (mn ) (5 b) (mn + 5 b) (mn 5 b) 6

(v) 6 9 5 6 9 5 () 3 3 + 5 5 (vi) 0.6 6. Fctorise Ans. (0.8) 6 8 3 5 0.8 + 0.8 8 8 ( ) + ( ) + + + 7. Fctorise Ans. 8. Fctorise Ans. () () + b b b b b b + + + b b + b b b b + b b b b b + + b b b 9. Simplify the following Ans. (i) 3 + 6y + 3m + 6my 3 ( + y) + 3m ( + y) ( + y) (3 + 3m) 3 ( + y) ( + m) Mth Clss VII 6 Question Bnk

+ b b Ans. + b b [ + b b ] [ b + b] [( ) b( )] ( ) ( b) (iii) 5 3 5 + 5 5 Ans. 5 3 5 + 5 5 5[ 3 3 3] 5[ ( 3) + ( 3)] 5 ( 3) ( + ) (iv) 5 0 3 0 + 0 Ans. 5 0 3 0 + 0 5[ 3 + 8] 5[ 3 ( ) ( )] 5( ) ( 3 ) 0. Simplify the following : (i) ( + b) 9 ( b) 6 8b Ans. (i) ( + b) 9 ( b) [ ( + b)] [3( b)] [( + b) 3( b) ] [ ( + b) + 3 ( b)] [ + b 3 + 3b] [ + b + 3 3b] ( + 5b) (5 b) (5b ) (5 b) 6 8b ( ) (9b ) ( + 9b ) ( 9b ) ( + 9b ) [() 3b) ] ( + 9b ) [( + 3b) ( 3b)] ( + 9b ) ( + 3b) ( 3b). Simplify.(i) p 3 p p p Ans. (i) p 3 p p (p ) p(p ) p(p + ) (p ) p p p (p ) p [(p) () ] p (p + ) (p ).Fctorise the following. (i) p 9 6 6 Ans. (i) Since b ( + b) ( b) 5 p (p) 6 6 5 ( ) 5 5 p + p + + ( ) + + ICSE Mth Clss VII 7 Question Bnk

3. Work out the vlue of : (i) (0000) (99998) Ans. Since, b ( + b) ( b) (i) (0000) (99998) 3 8 7 (0000 + 99998) (0000 99998) 00000 800000 3 8 7 3 3 8 + 7 8 7 35 9 35 9 + 35 + 9 35 9 6 6.Evlute : (i) (5) (9) (0.) (9.8) Ans. (i) (5) (9) (5 + 9) (5 9) [ b ( + b) ( b)] 00 00 (0.) (9.8) (0. + 9.8) (0. 9.8) 0.0 0. 8.00 8 5. Fctorise: ( + b) 9( b) Ans. ( + b) 9( b) [( + b)] [3( b)] [Since, b ( + b) ( b)] [ ( + b) + 3( b)] [( + b) 3( b)] ( + b + 3 3b) ( + b 3 + 3b) (5 b) (5b ) 6. Epress ech of the following s perfect squre : (i) 8p + 90pq + 5q 5 60y + 36y (iii) + + (iv) 9 30 + 5 Ans. (i) 8p + 90pq + 5q (9p) + 9p 5q + (5q) (9p + 5q) which is perfect squre 5 60y + 36y (5) 5 6y + (6y) (5 6y) which is perfect squre. Mth Clss VII 8 Question Bnk

(iii) + + () + + + Which is perfect squre. (iv) 9 30 + 5 (3) 3 5 + (5 ) (3 5 ) Which is perfect squre. 7. If +, evlute : (i) + + Ans. (i) Given, + Squring both sides, we hve, + + + 6 + + 6 + 6 + From (i), + squring both sides we get + ( ) + + 96 + + 96 + 96 + 9 8. If, find : (i) + + Ans. (i) Given, Squring both sides, we get ICSE Mth Clss VII 9 Question Bnk

+ + + + 6 From (i) + 6 Squring both sides, we get + 6 ( ) + + + 9. If + 3, evlute (i) Ans. (i) Given + 3 + 36 + 36 + 36 3 + + Squring both sides, we hve + 3 + + 9 + + 9 + 9 7...(i) + 7 Squring both sides, we hve + 7 ( ) + + 9 + + 9 + 9 7 Mth Clss VII 0 Question Bnk

30. If + 7, find the vlues of : (i) +, + Ans. Since, + 7 On squring both sides, we get + (7) + + 9 + + 9 + 9 7 (i) Squring both sides of eqution (i) we hve ( ) + + (7) + 09 + + 09 + 09 07 + 07 3. If, find the vlues of : (i) + + Ans. We hve ( b) + b s nd (given) (i) Squring both sides, we hve () ICSE Mth Clss VII Question Bnk

+ 6 + 6 + 6 + 8 (i) Squring both sides of + 8 we hve + (8) m 3. In P, m + n ( ) + + 3 + + 3 + 3 3 mke m s the subject m Ans. P, m + n P (m + n) m Pm + Pn m m Pm Pn m( P) Pn Pn m P 33. In s ut + Ans. s ut + ft, ft mke f s the subject. ft s ut ft (s ut) ( s ut) f t n 3. In S [ + ( n ) d], mke d s the subject. Mth Clss VII Question Bnk

n Ans. S [ + ( n ) d] S n[ + (n ) d] S n + n(n ) d S n n (n ) d n(n ) d S n d ( S n ) n( n ) RT 35. In A P +, 00 mke T s the subject. RT A RT Ans. A P +, 00 P + 00 RT A RT A + 00 P 00 P 36. In M Ans. In M 00 A T R P N( P R ), mke R s the subjects. 0 N ( P R) 0 M N (P R) 0 0 M P R N R P 0M N PRT 37. Given : I. Find P, when I 600, R nd T. 00 PRT Ans. I 00I PRT 00 PRT 00I P 00I RT 00 600 P 50 50 ( I 600, R nd T ) 38. Given : F 9 C + 3. Find C, when F 59. 5 ICSE Mth Clss VII 3 Question Bnk

Ans. F 9 3 5 C + 9 3 5 C F C 5 (F 3) 9 When, F 59, we hve C 5 (59 3) 9 C 5 7 9 C 5 39. Given : V Ans. V 3 π r h 3 r Putting V 35,π 7 35 π h. Find r when V 35, π nd h, we get r 3 7 35 3 7 r r 6 () l 0. Given : t π, Find l, when g 3, π g 7 Ans. t π l g Putting g 3, π nd t, we get 7 l 7 3 l 7 7 3 7 nd h nd t. l 9 (squring both sides) 3 6 9 l 3 98 6 + 3. Given:, Find, when 3. Mth Clss VII Question Bnk

Ans. + 3 ( ) + 3 + 3 + 3 + 3 ( ) + 3 Now put 3, we hve 3 + 3 6 3 3. Given : b +. Find b, when 5 nd. Ans. b + b b b Now put the vlue 5 nd we hve b (5) () b 5 b 8 b 9 3. Given : S n [ + (n ) d]. Find d, when S 30, n 6 nd 5 Ans. S n [ +(n ) d] S n[ +(n ) d] S n +(n ) d +(n ) d S n (n ) d S n s d n n Put S 30, n 6 nd 5, we hve 30 d 5 6 6 d [0 0] 30 5 5 ICSE Mth Clss VII 5 Question Bnk

d RT. Given A P +. 00 Find R, when A 5, P 650 nd T 3. RT Ans. A P + 00 A RT P + RT 00 00 + A P RT A 00 P Put A 5, P 650 nd T 3, we hve 00 5 R 3 650 00 5 650 R 3 650 00 95 00 65 R 3 650 650 R 0 5. Given : t Ans. rt g + r Find r, when g 00, t 3 nd T 7. rt t g + r t(g + r) rt tg + tr rt tg rt tr tg r [T t] r[t t] tg tg r T t Now put g 00, t 3 nd T 7, we hve 3 00 r 3 00 7 3 r 50 Mth Clss VII 6 Question Bnk