Expansion formulae
Class-8-Mathematics-Chapter-5-Maharashtra Board
Solutions
Practice Set 5.1
Question 1.1. Expand.
(1) (a + 2)(a − 1)
(a + 2)(a − 1) = a2 + (2 − 1)a + (2) × (−1) = a2 + a − 2 Answer : a2 + a − 2.
(2) (m − 4)(m + 6)
(m − 4)(m + 6) = m2 + (−4 + 6)m + (−4) × (6) = m2 + 2m − 24 Answer : m2 + 2m − 24.
(3) (p + 8)(p −3)
(p + 8)(p −3) = p2 + (8 − 3)p + (8) × (−3) = p2 + 5p − 24 Answer : p2 + 5p − 24.
(4) (13 + x)(13 − x)
(13 + x)(13 − x) = (13)2 + (x − x)13 + (x) × (−x) = 169 + (0) × 13 − x2 = 169 − x2 Answer : 169 − x2.
(5) (3x + 4y)(3x + 5y)
(3x + 4y)(3x + 5y) = (3x)2 + (4y + 5y)(3x) + (4y) × (5y) = 9x2 + 9y × 3x + 20y2 = 9x2 + 27xy + 20y2 Answer : 9x2 + 27xy + 20y2.
(6) (9x − 5t)(9x + 3t)
(9x − 5t)(9x + 3t) = (9x)2 + (−5t + 3t) × 9x + (−5t) × 3t = 81x2 − 2t × 9x − 15t2 = 81x2 −18xt − 15t2 Answer : 81x2 −18xt − 15t2.
(7) \((m+\frac{2}{3})\)\((m−\frac{7}{3})\)
\((m+\frac{2}{3})\)\((m−\frac{7}{3})\) = m2 + \((\frac{2}{3}−\frac{7}{3})\)m + \((\frac{2}{3})×(−\frac{7}{3})\) = m2 − \(\frac{5m}{3}\) − \(\frac{14}{9}\) Answer : m2 − \(\frac{5m}{3}\) − \(\frac{14}{9}\)
(8) \((x+\frac{1}{x})\)\((x−\frac{1}{x})\)
\((x+\frac{1}{x})\)\((x−\frac{1}{x})\) = x2 + \((\frac{1}{x}−\frac{1}{x})\)x + \((\frac{1}{x})×(−\frac{1}{x})\) = x2 + (0)x − \(\frac{1}{x^2}\) = x2 − \(\frac{1}{x^2}\) Answer : x2 − \(\frac{1}{x^2}\)
(9) \((\frac{1}{y}+4)\)\((\frac{1}{y}−9)\)
\((\frac{1}{y}+4)\)\((\frac{1}{y}−9)\) = \((\frac{1}{y})^2\) + (4 – 9)\((\frac{1}{y})\) + 4 × −9 = \(\frac{1}{y^2}\) − \(\frac{5}{y}\) – 36 Answer : \(\frac{1}{y^2}\) − \(\frac{5}{y}\) – 36
Practice Set 5.2
Question 2.1. Expand.
(1) (k + 4)3
(k + 4)3 = (k)3 + 3 x k2 x 4 + 3 × k x (4)2 + (4)3 = k3 + 12k2 + 48k + 64 Answer : k3 + 12k2 + 48k + 64.
(2) (7x + 8y)3
(7x + 8y)3 = (7x)3 + 3(7x)2(8y) + 3(7x)(8y)2 + (8y)3 = 343x3 + 3(49x2)(8y) + 3(7x)(64y2) + 512y3 = 343x3 + 1176x2y + 1344xy2 + 512y3 Answer : 343x3 + 1176x2y + 1344xy2 + 512y3
(3) (7 + m)3
(7 + m)3 = (7)3 + 3(7)2(m) +3(7)(m)2 + m3 = 343 + 3(49)(m) + 21m2 + m3 = 343 + 147m + 21m2 + m3 Answer : 343 + 147m + 21m2 + m3.
(4) (52)3
(52)3 = (50 + 2)3 = (50)3 + 3(50)2(2) + 3(50)(2)2 + (2)3 = 125000 + 3(2500)(2) + 3(50)(4) + 8 = 125000 + 15000 + 600 + 8 = 140608 Answer : 140608.
(5) (101)3
(101)3 = (100)3 + 3(100)2(1) + 3(100)(1)2 + (1)3 = 1000000 + 3(10000) + 300 + 1 = 1000000 + 30000 + 300 + 1 = 1030301 Answer : 1030301.
(6) \((x+\frac{1}{x})^3\)
\((x+\frac{1}{x})^3\) = x3 + 3(x)2\((\frac{1}{x})\) + 3(x)\((\frac{1}{x})^2\) + \((\frac{1}{x})^3\) = x3 + 3x + \(\frac{3}{x}\) + \(\frac{1}{x^3}\) Answer : x3 + 3x + \(\frac{3}{x}\) + \(\frac{1}{x^3}\)
(7) \((2m+\frac{1}{5})^3\)
\((2m+\frac{1}{5})^3\) = (2m)3 + 3(2m)2\((\frac{1}{5})\)+ 3(2m)\((\frac{1}{5})^2\) + \((\frac{1}{5})^3\) = 8m3 + \(\frac{12m}{5}\) + \(\frac{6m}{25}\) + \(\frac{1}{125}\) Answer : 8m3 + \(\frac{12m}{5}\) + \(\frac{6m}{25}\) + \(\frac{1}{125}\)
(8) \((\frac{5x}{y}+\frac{y}{5x})^3\)
\((\frac{5x}{y}+\frac{y}{5x})^3\) = \((\frac{5x}{y})^3\) + 3\((\frac{5x}{y})^2\)\(\frac{y}{5x})\) + 3\((\frac{5x}{y})\)\((\frac{y}{5x})^2\) + \((\frac{y}{5x})^3\) = \(\frac{125x^3}{y^3}\) + 3\((\frac{25x^2}{y^2})\)\(\frac{y}{5x})\) + 3\((\frac{5x}{y})\)\((\frac{y^2}{25x^2})\) + \(\frac{y^3}{125x^3}\) = \(\frac{125x^3}{y^3}\) + \(\frac{15x}{y}\) + \(\frac{3y}{x}\) + \(\frac{y^3}{125x^3}\) Answer : \(\frac{125x^3}{y^3}\) + \(\frac{15x}{y}\) + \(\frac{3y}{x}\) + \(\frac{y^3}{125x^3}\)
Practice Set 5.3
Question 3.1. Expand.
(1) (2m − 5)3
(2m − 5)3 = (2m)3 – 3 × (2m)2 × 5 + 3 × (2m) × (5)2 − (5)3 = 8m3 – 15 × (4m2) + 6m × (25) − 125 = 8m3 − 60m2 + 150m − 125 Answer : 8m3 − 60m2 + 150m − 125.
(2) (4 − p)3
(4 − p)3 = (4)3 – 3 × (4)2 x (p) + 3 x (4) x (p)2 −(p)3 = 64 − 48p + 12p2 − p3 Answer : 64 − 48p + 12p2 − p3.
(3) (7x − 9y)3
(7x − 9y)3 = (7x)3 – 3 × (7x)2 × (9y) + 3 × (7x) x (9y)2 − (9y)3 = 343x3 – 3 × (49x2) × 9y + 3 × (7x) x (81y2) − 729y3 = 343x3 − 1323x2y + 1701xy2 − 729y3 Answer : 343x3 − 1323x2y + 1701xy2 − 729y3.
(4) (58)3
(58)3 = (60 − 2)3 =(60)3 – 3 × (60)2 × (2) + 3 x (60) x (2)2 − (2)3 = 216000 – 6 × 3600 + 180 × 4 − 8 = 216000 – 21600 + 720 − 8 = 195112 Answer : 1,95,112.
(5) (198)3
(198)3 = (200 − 2)3 = (200)3 – 3 × (200)2 × (2) + 3 × (200) × (2)2 − (2)3 = 8000000 – 3 × (40000) × 2 + 600 × 4 − 8 = 8000000 – 240000 + 2400 − 8 = 7762392 Answer : 77,62,392.
(6) \((2p-\frac{1}{2p})^3\)
\((2p-\frac{1}{2p})^3\) = (2p)3 – 3 x (2p)2 x \(\frac{1}{2p}\) + 3 x (2p) x \((\frac{1}{2p})^2\) − \((\frac{1}{2p})^3\) = 8p3 – 6p + \(\frac{3}{2p}\) − \(\frac{1}{8p^3}\) Answer : 8p3 – 6p + \(\frac{3}{2p}\) − \(\frac{1}{8p^3}\)
(7) \((1-\frac{1}{a})^3\)
\((1-\frac{1}{a})^3\) = 13 – 3 x (1)2 x \(\frac{1}{a}\) + 3 x (1) x \((\frac{1}{a})^2\) − \((\frac{1}{a})^3\) = 1 – \(\frac{3}{a}\) + \(\frac{3}{a^2}\) − \(\frac{1}{a^3}\) Answer : 1 – \(\frac{3}{a}\) + \(\frac{3}{a^2}\) − \(\frac{1}{a^3}\)
(8) \((\frac{x}{3}-\frac{3}{x})^3\)
\((\frac{x}{3}-\frac{3}{x})^3\) = \((\frac{x}{3})^3\) – 3 x \((\frac{x}{3})^2\) x \((\frac{3}{x})\) + 3 x \((\frac{x}{3})\) x \((\frac{3}{x})^2\) − \((\frac{3}{x})^3\) = \(\frac{x^3}{27}\) – x + \(\frac{9}{x}\) − \(\frac{27}{x^3}\) Answer : \(\frac{x^3}{27}\) – x + \(\frac{9}{x}\) − \(\frac{27}{x^3}\)
Question 3.2. Simplify.
(1) (2a + b)3 − (2a − b)3
(2a + b)3 −(2a − b)3 = (2a)3 + 3(2a)2(b) + 3(2a)(b)2 + (b)3 − [(2a)3 − 3(2a)2(b) + 3(2a)(b)2 − (b)3] = 8a3 + 12a2b + 6ab2 + b3 − [8a3 − 12a2b + 6ab2 − b3] = 8a3 + 12a2b + 6ab2 + b3 − 8a3 + 12a2b − 6ab2 + b3 = 24a2b + 2b3 Answer : 24a2b + 2b3.
(2) (3r − 2k)3 + (3r + 2k)3
(3r − 2k)3 + (3r + 2k)3 = (3r)3 − 3(3r)2(2k) + 3(3r)(2k)2 − (2k)3 + (3r)3 + 3(3r)2(2k) + 3(3r)(2k)2 + (2k)3 = 27r3 −54r2k + 36rk2 − 8k3 + 27r3 + 54r2k + 36rk2+ 8k3 = 54r3 + 72rk2 Answer : 54r3 + 72rk2.
(3) (4a − 3)3 − (4a + 3)3
(4a − 3)3 − (4a + 3)3 = (4a)3 − 3(4a)2(3) + 3(4a)(3)2 − (3)3 − [(4a)3 + 3(4a)2(3) + 3(4a)(3)2 + (3)3] = 64a3 − 144a2 + 108a – 27 − 64a3 − 144a2 − 108a − 27 = − 288a2 − 54 Answer : − 288a2 − 54.
(4) (5x − 7y)3 + (5x + 7y)3
(5x − 7y)3 + (5x + 7y)3 = (5x)3 − 3(5x)2(7y) + 3(5x)(7y)2 − (7y)3 + (5x)3 + 3(5x)2(7y) + 3(5x)(7y)2 + (7y)3 = 125x3 + 735xy2 + 125x3 + 735xy2 = 250x3 + 1470xy2 Answer : 250x3 + 1470xy2.
Practice Set 5.4
Question 4.1. Expand.
(1) (2p + q + 5)2
(2p + q + 5)2 = (2p)2 + q2 + (5)2 + 2 x 2p x q + 2 x q x 5 + 2 × 5 x 2p = 4p2 + q2 + 25 + 4pq + 10q + 20p Answer : 4p2 + q2 + 25 + 4pq + 10q + 20p.
(2) (m + 2n + 3r)2
(m + 2n + 3r)2 = m2 + (2n)2 + (3r)2 + 2 x m x 2n + 2 x 2n x 3r + 2 × 3r x m = m2 + 4n2 + 9r2 + 4mn + 12nr + 6rm Answer : m2 + 4n2 + 9r2 + 4mn + 12nr + 6rm.
(3) (3x + 4y − 5p)2
(3x + 4y − 5p)2 = (3x)2 + (4y)2 + (−5p)2 + 2 × 3x × 4y + 2 × 4y × (−5p) + 2 × (−5p) × 3x = 9x2 + 16y2 + 25p2 + 24xy − 40yp − 30px Answer : 9x2 + 16y2 + 25p2 + 24xy − 40yp − 30px.
(4) (7m − 3n − 4k)2
(7m − 3n − 4k)2 = (7m)2 + (−3n)2 + (−4k)2 + 2 × 7m × (−3n) + 2 x (−3n) × (−4k) + 2 × (−4k) × 7m = 49m2 + 9n2 + 16k2 − 42mn + 24nk − 56km Answer : 49m2 + 9n2 + 16k2 − 42mn + 24nk − 56km.
Question 4.2. Simplify.
(1) (x − 2y + 3)2 + (x + 2y −3)2
(x − 2y + 3)2 + (x + 2y −3)2 = x2 + 4y2 + 9 − 4xy − 12y + 6x + x2 + 4y2 + 9 + 4xy − 12y − 6x = 2x2 + 8y2 + 18 − 24y Answer : 2x2 + 8y2 + 18 − 24y.
(2) (3k − 4r −2m)2 − (3k + 4r − 2m)2
(3k − 4r −2m)2 − (3k + 4r − 2m)2 = 9k2 + 16r2 + 4m2 + 2 x 3k x (−4r) + 2 x (−4r) x (−2m) + 2x (−2m) × 3k − [9k2 + 16r2 + 4m2 + 2 x 3k x 4r + 2 x 4r x (−2m) + 2 × (−2m) × 3k] = 9k2 + 16r2 + 4m2 − 24kr + 16rm − 12mk − 9k2 – 16r2 − 4m2 − 24kr + 16rm + 12mk = − 48kr + 32rm Answer : − 48kr + 32rm.
(3) (7a − 6b + 5c)2 + (7a + 6b − 5c)2
(7a − 6b + 5c)2 + (7a + 6b − 5c)2 = 49a2 + 36b2 + 25c2 + 2 x 7a x (−6b) + 2 x (−6b) x 5c + 2 × 5c × 7a + 49a2 + 36b2 + 25c2 + 2 x 7a x 6b + 2 x 6b x (−5c) + 2 x (−5c) x 7a = 49a2 + 36b2 + 25c2 − 84ab − 60bc + 70ca + 49a2 + 36b2 + 25c2 + 84ab − 60bc − 70ca = 98a2 + 72b2 + 50c2 − 120bc Answer : 98a2 + 72b2 + 50c2 − 120bc.
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