Selina Concise Solutions for Chapter 12 Algebraic Identities Class 8 ICSE Mathematics
Exercise 12A
1. Use direct method to evaluate the following products :
(i) (x + 8)(x + 3)
(ii) (y + 5)(y – 3)
(iii) (a – 8)(a + 2)
(iv) (b – 3)(b – 5)
(v) (3x – 2y)(2x + y)
(vi) (5a + 16)(3a – 7)
(vii) (8 – b) (3 + b)
Solution
(i) (x + 8)(x + 3) = (x × x) + (x × 3) + (8 × x) + (8 × 3)
= x2 + 3x + 8x + 24
= x2 + 11x + 24
(ii) (y + 5)(y – 3) = (y × y) + (y × -3) + (5 × y) + (5 × -3)
= y2 + (-3y) + (5y) – 15
= y2 – 3y + 5y – 15
= y2 + 2y – 15
(iii) (a – 8)(a + 2) = (a × a) + (a × 2) + (-8) × a + (-8)(2)
= a2 + 2a – 8a – 16
= a2 – 6a – 16
(iv) (b – 3)(b – 5) = (b × b) + (b × -5) + (-3) × b + (-3)(-5)
= b2 – 5b – 3b + 15
= b2 – 8b + 15
(v) (3x – 2y)(2x + y) = (3x × 2x) + (3x × y) + (-2y × 2x) + (-2y × y)
= 6x2 + 3xy – 4xy -2y2
= 6x2 – xy – 2y2
(vi) (5a + 16)(3a – 7) = (5a × 3a) + (5a × -7) + (16 × 3a) + 16 × -7
= 15a2 + (-35a) + 48a + (-112)
= 15a2 – 35a + 48a – 112
= 15a2 + 13a – 112
(vii) (8 – b)(3 + b) = (8 × 3) + (8 × b) + (-b × 3) + (-b × b)
= 24 + 8b – 3b – b2
= 24 + 5b – b2
2. Use direct method to evaluate :
(i) (x + 1) (x – 1)
(ii) (2 + a)(2 – a)
(iii) (3b – 1)(3b + 1)
(iv) (4 + 5x)(4 – 5x)
(v) (2a + 3)(2a – 3)
(vi) (xy + 4)(xy – 4)
(vii) (ab + x2)(ab –x2)
(viii) (3x2 + 5y2)(3x2 – 5y2)
(ix) (z – 2/3)(z + 2/3)
(x) (3/5a + ½)(3/5a - ½)
(xi) (0.5 – 2a)(0.5 + 2a)
(xii) (a/2 – b/3)(a/2 + b/3)
Solution
Note: (a + b)(a – b) = a2 – b2
(i) (x + 1)(x – 1)
= (x)2 – (1)2
= x2 – 1
(ii) (2 + a)/(2 – a)
= x2 – 1
(ii) (2 + a)/(2 – a)
= (2)2 – (a)2
= 4 – a2
(iii) (3b - 1)(3b + 1)
= 4 – a2
(iii) (3b - 1)(3b + 1)
= (3b)2 – (1)2
= 9b2 – 1
(iv) (4 + 5x)(4 – 5x)
= (4)2 – (5x)2
= 16 – 25x2
(v) (2a + 3)(2a – 3)
= (2a)2 – (3)2
= 4a2 – 9
(vi) (xy + 4)(xy – 4)
= (xy)2 – (4)2
= x2y2 – 16
(vii) (ab + x2) (ab – x2)
= (ab)2 – (x2)2
= a2b2 – x4
(viii) (3x2 + 5y2)(3x2 – 5y2)
= 9b2 – 1
(iv) (4 + 5x)(4 – 5x)
= (4)2 – (5x)2
= 16 – 25x2
(v) (2a + 3)(2a – 3)
= (2a)2 – (3)2
= 4a2 – 9
(vi) (xy + 4)(xy – 4)
= (xy)2 – (4)2
= x2y2 – 16
(vii) (ab + x2) (ab – x2)
= (ab)2 – (x2)2
= a2b2 – x4
(viii) (3x2 + 5y2)(3x2 – 5y2)
= (3x2)2 – (5y2)2
= 9x4 – 25y4
(ix) (z - 2/3)(z + 2/3)
= (z)2 – (2/3)2
= z2 – 4/9
(x) (3/5a + ½) (3/5a – ½)
= (3/5a)2 – (1/2)2
= 9/25a2 – ¼
(xi) (0.5 – 2a)(0.5 + 2a)
= (0.5)2 – (2a)2
= 0.25 – 4a2
(xii) (a/2 – b/3)(a/2 + b/3)
= (a/2)2 – (b/3)2
= a2/4 – b2/9
= 9x4 – 25y4
(ix) (z - 2/3)(z + 2/3)
= (z)2 – (2/3)2
= z2 – 4/9
(x) (3/5a + ½) (3/5a – ½)
= (3/5a)2 – (1/2)2
= 9/25a2 – ¼
(xi) (0.5 – 2a)(0.5 + 2a)
= (0.5)2 – (2a)2
= 0.25 – 4a2
(xii) (a/2 – b/3)(a/2 + b/3)
= (a/2)2 – (b/3)2
= a2/4 – b2/9
3. Evaluate:
(i) (a + 1)(a – 1)(a2 + 1)
(ii) (a + b)(a – b)(a2 + b2)
(iii) (2a – b)(2a + b)(4a2 + b2)
(iv) (3 – 2x)(3 + 2x)(9 + 4x2)
(v) (3x – 4y)(3x + 4y)(9x2 + 16y2)
Solution
(i) (a + 1)(a – 1)(a2 + 1)
= [(a)2 – (1)2] (a2 + 1)
= (a2 – 1)(a2 + 1)
= (a2)2 – (1)2
= a4 – 1
(ii) (a + b)(a – b)(a2 + b2)
= (a2 – b2)(a2 + b2)
= (a2 – 1)(a2 + 1)
= (a2)2 – (1)2
= a4 – 1
(iii) (2a – b)(2a + b)(4a2 + b2)
= [(2a)2 – (b)2] (4a2 + b2)
= (4a2 – b2)(4a2 + b2)
= (4a2)2 – (b2)2
= (16a4 – b4)
(iv) (3 – 2x)(3 + 2x)(9 + 4x2)
= [(3)2 – (2x)2] (9 + 4x2)
= (9 – 4x2)(9 + 4x2)
= (9)2 – (4x2)2
= 81 – 16x4
(v) (3x – 4y)(3x + 4y)(9x2 + 16y2)
= [(3x)2 – (4y)2](9x2 + 16y2)
= (9x2 – 16y2)(9x2 + 16y2)
= (9x2)2 – (16y2)2
= 81x4 – 256y4
4. Use the product (a + b) (a – b) = a2 – b2 to evaluate:
(i) 21 x 19
(ii) 33 x 27
(iii) 103 x 97
(iv) 9.8 x 10.2
(v) 7.7 x 8.3
(vi) 4.6 x 5.4
Solution
(i) 21 × 19 = (20 + 1)(20 – 1)
= (20)2 – (1)2
= 400 – 1
= 399
(ii) 33 × 27 = (30 + 3)(30 – 3)
= (30)2 – (3)2
= 900 – 9
= 891
(iii) 103 × 97 = (100 + 3)(100 - 3)
= (100)2 – (3)2
= 10000 – 9
= 9991
(iv) 9.8 × 10.2 = (10 – 0.2)(10 + 0.2)
= (10)2 – (0.2)2
= 100 – 0.4
= 99.96
(v) 7.7 × 8.3 = (8 – 0.3)(8 + 0.3)
= (8)2 – (0.3)2
= 64 – 0.09
= 63.91
(vi) 4.6 × 5.4 = (5 - 0.4)(5 + 0.4)
= (5)2 – (0.4)2
= 25 - 0.16
= 24.84
5. Evaluate:
(i) (6 – xy) (6 + xy)
(ii) (7x + 2/3y)(7x – 2/3y)
(iii) (a/2b + 2b/a) (a/2b – 2b/a)
(iv) (3x – 1/2y)(3x + 1/2y)
(v) (2a + 3)(2a – 3)(4a2 + 9)
(vi) (a + bc)(a – bc)(a2 + b2c2)
(vii) (5x + 8y)(3x + 5y)
(viii) (7x + 15y)(5x – 4y)
(ix) (2a – 3b)(3a + 4b)
(x) (9a – 7b)(3a – b)
Solution
(i) (6 – xy)(6 + xy) = 6(6 + xy) - xy(6 + xy)
= 36 + 6xy – 6xy + (xy)2
= 36 – x2y2
(ii) (7x + 2/3y)(7x – 2/3y)
= 7x(7x – 2/3y) + 2/3y(7x – 2/3y)
= 49x2 – 14/3xy + 14/3xy – 4/9y2
= 49x2– 4/9y2
(iii) (a/2b + 2b/a)(a/2b – 2b/a)
= a/2b(a/2b – 2b/a) + 2b/a(a/2b – 2b/a)
(i) (a + 1)(a – 1)(a2 + 1)
(ii) (a + b)(a – b)(a2 + b2)
(iii) (2a – b)(2a + b)(4a2 + b2)
(iv) (3 – 2x)(3 + 2x)(9 + 4x2)
(v) (3x – 4y)(3x + 4y)(9x2 + 16y2)
Solution
(i) (a + 1)(a – 1)(a2 + 1)
= [(a)2 – (1)2] (a2 + 1)
= (a2 – 1)(a2 + 1)
= (a2)2 – (1)2
= a4 – 1
(ii) (a + b)(a – b)(a2 + b2)
= (a2 – b2)(a2 + b2)
= (a2 – 1)(a2 + 1)
= (a2)2 – (1)2
= a4 – 1
(iii) (2a – b)(2a + b)(4a2 + b2)
= [(2a)2 – (b)2] (4a2 + b2)
= (4a2 – b2)(4a2 + b2)
= (4a2)2 – (b2)2
= (16a4 – b4)
(iv) (3 – 2x)(3 + 2x)(9 + 4x2)
= [(3)2 – (2x)2] (9 + 4x2)
= (9 – 4x2)(9 + 4x2)
= (9)2 – (4x2)2
= 81 – 16x4
(v) (3x – 4y)(3x + 4y)(9x2 + 16y2)
= [(3x)2 – (4y)2](9x2 + 16y2)
= (9x2 – 16y2)(9x2 + 16y2)
= (9x2)2 – (16y2)2
= 81x4 – 256y4
4. Use the product (a + b) (a – b) = a2 – b2 to evaluate:
(i) 21 x 19
(ii) 33 x 27
(iii) 103 x 97
(iv) 9.8 x 10.2
(v) 7.7 x 8.3
(vi) 4.6 x 5.4
Solution
(i) 21 × 19 = (20 + 1)(20 – 1)
= (20)2 – (1)2
= 400 – 1
= 399
(ii) 33 × 27 = (30 + 3)(30 – 3)
= (30)2 – (3)2
= 900 – 9
= 891
(iii) 103 × 97 = (100 + 3)(100 - 3)
= (100)2 – (3)2
= 10000 – 9
= 9991
(iv) 9.8 × 10.2 = (10 – 0.2)(10 + 0.2)
= (10)2 – (0.2)2
= 100 – 0.4
= 99.96
(v) 7.7 × 8.3 = (8 – 0.3)(8 + 0.3)
= (8)2 – (0.3)2
= 64 – 0.09
= 63.91
(vi) 4.6 × 5.4 = (5 - 0.4)(5 + 0.4)
= (5)2 – (0.4)2
= 25 - 0.16
= 24.84
5. Evaluate:
(i) (6 – xy) (6 + xy)
(ii) (7x + 2/3y)(7x – 2/3y)
(iii) (a/2b + 2b/a) (a/2b – 2b/a)
(iv) (3x – 1/2y)(3x + 1/2y)
(v) (2a + 3)(2a – 3)(4a2 + 9)
(vi) (a + bc)(a – bc)(a2 + b2c2)
(vii) (5x + 8y)(3x + 5y)
(viii) (7x + 15y)(5x – 4y)
(ix) (2a – 3b)(3a + 4b)
(x) (9a – 7b)(3a – b)
Solution
(i) (6 – xy)(6 + xy) = 6(6 + xy) - xy(6 + xy)
= 36 + 6xy – 6xy + (xy)2
= 36 – x2y2
(ii) (7x + 2/3y)(7x – 2/3y)
= 7x(7x – 2/3y) + 2/3y(7x – 2/3y)
= 49x2 – 14/3xy + 14/3xy – 4/9y2
= 49x2– 4/9y2
(iii) (a/2b + 2b/a)(a/2b – 2b/a)
= a/2b(a/2b – 2b/a) + 2b/a(a/2b – 2b/a)
= (a/2b)2 – (2b/a)2
= a2/4b2 – 4b2/a2
(iv) (3x – 1/2y)(3x + 1/2y)
= 3x(3x + 1/2y) – 1/2y(3x + 1/2y)
= 9x2 + 3x/2y – 3x/2y – 1/4y2
= 9x2 – 1/4y2
(v) (2a + 3)(2a – 3)(4a2 + 9)
= [(2a)2 – (3)2] (4a2 + 9) [∵ (a + b)(a – b) = a2 – b2]
(iv) (3x – 1/2y)(3x + 1/2y)
= 3x(3x + 1/2y) – 1/2y(3x + 1/2y)
= 9x2 + 3x/2y – 3x/2y – 1/4y2
= 9x2 – 1/4y2
(v) (2a + 3)(2a – 3)(4a2 + 9)
= [(2a)2 – (3)2] (4a2 + 9) [∵ (a + b)(a – b) = a2 – b2]
= (4a2 – 9)(4a2 + 9)
= (4a2)2 – (9)2
= 16a4 – 81
(vi) (a + bc)(a – bc)(a2 + b2c2)
= [(a)2 – (bc)2](a2 + b2c2) [∵ (a + b)(a – b) = a2 – b2]
= (4a2)2 – (9)2
= 16a4 – 81
(vi) (a + bc)(a – bc)(a2 + b2c2)
= [(a)2 – (bc)2](a2 + b2c2) [∵ (a + b)(a – b) = a2 – b2]
= (a2 – b2c2)(a2 + b2c2)
= (a2)2 – (b2c2)2 [∵ (a + b)(c – b) = a2 – b2]
= a4 – b4c4
(vii) (5x + 8y)(3x + 5y)
= 5x(3x + 5y) + 8y(3x + 5y)
= 15x2 + 25xy + 24xy + 40y2
= 15x2 + 49xy + 40y2
(viii) (7x + 15y)(5x – 4y)
= 7x(5x – 4y) + 15y (5x – 4y)
= 35x2 – 28xy + 75xy – 60y2
= 35x2 + 47xy – 60y2
(ix) (2a – 3b)(3a + 4b)
= 2a(3a + 4b) – 3b(3a + 4b)
= 6a2 + 8ab – 9ab – 12b2
= 6a2 – ab – 12b2
(x) (9a – 7b)(3a – b)
= 9a(3a – b) – 7b(3a – b)
= 27a2 – 9ab – 21ab + 7b2
= 27a2 – 30ab + 7b2
= (a2)2 – (b2c2)2 [∵ (a + b)(c – b) = a2 – b2]
= a4 – b4c4
(vii) (5x + 8y)(3x + 5y)
= 5x(3x + 5y) + 8y(3x + 5y)
= 15x2 + 25xy + 24xy + 40y2
= 15x2 + 49xy + 40y2
(viii) (7x + 15y)(5x – 4y)
= 7x(5x – 4y) + 15y (5x – 4y)
= 35x2 – 28xy + 75xy – 60y2
= 35x2 + 47xy – 60y2
(ix) (2a – 3b)(3a + 4b)
= 2a(3a + 4b) – 3b(3a + 4b)
= 6a2 + 8ab – 9ab – 12b2
= 6a2 – ab – 12b2
(x) (9a – 7b)(3a – b)
= 9a(3a – b) – 7b(3a – b)
= 27a2 – 9ab – 21ab + 7b2
= 27a2 – 30ab + 7b2
Exercise 12 B
1. Expand :
(i) (2a + b)2
(ii) (a – 2b)2
(iii) (a + 1/2a)2
(iv) (2a – 1/a)2
(v) (a + b – c)2
(vi) (a – b + c)2
(vii) (3x + 1/3x)2
(viii) (2x – 1/2x)2
Solution
(i) (2a + b)2
(i) (2a + b)2
(ii) (a – 2b)2
(iii) (a + 1/2a)2
(iv) (2a – 1/a)2
(v) (a + b – c)2
(vi) (a – b + c)2
(vii) (3x + 1/3x)2
(viii) (2x – 1/2x)2
Solution
(i) (2a + b)2
= (2a)2 + (b)2 + 2 × 2a × b [(a + b)2 = a2 + b2 + 2ab]
= 4a2 + b2 + 4ab
(ii) (a – 2b)2
= 4a2 + b2 + 4ab
(ii) (a – 2b)2
= (a)2 + (2b)2 – 2×a×2b [(a – b)2 = a2 + b2 – 2ab]
= a2 + 4b2 – 4ab
(iii) (a + 1/2a)2
= (a)2 + (1/2a)2 + 2×a× 1/2a
= a2 + 1/4a2 + 2a/2a
= a2 + 1/4a2 + 1
(iv) (2a – 1/a)2
= (2a)2 + (1/a)2 – 2×2a× 1/a
= 4a2 + 1/a2 – 4
(v) (a + b – c)2
= (a)2 + (b)2 + (-c)2 + 2×a× b + 2×b×(-c) + 2 × (-c) × (a)
= a2 + b2 + c2 + 2ab – 2bc – 2ca
Note: (a + b + c)2 = a2 + b2 + c2 + 2ab – 2bc – 2ca
(vi) (a – b + c)2
= a2 + 4b2 – 4ab
(iii) (a + 1/2a)2
= (a)2 + (1/2a)2 + 2×a× 1/2a
= a2 + 1/4a2 + 2a/2a
= a2 + 1/4a2 + 1
(iv) (2a – 1/a)2
= (2a)2 + (1/a)2 – 2×2a× 1/a
= 4a2 + 1/a2 – 4
(v) (a + b – c)2
= (a)2 + (b)2 + (-c)2 + 2×a× b + 2×b×(-c) + 2 × (-c) × (a)
= a2 + b2 + c2 + 2ab – 2bc – 2ca
Note: (a + b + c)2 = a2 + b2 + c2 + 2ab – 2bc – 2ca
(vi) (a – b + c)2
= (a)2 + (-b)2 + (c)2 + 2×a×-b + 2(-b)(c) + 2×c×a
= a2 + b2 + c2 – 2ab – 2bc + 2ca
(vii) (3x + 1/3x)2
= (3x)2 + (1/3x)2 + 2×3x× 1/3x
= 9x2 + 1/9x2 + 2
(viii) (2x – 1/2x)2
= (2x)2 + (1/2x)2 – 2×2x× 1/2x
= 4x2 + 1/4x2 - 2
2. Find the square of :
(i) x + 3y
(ii) 2x – 5y
(iii) a + 1/5a
(iv) 2a – 1/a
(v) x – 2y + 1
(vi) 3a – 2b – 5c
(vii) 2x + 1/x +1
(viii) 5 – x + 2/x
(ix) 2x – 3y + z
(x) x + 1/x – 1
Solution
(i) (x + 3y)2
= (x)2 + (3y)2 + 2×x×3y
= x + 9y2 + 6xy
(ii) (2x – 5y)2
= (2x)2 + (5y)2 – 2×2x×5y
= 4x2 + 25y2 – 20xy
(iii) (a + 1/5a)2
= (a)2 + (1/5a)2 + 2×a× 1/5a
= a2 + 1/25a2 + 2/5
(iv) (2a – 1/a)2
= (2a)2 + (1/a)2 – 2×2a× 1/a
= 4a2 + 1/a2 – 4
(v) (x – 2y + 1)2
= (x)2 + (-2y)2 + (1)2 + 2×x×-2y + 2×(-2y)×1 + 2×1×x
= x2 + 4y2 + 1 – 4xy – 4y + 2x
(vi) (3a – 2b – 5c)2
= (3a)2 + (-2b)2 + (-5c)2 + 2×3a×-2b + 2×(-2b) (-5c) + 2×-5c×3a
= 9a2 + 4b2 + 25c2 – 12ab + 20bc - 30ca
(vii) (2x + 1/x + 1)
= a2 + b2 + c2 – 2ab – 2bc + 2ca
(vii) (3x + 1/3x)2
= (3x)2 + (1/3x)2 + 2×3x× 1/3x
= 9x2 + 1/9x2 + 2
(viii) (2x – 1/2x)2
= (2x)2 + (1/2x)2 – 2×2x× 1/2x
= 4x2 + 1/4x2 - 2
2. Find the square of :
(i) x + 3y
(ii) 2x – 5y
(iii) a + 1/5a
(iv) 2a – 1/a
(v) x – 2y + 1
(vi) 3a – 2b – 5c
(vii) 2x + 1/x +1
(viii) 5 – x + 2/x
(ix) 2x – 3y + z
(x) x + 1/x – 1
Solution
(i) (x + 3y)2
= (x)2 + (3y)2 + 2×x×3y
= x + 9y2 + 6xy
(ii) (2x – 5y)2
= (2x)2 + (5y)2 – 2×2x×5y
= 4x2 + 25y2 – 20xy
(iii) (a + 1/5a)2
= (a)2 + (1/5a)2 + 2×a× 1/5a
= a2 + 1/25a2 + 2/5
(iv) (2a – 1/a)2
= (2a)2 + (1/a)2 – 2×2a× 1/a
= 4a2 + 1/a2 – 4
(v) (x – 2y + 1)2
= (x)2 + (-2y)2 + (1)2 + 2×x×-2y + 2×(-2y)×1 + 2×1×x
= x2 + 4y2 + 1 – 4xy – 4y + 2x
(vi) (3a – 2b – 5c)2
= (3a)2 + (-2b)2 + (-5c)2 + 2×3a×-2b + 2×(-2b) (-5c) + 2×-5c×3a
= 9a2 + 4b2 + 25c2 – 12ab + 20bc - 30ca
(vii) (2x + 1/x + 1)
= (2x)2 + (1/x)2 + (1)2 + 2×2x× 1/x + 2× 1/x ×1 + 2×1×2x
= 4x2 + 1/x2 + 1 + 4 + 2/x + 4x
= 4x2 + 1/x2 + 5 + 2/x + 4x
(viii) (5 – x + 2/x)2
= (5)2 + (-x)2 + (2/x)2 + 2×5×(-x) + 2(-x)× 2/x + 2× 2/x ×5
= 25 + x2 + 4/x2 – 10x – 4 + 20x
= 21 + x2 + 4/x2 – 10x + 20/x
(ix) (2x – 3y + z)2
= (2x)2 + (-3y)2 + (z)2 + 2×2x×-3y + 2(-3y)×z + 2×z×2x
= 4x2 + 9y2 + z2 – 12xy – 6yz + 4zx
(x) (x + 1/x – 1)2
= (x)2 + (1/x)2 + (-1)2 + 2× x× 1/x + 2× 1/x ×(-1) + 2(-1)×x
= x2 + 1/x2 + 1 + 2 – 2/x – 2x
= x2 + 1/x2 + 3 – 2/x – 2x
3. Evaluate:
Using expansion of (a + b)2 or (a – b)2
(i) (208)2
(ii) (92)2
(iii) (415)2
(iv) (188)2
(v) (9.4)2
(vi) (20.7)2
Solution
(i) (208)2
= 4x2 + 1/x2 + 1 + 4 + 2/x + 4x
= 4x2 + 1/x2 + 5 + 2/x + 4x
(viii) (5 – x + 2/x)2
= (5)2 + (-x)2 + (2/x)2 + 2×5×(-x) + 2(-x)× 2/x + 2× 2/x ×5
= 25 + x2 + 4/x2 – 10x – 4 + 20x
= 21 + x2 + 4/x2 – 10x + 20/x
(ix) (2x – 3y + z)2
= (2x)2 + (-3y)2 + (z)2 + 2×2x×-3y + 2(-3y)×z + 2×z×2x
= 4x2 + 9y2 + z2 – 12xy – 6yz + 4zx
(x) (x + 1/x – 1)2
= (x)2 + (1/x)2 + (-1)2 + 2× x× 1/x + 2× 1/x ×(-1) + 2(-1)×x
= x2 + 1/x2 + 1 + 2 – 2/x – 2x
= x2 + 1/x2 + 3 – 2/x – 2x
3. Evaluate:
Using expansion of (a + b)2 or (a – b)2
(i) (208)2
(ii) (92)2
(iii) (415)2
(iv) (188)2
(v) (9.4)2
(vi) (20.7)2
Solution
(i) (208)2
= (200 + 8)2
= (200)2 + (8)2 + 2(200) (8)
= 40000 + 64 + 3200
= 43264
(ii) (92)2
= (100 – 8)2
= (100)2 + (8)2 – 2(100)(8)
= 10000 + 64 – 1600
= 10064 – 1600
= 8464
(iii) (415)2
= (200)2 + (8)2 + 2(200) (8)
= 40000 + 64 + 3200
= 43264
(ii) (92)2
= (100 – 8)2
= (100)2 + (8)2 – 2(100)(8)
= 10000 + 64 – 1600
= 10064 – 1600
= 8464
(iii) (415)2
= (400 + 15)2
= (400)2 + (15)2 + 2(400)(15)
= 160000 + 225 + 12000
= 172225
(iv) (188)2
= (400)2 + (15)2 + 2(400)(15)
= 160000 + 225 + 12000
= 172225
(iv) (188)2
= (200 – 12)2
= (200)2 + (12)2 – 2(200)(12)
= 4000 + 144 – 4800
= 40144 – 4800
= 35344
(v) (9.4)2
= (200)2 + (12)2 – 2(200)(12)
= 4000 + 144 – 4800
= 40144 – 4800
= 35344
(v) (9.4)2
= (10 - 0.6)2
= (10)2 + (0.6)2 - 2(10)(0.6)
= 100 + 0.36 – 12
= 88 + 0.36
= 88.36
(vi) (20.7)2
= (10)2 + (0.6)2 - 2(10)(0.6)
= 100 + 0.36 – 12
= 88 + 0.36
= 88.36
(vi) (20.7)2
= (20 + 0.7)2
= (20)2 + (0.7)2 + 2(20)(0.7)
= 400 + 0.49 + 28
= 428 + 0.49
= 428.49
4. Expand :
(i) (2a + b)3
(ii) (a – 2b)3
(iii) (3x – 2y)3
(iv) (x + 5y)3
(v) (a + 1/a)3
(vi) (2a – 1/2a)3
Solution
(i) (2a + b)3
= (2a)3 + (b)3 + 3×2a×b(2a + b) [(a + b)3 = a3 + b3 + 3ab (a + b)]
= 8a3 + b3 + 6ab(2a + b)
= 8a3 + b3 + 12a2b + 6ab2
(ii) (a – 2b)3
= (a)3 – (2b)3 -3×a×2b(a – 2b) [(a – b)3 = a3 – b3 – 3ab (a – b)]
= a3 – 8b3 - 6ab(a - 2b)
= a3 – 8b3 – 6a2b + 12ab2
(iii) (3x – 2y)3
= (3x)3 – (2y)3 - 3×3x×2y(3x – 2y)
= 27x3 – 8y3 – 18xy (3x – 2y)
= 27x3 – 8y3 – 54x2y + 36xy2
(iv) (x + 5y)3
= (x)3 + (5y)3 + 3×x×5y(x + 5y)
= x3 + 125y3 + 15xy(x + 5y)
= x3 + 125y3 + 15x2y + 75xy2
(v) (a + 1/a)3
= a3 + (1/a)3 + 3×a× 1/a ×(a + 1/a)
= a3 + 1/a3 + 3(a + 1/a)
= a3 + 1/a3 + 3a + 3/a
(vi) (2a – 1/2a)3
= (2a)3 – (1/2a)3 – 3×2a× 1/2a (2a – 1/2a)
= 8a3 – 1/8a3 – 3(2a – 1/2a)
= 8a3 – 1/8a3 – 6a + 3/2a
5. Find the cube of :
(i) a + 2
(ii) 2a – 1
(iii) 2a + 3b
(iv) 3b – 2a
(v) 2x + 1/x
(v) x – ½
Solution
(i) (a + 2)3
= (a)3 + (2)3 + 3 × a × 2(a + 2)
= a3 + 8 + 6a(a + 2)
= a3 + 8 + 6a2 + 12a
= a3 + 6a2 + 12a + 8
(ii) (2a – 1)3
= (2a)3 – (1)3 - 3 × 2a × 1(2a – 1)
= 8a3 – 1 – 6a(2a – 1)
= 8a3 – 1 – 12a2 + 6a
= 8a3 – 12a2 + 6a – 1
(iii) (2a + 3b)3
= (2a)3 + (3b)3 + 3 × 2a × 3b(2a + 3b)
= 8a3 + 27b3 + 18ab (2a + 3b)
= 8a3 + 27b3 + 36a2b + 54 ab2
= 8a3 + 36a2b + 54ab2 + 27b3
(iv) (3b – 2a)3
= (3b)3 – (2a)3 – 3 × 3b × 2a(3b – 2a)
= 27b3 – 8a3 – 18ab(3b – 2a)
= 27b3 – 8a3 – 54ab2 + 36a2b
= 27b3 – 54b2a + 36ba2 – 8a3
(v) (2x + 1/x)3
= (2x)3 + (1/x)3 + 3 × 2x × 1/x(2x + 1/x)
= 8x3 + 1/x3 + 6 (2x + 1/x)
= 8x3 + 1/x3 + 12x + 6/x
= 8x3 + 12x + 6/x + 1/x3
(vi) (x – 1/2)3
= (x)3 – (1/2)3 – 3 × x × 1/2 (x – ½)
= x3 – 1/8 – 3x/2(x – ½)
= x3 – 1/8 - 3x2/2 + 3x/4
= x3 – 3x2/2 + 3x/4 – 1/8
= (20)2 + (0.7)2 + 2(20)(0.7)
= 400 + 0.49 + 28
= 428 + 0.49
= 428.49
4. Expand :
(i) (2a + b)3
(ii) (a – 2b)3
(iii) (3x – 2y)3
(iv) (x + 5y)3
(v) (a + 1/a)3
(vi) (2a – 1/2a)3
Solution
(i) (2a + b)3
= (2a)3 + (b)3 + 3×2a×b(2a + b) [(a + b)3 = a3 + b3 + 3ab (a + b)]
= 8a3 + b3 + 6ab(2a + b)
= 8a3 + b3 + 12a2b + 6ab2
(ii) (a – 2b)3
= (a)3 – (2b)3 -3×a×2b(a – 2b) [(a – b)3 = a3 – b3 – 3ab (a – b)]
= a3 – 8b3 - 6ab(a - 2b)
= a3 – 8b3 – 6a2b + 12ab2
(iii) (3x – 2y)3
= (3x)3 – (2y)3 - 3×3x×2y(3x – 2y)
= 27x3 – 8y3 – 18xy (3x – 2y)
= 27x3 – 8y3 – 54x2y + 36xy2
(iv) (x + 5y)3
= (x)3 + (5y)3 + 3×x×5y(x + 5y)
= x3 + 125y3 + 15xy(x + 5y)
= x3 + 125y3 + 15x2y + 75xy2
(v) (a + 1/a)3
= a3 + (1/a)3 + 3×a× 1/a ×(a + 1/a)
= a3 + 1/a3 + 3(a + 1/a)
= a3 + 1/a3 + 3a + 3/a
(vi) (2a – 1/2a)3
= (2a)3 – (1/2a)3 – 3×2a× 1/2a (2a – 1/2a)
= 8a3 – 1/8a3 – 3(2a – 1/2a)
= 8a3 – 1/8a3 – 6a + 3/2a
5. Find the cube of :
(i) a + 2
(ii) 2a – 1
(iii) 2a + 3b
(iv) 3b – 2a
(v) 2x + 1/x
(v) x – ½
Solution
(i) (a + 2)3
= (a)3 + (2)3 + 3 × a × 2(a + 2)
= a3 + 8 + 6a(a + 2)
= a3 + 8 + 6a2 + 12a
= a3 + 6a2 + 12a + 8
(ii) (2a – 1)3
= (2a)3 – (1)3 - 3 × 2a × 1(2a – 1)
= 8a3 – 1 – 6a(2a – 1)
= 8a3 – 1 – 12a2 + 6a
= 8a3 – 12a2 + 6a – 1
(iii) (2a + 3b)3
= (2a)3 + (3b)3 + 3 × 2a × 3b(2a + 3b)
= 8a3 + 27b3 + 18ab (2a + 3b)
= 8a3 + 27b3 + 36a2b + 54 ab2
= 8a3 + 36a2b + 54ab2 + 27b3
(iv) (3b – 2a)3
= (3b)3 – (2a)3 – 3 × 3b × 2a(3b – 2a)
= 27b3 – 8a3 – 18ab(3b – 2a)
= 27b3 – 8a3 – 54ab2 + 36a2b
= 27b3 – 54b2a + 36ba2 – 8a3
(v) (2x + 1/x)3
= (2x)3 + (1/x)3 + 3 × 2x × 1/x(2x + 1/x)
= 8x3 + 1/x3 + 6 (2x + 1/x)
= 8x3 + 1/x3 + 12x + 6/x
= 8x3 + 12x + 6/x + 1/x3
(vi) (x – 1/2)3
= (x)3 – (1/2)3 – 3 × x × 1/2 (x – ½)
= x3 – 1/8 – 3x/2(x – ½)
= x3 – 1/8 - 3x2/2 + 3x/4
= x3 – 3x2/2 + 3x/4 – 1/8
Exercise 12 C
1. If a + b = 5 and ab = 6; find a2 + b2
Solution
(a + b)2 = a2 + b2 + 2ab
⇒ (5)2 = a2 + b2 + 2 × 6
⇒ 25 = a2 + b2 + 12
⇒ 25 – 12 = a2 + b2
⇒ 13 = a2 + b2
∴ a2 + b2 = 13
2. If a – b = 6 and ab = 16; find a2 + b2
Solution
(a – b)2 = a2 + b2 – 2ab
⇒ (6)2 = a2 + b2 – 2 × 16
⇒ 36 = a2 + b2 – 32
⇒ 36 + 32 = a2 + b2
⇒ 68 = a2 + b2
∴ a2 + b2 = 68
3. If a2 + b2 = 29 and ab = 10 ; find :
(i) a + b
(ii) a – b
Solution
(i) (a + b)2 = a2 + b2 + 2ab
⇒ (a + b)2 = 29 + 2 × 10
⇒ (a + b)2 = 29 + 20
⇒ (a + b)2 = 49
⇒ a + b = √49
⇒ a + b = 7
(ii) (a – b)2 = a2 + b2 – 2ab
⇒ (a – b)2 = 29 – 2 × 10
⇒ (a – b)2 = 29 – 20
⇒ (a – b)2 = 9
⇒ a – b = √9
⇒ a – b = 3
4. If a2 + b2 = 10 and ab = 3; find :
(i) a – b
(ii) a + b
Solution
(i) (a – b)2 = a2 + b2 – 2ab
⇒ (a – b)2 = 10 – 2 × 3
⇒ (a – b)2 = 10 – 6
⇒ (a – b)2 = 4
⇒ (a – b) = √4
⇒ a – b = 2
(ii) (a + b)2 = a2 + b2 + 2ab
⇒ (a + b)2 = 10 + 2 × 3
⇒ (a + b)2 = 10 + 6
⇒ (a + b)2 = 16
⇒ (a + b) = √16
⇒ (a + b) = 4
5. If a + 1/a = 3; find a2 + 1/a2
Solution
(a + 1/a)2 = a2 + 1/a2 + 2
⇒ (3)2 = a2 + 1/a2 + 2
⇒ 9 = a2 + 1/a2 + 2
⇒ 9 – 2 = a2 + 1/a2
⇒ 7 = a2 + 1/a2
∴ a2 + 1/a2 = 7
Alternative Method:
a + 1/a = 3
⇒ (a + 1/a)2 = (3)2
⇒ a2 + 1/a2 + 2 = 9
⇒ a2 + 1/a2 = 9 – 2
⇒ a2 + 1/a2 = 7
6. If a – 1/a = 4; find a2 + 1/a2
Solution
(a - 1/a)2 = a2 + 1/a2 – 2
⇒ (4)2 = a2 + 1/a2 – 2
⇒ 16 = a2 + 1/a2 – 2
⇒ 16 + 2 = a2 + 1/a2
⇒ 18 = a2 + 1/a2
∴ a2 + 1/a2 = 18
Alternative method:
a – 1/a = 4
⇒ (a – 1/a)2 = (4)2
⇒ a2 + 1/a2 – 2 = 16
⇒ a2 + 1/a2 = 16 + 2
⇒ a2 + 1/a2 = 18
7. If a2 + 1/a2 = 23; find a + 1/a
Solution
(a + 1/a)2 = a2 + 1/a2 + 2
⇒ (a + 1/a)2 = 23 + 2
⇒ (a + 1/a)2 = 25
⇒ a + 1/a = √25
⇒ a + 1/a = 5
8. If a2 + ½ = 11; find a2 – 1/a
Solution
(a – 1/a)b2 = a2 + 1/a2 – 2
⇒ (a – 1/a)2 = 11 – 2
⇒ (a – 1/a)2 = 9
⇒ a – 1/a = √9
⇒ a- 1/a = 3
9. If a + b + c = 10 and a2 + b2 + c2 = 38; find ab + bc + ca
Solution
a + b + c = 10
(a + b + c)2 = (10)2
⇒ a2 + b2 + c2 + 2(ab + bc + ca) = 100
⇒ 38 + 2(ab + bc + ca) = 100 - 38
⇒ 2(ab + bc + ca) = 62
⇒ (ab + bc + ca) = 62/2
⇒ ab + bc + ca = 31
Alternative Method:
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
⇒ (10)2 = 38 + 2(ab + bc + ca)
⇒ 100 = 38 + 2(ab + bc + ca)
⇒ 100 - 38 = 2(ab + bc + ca)
⇒ 62 = 2(ab + bc + ca)
⇒ 62/2 = ab + bc + ca
⇒ 31 = ab + bc + ca
∴ ab + bc + ca = 31
10. Find a2 + b2 + c2 ; if a + b + c = 9 and ab + bc + ca = 24
Solution
a + b + c = 9
⇒ (a + b + c)2 = (9)2
⇒ a2 + b2 + c2 + 2ab + 2bc + 2ca = 81
⇒ a2 + b2 + c2 + 2(ab + bc + ca) = 81
a2 + b2 + c2 + 2 × 24 = 81
a2 + b2 + c2 + 2 × 24 = 81
⇒ a2 + b2 + c2 = 81 - 48
⇒ a2 + b2 + c2 = 33
11. Find a + b + c; if a2 + b2 + c2 = 83 and ab + bc + ca = 71
Solution
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
⇒ (a + b + c)2 = 83 + 2(ab + bc + ca)
⇒ (a + b + c)2 = 83 + 2 × 71
⇒ (a + b + c)2 = 83 + 142
⇒ (a + b + c)2 = 225
⇒ a + b + c = √225
⇒ a + b + c = 15
12. If a + b = 6 and ab = 8; find a3 + b3
Solution
a + b = 6
⇒ (a + b)3 = (6)3
⇒ a3 + b3 + 3ab (a + b) = 216
⇒ a3 + b3 + 3 × 8(6) = 216
⇒ a3 + b3 + 144 = 216
⇒ a3 + b3 = 216 – 144
⇒ a3 + b3 = 72
Alternative method:
(a + b)3 = a3 + b3 + 3ab (a + b)
⇒ (6)3 = a3 + b3 + 3 × 8(6)
⇒ 216 = a3 + b3 + 144
⇒ 216 – 144 = a3 + b3
⇒ 72 = a3 + b3
⇒ a3 + b3 = 72
13. If a – b = 3 and ab = 10; find a3 – b3
Solution
a – b = 3
⇒ (a – b)3 = (3)3
⇒ a3 – b3 – 3ab(a – b) = 27
⇒ a3 – b3 – 3 × 10 (3) = 27
⇒ a3 – b3 – 90 = 27
⇒ a3 – b3 = 27 + 90
⇒ a3 – b3 = 117
Alternative method :
(a – b)3 = a3 – b3 – 3ab (a – b)
⇒ (3)3 = a3 – b3 – 3 × 10(3)
⇒ 27 = a3 – b3 – 90
⇒ 27 + 90 = a3 – b3
⇒ 117 = a3 – b3
⇒ a3 – b3 = 117
14. Find a3 + 1/a3 if a + 1/a = 5
Solution
a + 1/a = 5
⇒ (a + 1/a)3 = (5)3
⇒ a3 + 1/a3 + 3a × 1/a (a + 1/a) = 125
⇒ a3 + 1/a3 + 3(5) = 125 [a + 1/a = 5]
⇒ a3 + 1/a3 + 15 = 125
⇒ a3 + 1/a3 = 125 – 15
⇒ a3 + 1/a3 = 110
15. Find a3 – 1/a3 if a – 1/a = 4
Solution
a – 1/a = 4
⇒ (a – 1/a)3 = (4)3
⇒ a3 – 1/a3 – 3a × 1/a (a – 1/a) = 64
⇒ a3 – 1/a3 – 3(4) = 64 [∵ a – 1/a = 4]
⇒ a3 – 1/a3 – 12 = 64
⇒ a3 – 1/a3 = 64 + 12
⇒ a3 – 1/a3 = 76
16. If 2x – 1/2x = 4 ; find :
(i) 4x2+ 1/4x2
(ii) 8x3 - 1/8x3
Solution
(i) 2x – 1/2x = 4
⇒ (2x – 1/2x)2 = (4)2
⇒ (2x)2 + (1/2x)2 – 2 × 2x × 1/2x = 16
⇒ 4x2 + 1/4x2 – 2 = 16
⇒ 4x2 + 1/4x2 = 16 + 2
⇒ 4x2 + 1/42 = 18
(ii) 2x – 12/2x = 4
⇒ (2x - 1/2x)3 = (4)3
⇒ (2x)3 – (1/2x)3 – 3 × 2x × 1/2x (2x – 1/2x) = 64
⇒ 8x3– 1/8x3 – 3(4) = 64
⇒ 8x3 – 1/8x3 – 12 = 64
⇒ 8x3 – 1/8x3 = 64 + 12
⇒ 8x3 – 1/8x3 = 76
17. If 3x + 1/3x = 3; find :
(i) 9x2 + 1/9x2
(ii) 27x3 + 1/27x3
Solution
(i) 3x + 1/3x = 3
⇒ (3x + 1/3x)2 = (3)2
⇒ (3x)2 + (1/3x)2 + 2 × 3x × 1/3x = 9
⇒ 9x2 + 1/9x2 + 2 = 9
⇒ 9x2 + 1/9x2 = 9 – 2
⇒ 9x2 + 1/9x2 = 7
(ii) 3x + 1/3x = 3
⇒ (3x + 13x)3 = (3)3
⇒ (3x)3 + (1/3x)3 + 3 × 3x × 1/3x(3x + 1/3x) = 27
⇒ 27x3 + 1/27x3 + 3(3x + 1/3x) = 27
⇒ 27x3 + 1/27x3 + 3(3) = 27
⇒ 27x3 + 1/27x3 + 9 = 27
⇒ 27x3 + 1/27x3 = 27 – 9
⇒ 27x3 + 1/27x3 = 18
18. The sum of the squares of two numbers is 13 and their product is 6. Find:
(i) the sum of the two numbers.
(ii) the difference between them.
Solution
Let x and y be the two numbers, then
x2 + y2 = 13 and xy = 6
(i) (x + y)2 = x2 + y2 + 2xy
= 13 + 2 × 6
= 13 + 12
= 25
∴ x + y = ± √25
= ±5
(ii) (x – y)2 = x2 + y2 – 2xy
= 13 – 12
= 1
∴ x – y = ± 1
Solution
(a + b)2 = a2 + b2 + 2ab
⇒ (5)2 = a2 + b2 + 2 × 6
⇒ 25 = a2 + b2 + 12
⇒ 25 – 12 = a2 + b2
⇒ 13 = a2 + b2
∴ a2 + b2 = 13
2. If a – b = 6 and ab = 16; find a2 + b2
Solution
(a – b)2 = a2 + b2 – 2ab
⇒ (6)2 = a2 + b2 – 2 × 16
⇒ 36 = a2 + b2 – 32
⇒ 36 + 32 = a2 + b2
⇒ 68 = a2 + b2
∴ a2 + b2 = 68
3. If a2 + b2 = 29 and ab = 10 ; find :
(i) a + b
(ii) a – b
Solution
(i) (a + b)2 = a2 + b2 + 2ab
⇒ (a + b)2 = 29 + 2 × 10
⇒ (a + b)2 = 29 + 20
⇒ (a + b)2 = 49
⇒ a + b = √49
⇒ a + b = 7
(ii) (a – b)2 = a2 + b2 – 2ab
⇒ (a – b)2 = 29 – 2 × 10
⇒ (a – b)2 = 29 – 20
⇒ (a – b)2 = 9
⇒ a – b = √9
⇒ a – b = 3
4. If a2 + b2 = 10 and ab = 3; find :
(i) a – b
(ii) a + b
Solution
(i) (a – b)2 = a2 + b2 – 2ab
⇒ (a – b)2 = 10 – 2 × 3
⇒ (a – b)2 = 10 – 6
⇒ (a – b)2 = 4
⇒ (a – b) = √4
⇒ a – b = 2
(ii) (a + b)2 = a2 + b2 + 2ab
⇒ (a + b)2 = 10 + 2 × 3
⇒ (a + b)2 = 10 + 6
⇒ (a + b)2 = 16
⇒ (a + b) = √16
⇒ (a + b) = 4
5. If a + 1/a = 3; find a2 + 1/a2
Solution
(a + 1/a)2 = a2 + 1/a2 + 2
⇒ (3)2 = a2 + 1/a2 + 2
⇒ 9 = a2 + 1/a2 + 2
⇒ 9 – 2 = a2 + 1/a2
⇒ 7 = a2 + 1/a2
∴ a2 + 1/a2 = 7
Alternative Method:
a + 1/a = 3
⇒ (a + 1/a)2 = (3)2
⇒ a2 + 1/a2 + 2 = 9
⇒ a2 + 1/a2 = 9 – 2
⇒ a2 + 1/a2 = 7
6. If a – 1/a = 4; find a2 + 1/a2
Solution
(a - 1/a)2 = a2 + 1/a2 – 2
⇒ (4)2 = a2 + 1/a2 – 2
⇒ 16 = a2 + 1/a2 – 2
⇒ 16 + 2 = a2 + 1/a2
⇒ 18 = a2 + 1/a2
∴ a2 + 1/a2 = 18
Alternative method:
a – 1/a = 4
⇒ (a – 1/a)2 = (4)2
⇒ a2 + 1/a2 – 2 = 16
⇒ a2 + 1/a2 = 16 + 2
⇒ a2 + 1/a2 = 18
7. If a2 + 1/a2 = 23; find a + 1/a
Solution
(a + 1/a)2 = a2 + 1/a2 + 2
⇒ (a + 1/a)2 = 23 + 2
⇒ (a + 1/a)2 = 25
⇒ a + 1/a = √25
⇒ a + 1/a = 5
8. If a2 + ½ = 11; find a2 – 1/a
Solution
(a – 1/a)b2 = a2 + 1/a2 – 2
⇒ (a – 1/a)2 = 11 – 2
⇒ (a – 1/a)2 = 9
⇒ a – 1/a = √9
⇒ a- 1/a = 3
9. If a + b + c = 10 and a2 + b2 + c2 = 38; find ab + bc + ca
Solution
a + b + c = 10
(a + b + c)2 = (10)2
⇒ a2 + b2 + c2 + 2(ab + bc + ca) = 100
⇒ 38 + 2(ab + bc + ca) = 100 - 38
⇒ 2(ab + bc + ca) = 62
⇒ (ab + bc + ca) = 62/2
⇒ ab + bc + ca = 31
Alternative Method:
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
⇒ (10)2 = 38 + 2(ab + bc + ca)
⇒ 100 = 38 + 2(ab + bc + ca)
⇒ 100 - 38 = 2(ab + bc + ca)
⇒ 62 = 2(ab + bc + ca)
⇒ 62/2 = ab + bc + ca
⇒ 31 = ab + bc + ca
∴ ab + bc + ca = 31
10. Find a2 + b2 + c2 ; if a + b + c = 9 and ab + bc + ca = 24
Solution
a + b + c = 9
⇒ (a + b + c)2 = (9)2
⇒ a2 + b2 + c2 + 2ab + 2bc + 2ca = 81
⇒ a2 + b2 + c2 + 2(ab + bc + ca) = 81
a2 + b2 + c2 + 2 × 24 = 81
a2 + b2 + c2 + 2 × 24 = 81
⇒ a2 + b2 + c2 = 81 - 48
⇒ a2 + b2 + c2 = 33
11. Find a + b + c; if a2 + b2 + c2 = 83 and ab + bc + ca = 71
Solution
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
⇒ (a + b + c)2 = 83 + 2(ab + bc + ca)
⇒ (a + b + c)2 = 83 + 2 × 71
⇒ (a + b + c)2 = 83 + 142
⇒ (a + b + c)2 = 225
⇒ a + b + c = √225
⇒ a + b + c = 15
12. If a + b = 6 and ab = 8; find a3 + b3
Solution
a + b = 6
⇒ (a + b)3 = (6)3
⇒ a3 + b3 + 3ab (a + b) = 216
⇒ a3 + b3 + 3 × 8(6) = 216
⇒ a3 + b3 + 144 = 216
⇒ a3 + b3 = 216 – 144
⇒ a3 + b3 = 72
Alternative method:
(a + b)3 = a3 + b3 + 3ab (a + b)
⇒ (6)3 = a3 + b3 + 3 × 8(6)
⇒ 216 = a3 + b3 + 144
⇒ 216 – 144 = a3 + b3
⇒ 72 = a3 + b3
⇒ a3 + b3 = 72
13. If a – b = 3 and ab = 10; find a3 – b3
Solution
a – b = 3
⇒ (a – b)3 = (3)3
⇒ a3 – b3 – 3ab(a – b) = 27
⇒ a3 – b3 – 3 × 10 (3) = 27
⇒ a3 – b3 – 90 = 27
⇒ a3 – b3 = 27 + 90
⇒ a3 – b3 = 117
Alternative method :
(a – b)3 = a3 – b3 – 3ab (a – b)
⇒ (3)3 = a3 – b3 – 3 × 10(3)
⇒ 27 = a3 – b3 – 90
⇒ 27 + 90 = a3 – b3
⇒ 117 = a3 – b3
⇒ a3 – b3 = 117
14. Find a3 + 1/a3 if a + 1/a = 5
Solution
a + 1/a = 5
⇒ (a + 1/a)3 = (5)3
⇒ a3 + 1/a3 + 3a × 1/a (a + 1/a) = 125
⇒ a3 + 1/a3 + 3(5) = 125 [a + 1/a = 5]
⇒ a3 + 1/a3 + 15 = 125
⇒ a3 + 1/a3 = 125 – 15
⇒ a3 + 1/a3 = 110
15. Find a3 – 1/a3 if a – 1/a = 4
Solution
a – 1/a = 4
⇒ (a – 1/a)3 = (4)3
⇒ a3 – 1/a3 – 3a × 1/a (a – 1/a) = 64
⇒ a3 – 1/a3 – 3(4) = 64 [∵ a – 1/a = 4]
⇒ a3 – 1/a3 – 12 = 64
⇒ a3 – 1/a3 = 64 + 12
⇒ a3 – 1/a3 = 76
16. If 2x – 1/2x = 4 ; find :
(i) 4x2+ 1/4x2
(ii) 8x3 - 1/8x3
Solution
(i) 2x – 1/2x = 4
⇒ (2x – 1/2x)2 = (4)2
⇒ (2x)2 + (1/2x)2 – 2 × 2x × 1/2x = 16
⇒ 4x2 + 1/4x2 – 2 = 16
⇒ 4x2 + 1/4x2 = 16 + 2
⇒ 4x2 + 1/42 = 18
(ii) 2x – 12/2x = 4
⇒ (2x - 1/2x)3 = (4)3
⇒ (2x)3 – (1/2x)3 – 3 × 2x × 1/2x (2x – 1/2x) = 64
⇒ 8x3– 1/8x3 – 3(4) = 64
⇒ 8x3 – 1/8x3 – 12 = 64
⇒ 8x3 – 1/8x3 = 64 + 12
⇒ 8x3 – 1/8x3 = 76
17. If 3x + 1/3x = 3; find :
(i) 9x2 + 1/9x2
(ii) 27x3 + 1/27x3
Solution
(i) 3x + 1/3x = 3
⇒ (3x + 1/3x)2 = (3)2
⇒ (3x)2 + (1/3x)2 + 2 × 3x × 1/3x = 9
⇒ 9x2 + 1/9x2 + 2 = 9
⇒ 9x2 + 1/9x2 = 9 – 2
⇒ 9x2 + 1/9x2 = 7
(ii) 3x + 1/3x = 3
⇒ (3x + 13x)3 = (3)3
⇒ (3x)3 + (1/3x)3 + 3 × 3x × 1/3x(3x + 1/3x) = 27
⇒ 27x3 + 1/27x3 + 3(3x + 1/3x) = 27
⇒ 27x3 + 1/27x3 + 3(3) = 27
⇒ 27x3 + 1/27x3 + 9 = 27
⇒ 27x3 + 1/27x3 = 27 – 9
⇒ 27x3 + 1/27x3 = 18
18. The sum of the squares of two numbers is 13 and their product is 6. Find:
(i) the sum of the two numbers.
(ii) the difference between them.
Solution
Let x and y be the two numbers, then
x2 + y2 = 13 and xy = 6
(i) (x + y)2 = x2 + y2 + 2xy
= 13 + 2 × 6
= 13 + 12
= 25
∴ x + y = ± √25
= ±5
(ii) (x – y)2 = x2 + y2 – 2xy
= 13 – 12
= 1
∴ x – y = ± 1
Exercise 12 D
1. Evaluate :
(i) (3x + ½) (2x + 1/3)
(ii) (2a + 0.5) (7a – 0.3)
(iii) (9 – y)(7 + y)
(iv) (2 – z)(15 – z)
(v) (a2 + 5)(a2 – 3)
(vi) (4 – ab)(8 + ab)
(vii) (5xy – 7)(7xy + 9)
(viii) (3a2 – 4b2)(8a2 – 3b2)
Solution
(i) (3x + 1/2 )(2x + 1/3)
= 3x(2x + 1/3) + ½(2x + 1/3)
= 6x2 + x + x + 1/6
= 6x2 + 2x + 1/6
(ii) (2a + 0.5)(7a – 0.3)
= 2a(7a – 0.3) + 0.5(7a – 0.3)
= 14a2 – 0.6a + 3.5a – 0.15
= 14a2 + 2.9a – 0.15
(iii) (9 – y)(7 + y)
(i) (3x + ½) (2x + 1/3)
(ii) (2a + 0.5) (7a – 0.3)
(iii) (9 – y)(7 + y)
(iv) (2 – z)(15 – z)
(v) (a2 + 5)(a2 – 3)
(vi) (4 – ab)(8 + ab)
(vii) (5xy – 7)(7xy + 9)
(viii) (3a2 – 4b2)(8a2 – 3b2)
Solution
(i) (3x + 1/2 )(2x + 1/3)
= 3x(2x + 1/3) + ½(2x + 1/3)
= 6x2 + x + x + 1/6
= 6x2 + 2x + 1/6
(ii) (2a + 0.5)(7a – 0.3)
= 2a(7a – 0.3) + 0.5(7a – 0.3)
= 14a2 – 0.6a + 3.5a – 0.15
= 14a2 + 2.9a – 0.15
(iii) (9 – y)(7 + y)
= 9(7 + y) – y(7 + y)
= 63 + 9y - 7y – y2
= 63 + 2y – y2
(iv) (2 – z)(15 – z) = 2(15 – z) – z(15 – z)
= 30 – 2z – 15z + z2
= 30 – 17z + z2
(v) (a2 + 5)(a2 - 3)
= 63 + 9y - 7y – y2
= 63 + 2y – y2
(iv) (2 – z)(15 – z) = 2(15 – z) – z(15 – z)
= 30 – 2z – 15z + z2
= 30 – 17z + z2
(v) (a2 + 5)(a2 - 3)
= a2(a2 – 3) + 5(a2 – 3)
= a4 – 3a2 + 5a2 – 15
= a4 + 2a2 – 15
(vi) (4 – ab)(8 + ab)
= a4 – 3a2 + 5a2 – 15
= a4 + 2a2 – 15
(vi) (4 – ab)(8 + ab)
= 4(8 + ab) – ab(8 + ab)
= 32 + 4ab – 8ab – a2b2
= 32 – 4ab – a2b2
(vii) (5xy – 7)(7xy + 9)
= 32 + 4ab – 8ab – a2b2
= 32 – 4ab – a2b2
(vii) (5xy – 7)(7xy + 9)
= 5xy(7xy + 9) – 7(7xy + 9)
= 35x2y2 + 45xy – 49xy - 63
= 35x2y2 – 4xy – 63
(viii) (3a2 – 4b2)(8a2 – 3b2)
= 3a2 (8a2 – 3b2) – 4b2(8a2 – 3b2)
= 24a4 – 9a2b2 – 32a2b2 + 12b4
= 24a4 – 41a2b2 + 12b4
2. Evaluate :
(i) (2x – 3/5)(2x + 3/5)
(ii) (4/7a + 3/4b) (4/7 a – 3/4b)
(iii) (6 – 5xy)(6 + 5xy)
(iv) (2a + 1/2a) (2a – 1/2a)
(v) (4x2 – 5y2)(4x2 + 5y2)
(vi) (1.6x + 0.7y)(1.6x – 0.7y)
(vii) (m + 3)(m – 3)(m2 + 9)
(viii) (3x + 4y)(3x – 4y)(9x2 + 16y2)
(ix) (a + bc)(a – bc)(a2 + b2c2)
(x) 203 × 197
(xi) 20.8 × 19.2
Solution
(i) (2x – 3/5)(2x + 3/5)
= (2x)2 – (3/5)2 [∵ (a – b)(a + b) = a2-b2]
= 4x2 – 9/25
(ii) (4/7a + 3/4b)(4/7a – 3/4b)
= (4/7a)2 – (3/4b)2 [∵ (a – b)(a + b) = a2-b2]
= 16/49a2 – 9/16b2
(iii) (6 – 5xy) (6 + 5xy)
= (6)2 – (5xy)2
= 36 – 25x2y2 [∵ (a – b)(a + b) = a2-b2]
(iv) (2a + 1/2a)(2a – 1/2a)
= (2a)2 – (1/2a)2 [∵ (a – b)(a + b) = a2-b2]
= 4a2 – 1/4a2
(v) (4x2 – 5y2)(4x2 + 5y2)
= (4x2)2 – (5y2)2
= 16x4 – 25y4 [∵ (a – b)(a + b) = a2-b2]
(vi) (1.6x + 0.7y)(1.6x – 0.7y)
= (1.6x)2 – (0.7y)2 [∵ (a – b)(a + b) = a2-b2]
= 2.56x2 – 0.49y2
(vii) (m + 3)(m – 3)(m2 + 9)
= (m)2 – (3)2 (m2 + 9) [∵ (a – b)(a + b) = a2-b2]
= (m2 – 9)(m2 + 9)
= (m2)2 – 92
= m4 – 81
(viii) (3x + 4y)(3x – 4y)(9x2 + 16y2)
= [(3x)2 – (4y)2 (9x2 + 16y2) [∵ (a – b)(a + b) = a2-b2]
= (9x2 – 16y2)(9x2 + 16y2)
= (9x2)2 – (16y2)2 [∵ (a – b)(a + b) = a2-b2]
= 81x4 – 256y4
(ix) (a + bc)(a – bc)(a2 + b2c2)
= [a2 – (bc)2] (a2 + b2c2) [∵ (a – b)(a + b) = a2-b2]
= (a2 – b2c2)(a2 + b2c2)
= (a2)2 – (b2c2)2 [∵ (a – b)(a + b) = a2-b2]
= a4 – b4c4
(x) 203 × 197
= (200 + 3)(200 – 3)
= (200)2 – (3)2
= 40000 – 9 [∵ (a – b)(a + b) = a2-b2]
= 39991
(xi) 20.8 × 19.2
= (20 + 0.8)(20 - 0.8)
= (20)2 – (0.8)2 [∵ (a – b)(a + b) = a2-b2]
= 400 - 0.64
= 399.36
3. Find the square of :
(i) 3x + 2/y
(ii) 5a/6b – 6b/5a
(iii) 2m2 – 2/3m2
(iv) 5x + 1/5x
(v) 8x + 3/2y
(vi) 607
(vii) 391
(viii) 9.7
Solution
(i) 3x + 2/y
(3x + 2/y)2
= (3x)2 + (2/y)2 + 2(3x)(2/y)
= 9x2 + 4/y2 + 12x/y
(ii) (5a/6b - 6b/5a)2
= (5a/6b)2 + (6b/5a)2 - 2 × 5a/6b × 6b/5a
= 25a2/36b2 - 2 + 36b2/25a2
(iii) 2m2 – 2/3n2
= (2m2 – 2/n2)
= (2m2)2 + (2/3n2)2 – 2 × 2m2 × 2/3n2
= 4m4 + 4/9n4 – 8/3m2n2
= 4m4– 8/3m2n2 + 4/9n2
(iv) (5x + 1/5x)2
= (5x)2 + 1/(5x)2 + 2 × 5x × 1/5x
= 25x2 + 1/25x2 + 2
= 25x2 + 2 + 1/25x2
(v) (8x + 3/2y)2
= (8x)2 + (3/2y)2 + 2 × 8x × 3/2y
= 64x2 + 9/4y2 + 24xy
= 64x2 + 24xy + 9/4y2
(vi) (607)2 = (600 + 7)2
= (600)2 + (7)2 + 2(600)(7)
= 360000 + 49 + 8400
= 368449
(vii) (391)2 = (400 – 9)2
= (400)2 + 92 – 2(400)(9)
= 16000 + 81 – 7200
= 152881
(viii) (9.7)2 = (10 - 0.3)2
= (10)2 – (0.3)2 – 2(10)(0.3)
= 100 + 0.09 – 6
= 100.09 – 6.00
= 94.09
4. If a + 1/a = 2, find:
(i) a2 + 1/a2
(ii) a2 + 1/a4
Solutio
(i) a2 + 1/a2
= (a + 1/a)2 – 2
= (2)3 – 2
= 4 – 2
= 2
(ii) a4 + 1/a4
= 35x2y2 + 45xy – 49xy - 63
= 35x2y2 – 4xy – 63
(viii) (3a2 – 4b2)(8a2 – 3b2)
= 3a2 (8a2 – 3b2) – 4b2(8a2 – 3b2)
= 24a4 – 9a2b2 – 32a2b2 + 12b4
= 24a4 – 41a2b2 + 12b4
2. Evaluate :
(i) (2x – 3/5)(2x + 3/5)
(ii) (4/7a + 3/4b) (4/7 a – 3/4b)
(iii) (6 – 5xy)(6 + 5xy)
(iv) (2a + 1/2a) (2a – 1/2a)
(v) (4x2 – 5y2)(4x2 + 5y2)
(vi) (1.6x + 0.7y)(1.6x – 0.7y)
(vii) (m + 3)(m – 3)(m2 + 9)
(viii) (3x + 4y)(3x – 4y)(9x2 + 16y2)
(ix) (a + bc)(a – bc)(a2 + b2c2)
(x) 203 × 197
(xi) 20.8 × 19.2
Solution
(i) (2x – 3/5)(2x + 3/5)
= (2x)2 – (3/5)2 [∵ (a – b)(a + b) = a2-b2]
= 4x2 – 9/25
(ii) (4/7a + 3/4b)(4/7a – 3/4b)
= (4/7a)2 – (3/4b)2 [∵ (a – b)(a + b) = a2-b2]
= 16/49a2 – 9/16b2
(iii) (6 – 5xy) (6 + 5xy)
= (6)2 – (5xy)2
= 36 – 25x2y2 [∵ (a – b)(a + b) = a2-b2]
(iv) (2a + 1/2a)(2a – 1/2a)
= (2a)2 – (1/2a)2 [∵ (a – b)(a + b) = a2-b2]
= 4a2 – 1/4a2
(v) (4x2 – 5y2)(4x2 + 5y2)
= (4x2)2 – (5y2)2
= 16x4 – 25y4 [∵ (a – b)(a + b) = a2-b2]
(vi) (1.6x + 0.7y)(1.6x – 0.7y)
= (1.6x)2 – (0.7y)2 [∵ (a – b)(a + b) = a2-b2]
= 2.56x2 – 0.49y2
(vii) (m + 3)(m – 3)(m2 + 9)
= (m)2 – (3)2 (m2 + 9) [∵ (a – b)(a + b) = a2-b2]
= (m2 – 9)(m2 + 9)
= (m2)2 – 92
= m4 – 81
(viii) (3x + 4y)(3x – 4y)(9x2 + 16y2)
= [(3x)2 – (4y)2 (9x2 + 16y2) [∵ (a – b)(a + b) = a2-b2]
= (9x2 – 16y2)(9x2 + 16y2)
= (9x2)2 – (16y2)2 [∵ (a – b)(a + b) = a2-b2]
= 81x4 – 256y4
(ix) (a + bc)(a – bc)(a2 + b2c2)
= [a2 – (bc)2] (a2 + b2c2) [∵ (a – b)(a + b) = a2-b2]
= (a2 – b2c2)(a2 + b2c2)
= (a2)2 – (b2c2)2 [∵ (a – b)(a + b) = a2-b2]
= a4 – b4c4
(x) 203 × 197
= (200 + 3)(200 – 3)
= (200)2 – (3)2
= 40000 – 9 [∵ (a – b)(a + b) = a2-b2]
= 39991
(xi) 20.8 × 19.2
= (20 + 0.8)(20 - 0.8)
= (20)2 – (0.8)2 [∵ (a – b)(a + b) = a2-b2]
= 400 - 0.64
= 399.36
3. Find the square of :
(i) 3x + 2/y
(ii) 5a/6b – 6b/5a
(iii) 2m2 – 2/3m2
(iv) 5x + 1/5x
(v) 8x + 3/2y
(vi) 607
(vii) 391
(viii) 9.7
Solution
(i) 3x + 2/y
(3x + 2/y)2
= (3x)2 + (2/y)2 + 2(3x)(2/y)
= 9x2 + 4/y2 + 12x/y
(ii) (5a/6b - 6b/5a)2
= (5a/6b)2 + (6b/5a)2 - 2 × 5a/6b × 6b/5a
= 25a2/36b2 - 2 + 36b2/25a2
(iii) 2m2 – 2/3n2
= (2m2 – 2/n2)
= (2m2)2 + (2/3n2)2 – 2 × 2m2 × 2/3n2
= 4m4 + 4/9n4 – 8/3m2n2
= 4m4– 8/3m2n2 + 4/9n2
(iv) (5x + 1/5x)2
= (5x)2 + 1/(5x)2 + 2 × 5x × 1/5x
= 25x2 + 1/25x2 + 2
= 25x2 + 2 + 1/25x2
(v) (8x + 3/2y)2
= (8x)2 + (3/2y)2 + 2 × 8x × 3/2y
= 64x2 + 9/4y2 + 24xy
= 64x2 + 24xy + 9/4y2
(vi) (607)2 = (600 + 7)2
= (600)2 + (7)2 + 2(600)(7)
= 360000 + 49 + 8400
= 368449
(vii) (391)2 = (400 – 9)2
= (400)2 + 92 – 2(400)(9)
= 16000 + 81 – 7200
= 152881
(viii) (9.7)2 = (10 - 0.3)2
= (10)2 – (0.3)2 – 2(10)(0.3)
= 100 + 0.09 – 6
= 100.09 – 6.00
= 94.09
4. If a + 1/a = 2, find:
(i) a2 + 1/a2
(ii) a2 + 1/a4
Solutio
(i) a2 + 1/a2
= (a + 1/a)2 – 2
= (2)3 – 2
= 4 – 2
= 2
(ii) a4 + 1/a4
= (a2 + 1/a2)2 – 2
= (2)2 – 2
= 4 – 2
= 2
5. If m – 1/m = 5, find:
(i) m2 + 1/m2
(ii) m4 + 1/m4
(iii) m2 – 1/m2
Solution
(i) m2 + 1/m2 = (m – 1/m)2 + 2
= (5)2 + 2
= 25 + 2
= 27
(ii) m4 + 1/m4
= (m2 – 1/m2)2 – 2
= (27)2 – 2
= 729 – 2
= 727
(iii) m2 – 1/m2
= (m + 1/m)(m – 1/m)
= 5(m + 1/m)
Now (m + 1/m)2 = (m – 1/m)2 + 4
= (5)2 + 4
= 25 + 4
= 29
∴ m + 1/m = √29
∴ m2 - 1/m2 = (5)(√29)
= 5√29
6. If a2 + b2 = 41 and ab = 4, find :
(i) a – b
(ii) a + b
Solution
(i) (a – b)2 = a2 + b2 – 2ab
= 41 – 2(4)
= 41 – 8
= 33
∴ a – b = √33
(ii) (a + b)2 = a2 + b2 + 2ab
= 41 + 2(4)
= 41 + 8
= 49
⇒ (a + b)2 = 49
∴ a + b = 7
7. If 2a + 1/2a = 8, find:
(i) 4a2 + 1/4a2
(ii) 16a4 + 1/16a4
Solution
(i) 4a2 + 1/4a2
= (2a + 1/2a)2 – 2×2a× 1/2a
(2a + 1/2a)2 – 2
= (8)2 – 2
= 64 – 2
= 62
(ii) 16a4 + 1/16a4
= (4a2 + 1/4a2)2 – 2×4a2× 1/4a2
= (2)2 – 2
= 4 – 2
= 2
5. If m – 1/m = 5, find:
(i) m2 + 1/m2
(ii) m4 + 1/m4
(iii) m2 – 1/m2
Solution
(i) m2 + 1/m2 = (m – 1/m)2 + 2
= (5)2 + 2
= 25 + 2
= 27
(ii) m4 + 1/m4
= (m2 – 1/m2)2 – 2
= (27)2 – 2
= 729 – 2
= 727
(iii) m2 – 1/m2
= (m + 1/m)(m – 1/m)
= 5(m + 1/m)
Now (m + 1/m)2 = (m – 1/m)2 + 4
= (5)2 + 4
= 25 + 4
= 29
∴ m + 1/m = √29
∴ m2 - 1/m2 = (5)(√29)
= 5√29
6. If a2 + b2 = 41 and ab = 4, find :
(i) a – b
(ii) a + b
Solution
(i) (a – b)2 = a2 + b2 – 2ab
= 41 – 2(4)
= 41 – 8
= 33
∴ a – b = √33
(ii) (a + b)2 = a2 + b2 + 2ab
= 41 + 2(4)
= 41 + 8
= 49
⇒ (a + b)2 = 49
∴ a + b = 7
7. If 2a + 1/2a = 8, find:
(i) 4a2 + 1/4a2
(ii) 16a4 + 1/16a4
Solution
(i) 4a2 + 1/4a2
= (2a + 1/2a)2 – 2×2a× 1/2a
(2a + 1/2a)2 – 2
= (8)2 – 2
= 64 – 2
= 62
(ii) 16a4 + 1/16a4
= (4a2 + 1/4a2)2 – 2×4a2× 1/4a2
= (62)2 – 2
= 3844 – 2
= 3842
8. If 3x – 1/3x = 5, find:
(i) 9x2 + 1/9x2
(ii) 81x4 + 1/81x4
Solution
(i) 9x2 + 1/9x2 = (3x – 1/3x)2 + 2
= (5)2 + 2
= 25 + 2
= 27
= (9x2 + 1/9x2)2 – 2
(ii) 81x4 + 1/81x4
= (9x2 + 1/9x2)2 - 2
= (27)2 – 2
= 729 – 2
= 727
9. Expand:
(i) (3x – 4y + 5x)2
(ii) (2a – 5b – 4c)2
(iii) (5x + 3y)3
(iv) (6a - 7b)3
Solution
(i) (3x – 4y + 5z)2
= (3x)2 + (-4y)2 + (5z)2 + 2(3x)(-4y) + 2(-4y)(5z) + 2(5z)(3x)
= 9x2 + 16y2 + 25z2 – 24xy – 40yz + 30zx
(ii) (2a – 5b – 4c)2
= (2a)2 + (-5b)2 + (-4c)2 + 2(2a)(-5b) + 2(-5b)(-4c) + 2(-4c)(2a)
= 4a2 + 25b2 + 16c2 – 20ab + 40bc – 16ca
(iii) (5x + 3y)3
= (5x)3 + (3y)3 + 3(5x)(3y)(5x + 3y)
= 125x3 + 27y3 + 45xy(5x + 3y)
= 125x3 + 27y3 + 225x2y + 135xy2
(iv) (6a – 7b)3
= (6a)3 – (7b)3 – 3(6a)(7b)(6a – 7b)
= 216a3 – 343b3 – 126ab(6a – 7b)
= 216a3 – 343b3 – 75a2b + 882ab2
= 216a3 – 756a2b + 882ab2 – 343b3
10. If a + b + c = 9 and ab + bc + ca = 15, find: a2 + b2 + c2.
Solution
Since, (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
∴ (9)2 = a2 + b2 + c2 + 2(15)
81 = a2 + b2 + c2 + 30
∴ a2 + b2 + c2 = 81 – 30 = 51
11. If a + b + c = 11 and a2 + b2 + c2 = 81, find ab + bc + ca.
Solution
Since, (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
∴ (11)2 = 81 + 2(ab + bc + ca)
∴ 2(ab + bc + ca) = 121 – 81 = 40
ab + bc + ca = 40/2
⇒ ab + bc + ca = 20
12. If 3x – 4y = 5 and xy = 3, find: 27x3 – 64y3
Solution
27x3 – 64x3
= 3844 – 2
= 3842
8. If 3x – 1/3x = 5, find:
(i) 9x2 + 1/9x2
(ii) 81x4 + 1/81x4
Solution
(i) 9x2 + 1/9x2 = (3x – 1/3x)2 + 2
= (5)2 + 2
= 25 + 2
= 27
= (9x2 + 1/9x2)2 – 2
(ii) 81x4 + 1/81x4
= (9x2 + 1/9x2)2 - 2
= (27)2 – 2
= 729 – 2
= 727
9. Expand:
(i) (3x – 4y + 5x)2
(ii) (2a – 5b – 4c)2
(iii) (5x + 3y)3
(iv) (6a - 7b)3
Solution
(i) (3x – 4y + 5z)2
= (3x)2 + (-4y)2 + (5z)2 + 2(3x)(-4y) + 2(-4y)(5z) + 2(5z)(3x)
= 9x2 + 16y2 + 25z2 – 24xy – 40yz + 30zx
(ii) (2a – 5b – 4c)2
= (2a)2 + (-5b)2 + (-4c)2 + 2(2a)(-5b) + 2(-5b)(-4c) + 2(-4c)(2a)
= 4a2 + 25b2 + 16c2 – 20ab + 40bc – 16ca
(iii) (5x + 3y)3
= (5x)3 + (3y)3 + 3(5x)(3y)(5x + 3y)
= 125x3 + 27y3 + 45xy(5x + 3y)
= 125x3 + 27y3 + 225x2y + 135xy2
(iv) (6a – 7b)3
= (6a)3 – (7b)3 – 3(6a)(7b)(6a – 7b)
= 216a3 – 343b3 – 126ab(6a – 7b)
= 216a3 – 343b3 – 75a2b + 882ab2
= 216a3 – 756a2b + 882ab2 – 343b3
10. If a + b + c = 9 and ab + bc + ca = 15, find: a2 + b2 + c2.
Solution
Since, (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
∴ (9)2 = a2 + b2 + c2 + 2(15)
81 = a2 + b2 + c2 + 30
∴ a2 + b2 + c2 = 81 – 30 = 51
11. If a + b + c = 11 and a2 + b2 + c2 = 81, find ab + bc + ca.
Solution
Since, (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
∴ (11)2 = 81 + 2(ab + bc + ca)
∴ 2(ab + bc + ca) = 121 – 81 = 40
ab + bc + ca = 40/2
⇒ ab + bc + ca = 20
12. If 3x – 4y = 5 and xy = 3, find: 27x3 – 64y3
Solution
27x3 – 64x3
= (3x)3 – (4y)3
= (3x – 4y)3(3x – 4y)3 + 3(3x)(4y)(3x – 4y) [∵ a3 – b3 = (a – b)3 + 3ab(a – b)]
= (5)3 + 36(xy)(3x – 4y)
= 125 + 36(3)(5)
= 125 + 540
= 665
13. If a + b = 8 and ab = 15, find : a3 + b3.
Solution
a3 + b3
= (3x – 4y)3(3x – 4y)3 + 3(3x)(4y)(3x – 4y) [∵ a3 – b3 = (a – b)3 + 3ab(a – b)]
= (5)3 + 36(xy)(3x – 4y)
= 125 + 36(3)(5)
= 125 + 540
= 665
13. If a + b = 8 and ab = 15, find : a3 + b3.
Solution
a3 + b3
= (a + b)3 – 3ab(a + b)
= (8)3 – 3(15)(8)
= 512 – 360
= 152
14. If 3x + 2y = 9 xy = 3, find: 27x3 + 8y3
Solution
27x3 + 8y3 = (3x)3 + (2y)3
= (3x + 2y)3 – 3×3x×2y(3x + 2y)
= (3x – 2y)3 – 18xy(3x + 2y)
= (9)3 – 18(3)(9)
= 729 – 486
= 243
15. If 5x – 4y = 7 and xy = 8, find : 125x3 – 64y3
Solution
125x3 – 64y3
= (5x)3 – (4y)3
= (5x – 4y)3 + 3(5x)(4y)(5x – 4y)
= (5x - 4y)3 + 60xy(5x - 4y)
= (7)3 + 60(8)(7)
= 343 + 3360
= 3703
16. The difference between two numbers is 5 and their products is 14. Find the difference between their cubes.
Solution
Let x and y be two numbers, then x – y = 5 and xy = 14
∴ x3– y3 = (x – y)3 + 3xy(x – y)
= (5)3 + 3 × 14 × 5
= 125 + 210
= 335
= (8)3 – 3(15)(8)
= 512 – 360
= 152
14. If 3x + 2y = 9 xy = 3, find: 27x3 + 8y3
Solution
27x3 + 8y3 = (3x)3 + (2y)3
= (3x + 2y)3 – 3×3x×2y(3x + 2y)
= (3x – 2y)3 – 18xy(3x + 2y)
= (9)3 – 18(3)(9)
= 729 – 486
= 243
15. If 5x – 4y = 7 and xy = 8, find : 125x3 – 64y3
Solution
125x3 – 64y3
= (5x)3 – (4y)3
= (5x – 4y)3 + 3(5x)(4y)(5x – 4y)
= (5x - 4y)3 + 60xy(5x - 4y)
= (7)3 + 60(8)(7)
= 343 + 3360
= 3703
16. The difference between two numbers is 5 and their products is 14. Find the difference between their cubes.
Solution
Let x and y be two numbers, then x – y = 5 and xy = 14
∴ x3– y3 = (x – y)3 + 3xy(x – y)
= (5)3 + 3 × 14 × 5
= 125 + 210
= 335