NCERT Solutions for Chapter 14 Factorisation Class 8 Mathematics
Page No: 220Exercise 14.1
1. Find the common factors of the given terms.
(i) 12x, 36
(ii) 2y, 22xy
(iii) 14pq, 28p2q2
(iv) 2x, 3x2, 4
(v) 6abc, 24ab2, 12a2b
(vi) 16x3, -4x2, 32x
(vii) 10 pq, 20qr, 30rp
(viii) 3x2y3, 10x3y2, 6x2y2z
Answer
(i) 12x = 2×2×3×x
36 = 2×2×3×3
Hence, the common factors are 2, 2 and 3 = 2×2×3 = 12
(ii) 2y = 2×y
22xy = 2×11×x×y
Hence, the common factors are 2 and y = 2×y = 2y
(iii) 14pq = 2×7×p×q
28p2q2 = 2×2×7×p×p×q×q
Hence, the common factors are 2×7×p×q = 14pq
(iv) 2x = 2×x×1
3x2 = 3×x×x×1
4 = 2×2×1
Hence, the common factor is 1.
(v) 6abc = 2×3×a×b×c
24ab2 = 2×2×2×3×a×b×b
12a2b = 2×2×3×a×a×b
Hence, the common factors are 2×3×a×b = 6ab
(vi) 16x3 = 2×2×2×x×x×x
-4x2 = (-1)×2×2×x×x
32x = 2×2×2×2×2×x
Hence the common factors are 2×2×x = 4x
(vii) 10pq = 2×5×p×q
20qr = 2×2×5×q×r
30rp = 2×3×5×r×p
Hence the common factors are 2×5 = 10
(viii) 3x2y3 = 3×x×x×y×y×y
10x3y2 = 2×5×x×x×x×y×y
6x2y2z = 2×3×x×x×y×y×z
Hence the common factors are x×x×y×y = x2y2
2. Factorize the following expressions.
(i) 7x - 42
(ii) 6p - 12q
(iii) 7a2 + 14a
(iv) -16z + 20z3
(v)20l2m + 30alm
(vi) 5x2y - 15xy2
(vii) 10a2 - 15b2 + 20c2
(viii) -4a2 + 4ab - 4ca
(ix) x2yz + xy2z + xyz2
(x) ax2y + bxy2 + cxyz
Answer
(i) 7x - 42 = 7×x - 2×3×7
Taking common factors from each term,
= 7(x - 2×3)
= 7(x - 6)
(ii) 6p - 12q = 2×3×p - 2×2×3×q
Taking common factors from each term,
= 2×3(p - 2q)
= 6(p - 2q)
(iii) 7a2 + 14a = 7×a×a + 2×7×a
Taking common factors from each term,
= 7×a(a + 2)
= 7a(a + 2)
(iv) -16z + 20z3
= (-1)×2×2×2×2×z + 2×2×5×z×z× z
Taking common factors from each term,
= 2×2×z (-2×2 + 5×z×z)
= 4z (-4 + 5z2)
(v) 20l2m + 30alm
= 2×2×5×l×l×m + 2×3×5×a×l×m
Taking common factors from each term,
= 2×5×l×m(2×l + 3×a)
= 10 lm(2l +3a)
(vi) 5x2y - 15xy2
= 5×x×x×y - 3×5×x×y×y (Taking common factors from each term)
= 5×x×y(x - 3y)
= 5xy(x - 3y)
(vii) 10a2 - 15b2 + 20c2
= 2×5×a×a - 3×5×b×b + 2×2×5×c×c
Taking common factors from each term,
= 5(2×a×a - 3×b×b + 2×2×c×c)
= 5(2a2 - 3b2 + 4c2)
(viii) -4a2 + 4ab - 4ca
= (-1)×2×2×a×a + 2×2×a×b - 2×2×c×a
Taking common factors from each term,
= 2×2×a(-a + b -c)
= 4a (-a + b - c)
(ix) x2yz + xy2z + xyz2
= x×x×y×z + x×y×y×z + z×y×z×z
Taking common factors from each term,
= x×y×z( x + y + z)
= xyz(x + y +z)
(x) ax2y + bxy2 + cxyz
= a×x×x×y + b×x×y×y + c×x×y×z
Taking common factors from each term,
= x×y(a×x + b×y + c×z)
= xy(ax + by +cz)
3. Factorize:
(i) x2 + xy + 8x + 8y
(ii) 15xy - 6x + 5y -2
(iii) ax + bx - ay - by
(iv) 15pq + 15 + 9q + 25p
(v) z - 7 + 7xy -xyz
Answer
(i) x2 + xy + 8x + 8y
= x(x + y) + 8(x + y)
= (x + y)(x + 8)
(ii) 15xy - 6x + 5y - 2
= 3x(5y - 2) + 1(5y - 2)
= (5y -2)(3x + 1)
(iii) ax + bx - ay - by
= (ax + bx) - (ay + by)
= x(a + b) - y(a + b)
= (a + b)(x - y)
(iv) 15pq + 15 + 9q + 25p
= 15pq + 25p + 9q + 15
= 5p(3q + 5) + 3(3q + 5)
= (3q + 5)(5p + 3)
(v) z -7 + 7xy - xyz = 7xy - 7 - xyz + z
= 7(xy - 1) - z(xy - 1)
= (xy -1)(7 - z) = (-1)(1 - xy)(-1)(z - 7)
= (1 - xy)(z - 7)
Page No. 223
Exercise 14.2
(ii) p2 - 10p + 25
(iii) 25m2 + 30m + 9
(iv) 49y2 + 84yz + 36z2
(v) 4x2 - 8x + 4
(vi) 121b2 - 88bc + 16c2
(vii) (l + m)2 - 4lm [Hint: Expand (l + m)2 first]
(viii) a4 + 2a2b2 + b4
Answer
(i) a2 + 8a + 16 = a2 + (4 + 4)a + 4 × 4
Using identity x2 + (a + b)x + ab = (x + a)(x + b),
Here x = a, a = 4 and b = 4
a2 + 8a + 16 = (a + 4)(a + 4) = (a + 4)2
(ii) p2 - 10p + 25 = p2 +(-5-5)p + (-5)(-5)
Using identity x2 + (a +b)x + ab = ( x + a)(x + b),
Here x = p, a = -5 and b = -5
p2 - 10p + 25 = (p -5)(p- 5) = (p - 5)2
(iii) 25m2 + 30m + 9 = (5m)2 + 2 × 5m × 3 + (3)2
Using identity a2 + 2ab + b2 = (a + b)2 , here a= 5m, b = 3
25m2 + 30m + 9 = (5m + 3)2
(iv) 49y2 + 84yz + 36z2 = (7y)2 + 2 × 7y × 6z + (6z)2
Using identity a2 + 2ab + b2 = (a + b)2 , here a = 7y, b = 6z
49y2 + 84yz + 36z2 = (7y + 6z)2
(v) 4x2 - 8x + 4 = (2x)2 - 2 × 2x ×2 + (2)2
Using identity a2 - 2ab + b2 = (a - b)2 , here a = 2x, b = 2
4x2 - 8x + 4 = (2x - 2)2
= (2)2 (x - 1)2 = 4( x - 1)2
(vi) 121b2 - 88bc + 16c2 = (11b)2 - 2 × 11b × 4c + (4c)2
Using identity a2 - 2ab + b2 = (a - b)2 , here a = 11b, b = 4c
121b2 - 88bc + 16c2 = (11b - 4c)2
(vii) (l + m)2 - 4lm
= l2 + 2 × l ×m + m2 - 4lm [ ∵ (a + b)2 = a2 + 2ab + b2 ]
= l2 + 2lm + m2 - 4lm
= l2 - 2lm + m2
= (l - m)2 [ ∵ (a- b)2 = a2 - 2ab + b2 ]
(viii) a4 + 2a2b2 + b4 = (a2)2 + 2 × a2 × b2 + (b2)2
= (a2 + b2)2 [∵ (a + b)2 = a2 + 2ab + b ]
(i) 4p2 - 9q2 (ii) 63a2 - 112b2
(iii) 49x2 - 36
(iv) 16x5 - 144x2
(v) (l + m)2 - (l -m)2
(vi) 9x2y2 - 16
(vii) (x2 - 2xy + y2) - z2
(viii) 25a2 - 4b2 + 28bc - 49c2
Answer
(i) 4p2 - 9q2 = (2p)2 - (3q)2
= (2p -3q)(2p + 3q) [∵ a2 - b2 = (a - b)(a +b)]
(ii) 63a2 - 112b2 = 7(9a2 - 16b2)
= 7 [ (3a)2 - (4b)2]
= 7(3a - 4b)(3a + 4b) [∵ a2 - b2 = (a - b)(a +b)]
(iii) 49x2 - 36 = (7x)2 - (6)2
= (7x - 6)(7x + 6) [∵ a2 - b2 = (a - b)(a +b)]
(iv) 16x5 - 144x3 = 16x3(x2 - 9)
= 16x3 [(x)2 - (3)2]
= 16x3 (x - 3)(x + 3) [∵ a2 - b2 = (a - b)(a +b)]
(v) (l + m)2 - (l - m)2
= [(l + m) + ( l - m)][(l + m)- (l - m)] [∵ a2 - b2 = (a - b)(a +b)]
= (l + m + l - m)(l + m - l +m)
= (2l) (2m) = 4lm
(vi) 9x2y2 - 16 = (3xy)2 - (4)2
= (3xy - 4)(3xy + 4) [ ∵ a2 - b2 = (a - b)(a +b)]
(vii) ( x2 - 2xy + y2) - z2 = ( x - y)2 - z2 [∵ (a -b)2 = a2 -2ab + b2]
= ( x - y - z)( x - y + z) [ ∵ a2 - b2 = (a - b)(a +b)]
(viii) 25a2 - 4b2 + 28bs - 49c2
= 25a2 - (4b2 - 28bc + 49c2)
= 25a2 - [ (2b)2 - 2 × 2b × 7c + (7c)2]
= 25a2 - (2b - 7c)2 [ ∵ (a -b)2 = a2 -2ab + b2]
= (5a)2 - (2b - 7c)2
= [5a - (2b - 7c)][5a + (2b - 7c)] [ ∵ a2 - b2 = (a - b)(a +b)]
= (5a - 2b + 7c)(5a + 2b - 7c)
(i) ax2 + bx(ii) 7p2 + 21q2
(iii) 2x3 + 2xy2 + 2xz2
(iv) am2 + bm2 + bn2 + an2
(v) (lm + l ) + m + 1
(vi) y( y + z) + 9 ( y + z)
(vii) 5y2 - 20y - 8z + 2yz
(viii) 10ab + 4a + 5b + 2
(ix) 6xy - 4y + 6 - 9x
Answer
(i) ax2 + bx = x(ax + b)
(ii) 7p2 + 21q2 = 7(p2 + 3q2)
(iii) 2x3 + 2xy2 + 2xz2 = 2x( x2 + y2 + z2)
(iv) am2 + bm2 + bn2 + an2
= m2( a + b) + n2(a + b)
= (a + b )(m2 + n2)
(v) (lm + l) + m + 1
= l(m + 1) + 1(m + 1)
= (m + 1)( l + 1)
(vi) y(y + z) + 9(y + z)
= (y + z)(y + 9)
(vii) 5y2 - 20y - 8z + 2yz
= 5y2 - 20y + 2yz - 8z
= 5y(y - 4) + 2z(y - 4)
= (y - 4)(5y + 2z)
(viii) 10ab + 4a + 5b + 2
= 2a(5b + 2) + 1 (5b + 2)
= (5b + 2)(2a + 1)
(ix) 6xy - 4y + 6 - 9x
= 6xy - 9x - 4y + 6
= 3x(2y - 3) - 2(2y - 3)
= (2y - 3) (3x - 2)
4. Factorize:
(i) a4 - b4
(ii) p4 - 81
(iii) x4 - (y + z)4
(iv)x4 - (x -z)4
(v) a4 - 2a2b2 + b4
Answer
(i) a4 - b4 = (a2)2 - (b2)2
= (a2 - b2)( a2 + b2) [ ∵ a2 - b2 = (a - b)(a +b)]
= (a - b)(a + b)(a2 + b2) [ ∵ a2 - b2 = (a - b)(a +b)]
(ii) p4 - 81 = (p2)2 - (9)2
= (p2 - 9)(p2 + 9) [∵ a2 - b2 = (a - b)(a +b)]
= (p2 - 32)(p2 + 9)
= ( p - 3)(p + 3)(p2 + 9) [∵ a2 - b2 = (a - b)(a +b)]
(iii) x4 - (y + z)4 = (x2)2 - [(y + z)2]2
= [x2 - (y + z)2][ x2 + (y + z)2] [∵ a2 - b2 = (a - b)(a +b)]
= [x -(y +z)][x + (y + z)][x2 + (y + z)2] [∵ a2 - b2 = (a - b)(a +b)]
= (x - y - z) (x + y + z) [x2 + (y + z)2]
(iv) x4 - (x - z)4 = (x2)2 - [(x - z)2]2
= [x2 -(x - z)2][x2 + (x - z)2] [ ∵ a2 - b2 = (a - b)(a +b)]
= [x - (x - z)][x + (x - z)] [x2 + (x - z)2] [∵ a2 - b2 = (a - b)(a +b)]
= [x - x + z] [x + x - z] [x2 + x2 - 2xz + z2] [∵ (a -b)2 = a2 -2ab + b2]
= z(2x - z) (2x2 - 2xz + z2)
(v) a4 - 2a2b2 + b4 = (a2)2 - 2a2b2 + (b2)2
= (a2 - b2)2 [∵ a2 - b2 = (a - b)(a +b)]
= [(a - b)(a + b)]2 [∵ a2 - b2 = (a - b)(a +b)]
= (a -b)2 (a + b)2 [ (xy)m = xmym]
5. Factorize the following expressions:
(i) p2 + 6p + 8
(ii) q2 - 10q + 21
(iii) p2 + 6p - 16
Answer
(i) p2 + 6p + 8 = p2 + ( 4 + 2)p + 4 × 2
= p2 + 4p + 2p + 4 ×2
= p(p + 4) + 2 ( p + 4)
= (p + 4)(p + 2)
(ii) q2 - 10q + 21 = q2 - ( 7 + 3)q + 7 × 3
= q2 - 7q - 3q + 7 × 3
= q(q - 7) - 3(q - 7)
= (q - 7)( q - 3)
(iii) p2 + 6p - 16
= p2 + (8 - 2)p - 8×2
= p2 + 8p - 2p - 8×2
= p(p + 8) - 2(p + 8)
= ( p + 8)(p -2)
Page No. 227
Exercise 14.3
1. Carry out the following divisions:
(i) 2x4 ÷ 56x
(ii) -36y3 ÷ 9y2
(iii) 66pq2r3 ÷ 11 qr2
(iv) 34x3y3x3 ÷ 51xy2z3
(v) 12a8b8 ÷ (-6a6b4)
Answer
(i) 2x4 ÷ 56x
= 28x4/56x
= 28/56 × x4/x
= 1/2 x3 [xm ÷ xn = xm-n]
(ii) -36y3 ÷ 9y2 = -36y3/9y2
= -36/9 × y3/y2
= -4y [xm ÷ xn = xm-n]
(iii) 66pq2r3 ÷ 11qr2
= 66pq2r3/11qr2
= 66/11 × pq2r3/qr2
= 6pqr [xm ÷ xn = xm-n]
(iv) 34x3y3z3 ÷ 51xy2z3
= 34x3y3z3/51xy2z3
= 34/51 ×x3y3z3/xy2z3
= 2/3x2y [xm ÷ xn = xm-n]
(v) 12a8b8 ÷ (- 6a6b4)
= 12a8b8/- 6a6b4
= 12/-6 × a8b8/a6b4
= -2a2b4 [xm ÷ xn = xm-n]
2. Divide the given polynomial by the given monomial:
(i) (5x2 - 6x) ÷ 3x
(ii) (3y8 - 4y6 + 5y4) ÷ y4
(iii) 8(x3y2z2 + x2y3z2 + x2y2z3) ÷ 4x2y2z2
(iv) (x3 + 2x2 + 3x) ÷2x
(v) (p3q6 - p6q3) ÷ p3q3
Answer
(i) (5x2 - 6x) ÷3x
= (5x2 - 6x)/3x
= 5x2/3x - 6x/3x = (5/3)x - 2 = 1/3 (5x - 6)
(ii) (3y8 - 4y6 + 5y4) ÷ y4
= (3y8 - 4y6 + 5y4)/ y4
= 3y8/y4 - 4y6/y4 + 5y4/y4 = 3y4 - 4y2 + 5
(iii) 8(x3y2z2 + x2y3z2 + x2y2z3) ÷ 4x2y2z2
= {8(x3y2z2 + x2y3z2 + x2y2z3)}/4 x2y2z2
= 8 x3y2z2/4 x2y2z2 + 8 x2y3z2/4x2y2z2 + 8 x2y2z3/4x2y2z2
= 2x + 2y + 2z
= 2(x + y + z)
(iv) (x3 + 2x2 + 3x) ÷ 2x
= (x3 + 2x2 + 3x)/2x
= x3/2x + 2x2/2x + 3x/2x = x2/2 + 2x/2 + 3/2
= 1/2( x2 + 2x + 3)
(v) (p3q6 - p6q3) ÷ p3q3
= (p3q6 - p6q3)/p3q3
= p3q6/p3q3 - p6q3/p3q3 = q3 - p3
3. Work out the following divisions:
(i) (10x - 25) ÷ 5
(ii) (10x - 25) ÷ (2x - 5)
(iii) 10y (6y + 21) ÷ 5(2y + 7)
(iv) 9x2y2(3z - 24) ÷ 27xy(z - 8)
(v) 96abc(3a - 12)(5b - 30) ÷ 144(a -4)(b - 6)
Answer
(i) (10x - 25) ÷ 5
= (10x - 25)/5
= {5(2x - 5)}/5
= 2x -5
(ii) (10x - 25) ÷ (2x - 5)
= (10x - 25)/(2x - 5)
= {5(2x - 5)/(2x - 5)
= 5
(iii) 10y(6y + 21) ÷ 5(2y + 7)
= {10y(6y + 21)}/5(2y + 7)
= {2×5×y× 3(2y + 7)}/5(2y + 7)
= 2×y×3
= 6y
(iv) 9x2y2(3z - 24) ÷ 27xy(z - 8)
= {9x2y2(3z - 24)}/27xy(z - 8)
= 9/27 × {xy × xy × 3(z - 8)}/xy(z - 8)
= xy
(v) 96abc(3a - 12)(5b - 30) ÷ 144(a- 4)(b - 6)
= {96abc(3a - 12)(5b - 30)}/144(a - 4)(b - 6)
= {12×4×2×abc× 3(a-4) × 5(b-6)}/{12×4×3 (a - 4)(b - 6)
= 10abc
4. Divide as directed:
(i) 5(2x + 1)(3x + 5) ÷ (2x + 1)
(ii) 26xy(x + 5)(y - 4) ÷ 13x(y - 4)
(iii) 52pqr(p + q)(q + r)(r + p) ÷ 104pq(q + r)(r + p)
(iv) 20(y + 4)(y2 + 5y + 3) ÷ 5(y + 4)
(v) x(x + 1)(x + 2)(x + 3) ÷ x(x + 1)
Answer
(i) 5(2x + 1)(3x + 5) ÷ (2x + 1)
= {5(2x + 1)(3x +5)}/(2x + 1)
= 5(3x + 5)
(ii) 26xy( x + 5)(y - 4) ÷ 13x(y - 4)
26xy( x + 5)(y -4) ÷ 13x(y - 4)
= {26xy(x + 5)(y - 4)}/13x(y - 4)
= {13×2×xy(x + 5)(y - 4)}/13x(y - 4)
= 2y(x + 5)
(iii) 52pqr( p + q)(q + r)( r + p) ÷ 104pq(q + r)(r + p)
= {52pqr(p + q)(q + r)( r + p)}/{52 × 2 × pq(q + r)(r + p)}
= (1/2)r (p + q)
(iv) 20( y + 4)(y2 + 5y + 3) ÷ 5(y + 4)
= {20(y + 4)(y2 + 5y + 3)}/5(y + 4)
= 4(y2 + 5y + 3)
(v) x( x + 1)(x + 2)(x + 3) ÷ x(x + 1)
= {x(x + 1)(x + 2)(x + 3)}/x(x + 1)
= (x + 2)(x + 3)
5. Factorize the expressions and divide them as directed:
(i) (y2 + 7y + 10) ÷ (y + 5)
(ii) (m2 - 14m - 32) ÷ (m + 2)
(iii) (5p2 - 25p + 20) ÷ (p - 1)
(iv) 4yz(z2 + 6z - 16) ÷ 2y( z + 8)
(v) 5pq(p2 - q2) ÷ 2p(p + q)
(vi) 12xy(9x2 - 16y2) ÷ 4xy(3x + 4y)
(vii) 39y3(50y2 - 98) ÷ 26y2(5y + 7)
Answer
(i) (y2 + 7y + 10) ÷ (y + 5)
= (y2 + 7y + 10)/(y + 5)
= {y2 + ( 2 + 5)y + 2 × 5}/(y +5)
= (y2 + 2y + 5y + 2 × 5)/(y + 5)
= {(y + 2)(y + 5)}/(y + 5) [∵ x2 + (a+b)x + ab = (x +a)(x+b)]
= y + 2
(ii) (m2 - 14m + 32) ÷ (m + 2)
= (m2 - 14m + 32)/(m +2)
= { m2 + (-16 + 2)m + (-16) × 2}/(m + 2)
= {(m - 16)(m + 2)}/(m +2) [∵ x2 + (a+b)x + ab = (x +a)(x+b)]
= (m - 16)
(iii) (5p2 - 25p + 20) ÷ (p -1)
= (5p2 - 25p + 20)/(p -1)
= (5p2 - 20p -5p + 20)/(p -1)
= {5p(p - 4) -5 (p - 4)}/(p -1)
= {(5p - 5)(p - 4)}/(p -1) = {5(p -1)(p -4)}/(p - 1)
= 5 (p - 4)
(iv) 4yz (z2 + 6z - 16) ÷ 2y(z + 8)
= {4yz(z2 + 6z - 16)}/2y(z + 8)
= [4yz{z2 + (8 - 2)z + 8 × (-2)}]/2y(z + 8)
= {4yz(z - 2)(z + 8)}/2y(z + 8) [∵ x2 + (a+b)x + ab = (x +a)(x+b)]
= 2z ( z -2)
(v) 5pq(p2 - q2) ÷ 2p( p + q)
= {5pq(p2 - q2)}/2p(p + q)
= {5pq(p - q)(p + q)}/2p( p + q) [∵ a2 - b2 = (a - b)(a + b)]
= (5/2)q (p - q)
(vi) 12xy(9x2 - 16y2) ÷ 4xy(3x + 4y)
= {12xy (9x2 - 16y2)}/4xy(3x + 4y)
= {12xy[(3x)2 - (4y)2]}/4xy(3x + 4y)
= {12xy(3x - 4y)(3x + 4y)}/4xy(3x + 4y) [∵ a2 - b2 = (a - b)(a + b)]
= 3(3x - 4y)
(vii) 39y3(50y2 - 98) ÷ 26y2(5y + 7)
= {39y3(50y2 - 98)}/26y2(5y + 7)
= {39y3 × 2(25y2 - 49)}/26y2(5y + 7)
= {39y2 × 2[(5y)2 - (7)2]}/26y2(5y + 7)
= {39y2 × 2(5y - 7)(5y + 7)}/26y2(5y + 7) [∵ a2 - b2 = (a - b)(a + b)]
= 3y(5y - 7)
Page No. 228
Exercise 14.4
1. Find and correct the errors in the following mathematical statements:
4(x-5) = 4x-5
Answer
L.H.S. = 4(x-5) = 4x- 20 ≠R.H.S.
Hence, the correct mathematical statement is 4(x-5) = 4x- 20.
2. x(3x+2) = 3x2+ 2
Answer
L.H.S. = x(3x+2) = 3x2+ 2 ≠ R.H.S.
Hence, the correct mathematical statement is x(3x+2) = 3x2+ 2
3. 2x + 3y = 5xy
Answer
L.H.S. = 2x + 3y ≠ R.H.S.
Hence, the correct mathematical statement is 2x+ 3y = 2x+ 3y
4. x+ 2x +3x = 5x
Answer
L.H.S. = x+ 2x + 3x = 6x ≠R.H.S.
Hence, the correct mathematical statement is x+ 2x + 3x = 6x.
5. 5y + 2y+ y-7y = 0
Answer
L.H.S. = 5y + 2y+ y - 7y = 8y-7y = y ≠ R.H.S.
Hence, the correct mathematical statement is 5y+ 2y+y- 7y = 4
6. 3x+2x = 5 x2
Answer
L.H.S. = 3x+ 2x = 5x ≠ R.H.S.
Hence the correct mathematical statement is 3x+ 2x = 5x
7. (2x)2+ 4(2x) + 7 = 2x2+ 8x+ 7
Answer
L.H.S. = (2x)2 + 4(2x) + 7 = 4x2 + 8x+ 7 ≠ R.H.S.
Hence, the correct mathematical statement is (2x)2 + 4(2x) + 7 = 4x2 + 8x+ 7
8. (2x)2+ 5x = 4x+ 5x = 9x
Answer
L.H.S. = (2x)2 + 5x = 4x2+ 5x ≠ R.H.S.
Hence the correct mathematical statement is (2x)2 + 5x = 4x2+ 5x.
9. (3x + 2)2= 3x2 + 6x + 4
Answer
L.H.S. = (3x + 2)2 = 3x2 + 2 × 3x × 2+ (2)2
= 9x2 + 12x + 4 ≠ RHS
Hence, the correct mathematical statementsis (3x + 2)2 = 9x2 + 12X + 4 × 3x
10. Substituting x = -3 in:
(a) x2 + 5X + 4 gives (-3)2 + 5(-3) + 4 = 9+ 2+4 = 15
(b) x2 - 5X + 4 gives (-3)2 - 5(-3) + 4 = 9 - 15 + 4 = -2
(c) x2 + 5X gives (-3)2 + 5(-3) = -9 - 15 = -24
Answer
(a) L.H.S. = x2 + 5x + 4
Putting x = -3 in given expression,
= (-3)2 + 5(-3) + 4 = 9 - 15 + 4 = -2 R.H.S.
Hence, x2 + 5x + 4 gives (-3)2 + 5(-3) + 4 = 9 - 15 + 4 = -2
(b) L.H.S. = x2 - 5X + 4
Putting x = -3 cin given expression,
= (-3)2 - 5(-3) + 4 = 9 + 15 + 4 = 28 ≠ R.H.S.
Hence x2 – 5x + 4 gives (-3)2 - 5(-3) + 4 = 9 + 15 + 4 = 28
(c) L.H.S. = x2 + 5X
Putting x= -3 in given expression,
= (-3)2 + 5(-3) = 9 - 15 = -6 ≠ R.H.S.
Hence, x2 + 5X gives (-3)2 + 5(-3) = 9 - 15 = -6
11. (y-3)2= y2 - 9
Answer
L.H.S. = (y-3)2 = y2 - 2 × y × 3 +(3)2 [ (a-b)2 = a2 - 2ab + b2]
= y2 - 6y + 9 ≠ R.H.S.
Hence, the correct statement is (y-3)2 = y2 - 6y + 9
12. (z+5)2 = z2 + 25
Answer
L.H.S. = (z+5)2 = z2 + 2 × z×5+ (5)2
= z2 + 10z +25 [ (a-b)2 = a2 - 2ab + b2]
Hence, the correct statement is (z+5)2 = z2 + 10z + 25
13. (2a +3b)(a-b) = 2a2- 3b2
Answer
L.H.S. = (2a + 3b)(a-b) = 2a(a-b) + 3b(a-b)
= 2a2 - 2ab + 3ab - 3b2
= 2a2 + ab - 3b2≠ R.H.S.
Hence, the correct statement is (2a +3b)(a-b) = 2a2 + ab - 3b2
14. (a + 4) (a + 2) = a2+ 8
Answer
L.H.S. = (a+4)(a+2) =a(a+2) + 4(a+2)
= a2 + 2a + 4a + 8 = a2 + 6a + 8 ≠ R.H.S.
Hence, the correct statement is (a+4)(a+2) = a2+6a+ 8
15. (a-4)(a-2) = a2- 8
Answer
L.H.S. = (a-4)(a-2) = a(a-2)-4(a-2)
= a2 - 2a -4a+8 = a2- 6a + 8 ≠ R.H.S
Hence, the correct statement is (a-4)(a-2) = a2- 6a + 8
16. 3x2/3x2= 0
Answer
L.H.S. = 3x2/3x2 =1/1 = 1 ≠ R.H.S.
Hence, the correct statement is 3x2/3x2 =1
17. 3x2 + 1 / 3x2 = 1+ 1 = 2
Answer
L.H.S. = 3x2 + 1 / 3x2 = 3x2/ 3x2 + 1/3x2
= 1 + 1 / 3x2 R.H.S.
Hence, the correct statement is 3x2 + 1 / 3x2 = 1 + 1/3x2
18. 3x/(3x+2) = 1/2
Answer
L.H.S. = 3x/(3x+2) ≠ R.H.S.
Hence, the correct statement is 3x/(3x+2) = 3x/(3x+2)
19. 3/(4x+3) = 1/4x
Answer
L.H.S. = 3/(4x+3) ≠ R.H.S.
Hence, the correct statement is 3/(4x+3) = 3/(4x+3)
20. (4x+5)/4x = 5
Answer
L.H.S. = (4x+5)/4x = 4x/4x + 5/4x = 1 + 5/4x ≠R.H.S.
Hence the correct statement is (4x+ 5)/4x = 1 + 5/4x
21. (7x+5)/5 = 7x
Answer
L.H.S. = (7x+5)/5 = 7x/5 + 5/5 = 7x/5 + 1 ≠ R.H.S.
Hence, the correct statement is (7x+5)/5 = 7x/5 +1
(iii) 2x3 + 2xy2 + 2xz2
(iv) am2 + bm2 + bn2 + an2
(v) (lm + l ) + m + 1
(vi) y( y + z) + 9 ( y + z)
(vii) 5y2 - 20y - 8z + 2yz
(viii) 10ab + 4a + 5b + 2
(ix) 6xy - 4y + 6 - 9x
Answer
(i) ax2 + bx = x(ax + b)
(ii) 7p2 + 21q2 = 7(p2 + 3q2)
(iii) 2x3 + 2xy2 + 2xz2 = 2x( x2 + y2 + z2)
(iv) am2 + bm2 + bn2 + an2
= m2( a + b) + n2(a + b)
= (a + b )(m2 + n2)
(v) (lm + l) + m + 1
= l(m + 1) + 1(m + 1)
= (m + 1)( l + 1)
(vi) y(y + z) + 9(y + z)
= (y + z)(y + 9)
(vii) 5y2 - 20y - 8z + 2yz
= 5y2 - 20y + 2yz - 8z
= 5y(y - 4) + 2z(y - 4)
= (y - 4)(5y + 2z)
(viii) 10ab + 4a + 5b + 2
= 2a(5b + 2) + 1 (5b + 2)
= (5b + 2)(2a + 1)
(ix) 6xy - 4y + 6 - 9x
= 6xy - 9x - 4y + 6
= 3x(2y - 3) - 2(2y - 3)
= (2y - 3) (3x - 2)
4. Factorize:
(i) a4 - b4
(ii) p4 - 81
(iii) x4 - (y + z)4
(iv)x4 - (x -z)4
(v) a4 - 2a2b2 + b4
Answer
(i) a4 - b4 = (a2)2 - (b2)2
= (a2 - b2)( a2 + b2) [ ∵ a2 - b2 = (a - b)(a +b)]
= (a - b)(a + b)(a2 + b2) [ ∵ a2 - b2 = (a - b)(a +b)]
(ii) p4 - 81 = (p2)2 - (9)2
= (p2 - 9)(p2 + 9) [∵ a2 - b2 = (a - b)(a +b)]
= (p2 - 32)(p2 + 9)
= ( p - 3)(p + 3)(p2 + 9) [∵ a2 - b2 = (a - b)(a +b)]
(iii) x4 - (y + z)4 = (x2)2 - [(y + z)2]2
= [x2 - (y + z)2][ x2 + (y + z)2] [∵ a2 - b2 = (a - b)(a +b)]
= [x -(y +z)][x + (y + z)][x2 + (y + z)2] [∵ a2 - b2 = (a - b)(a +b)]
= (x - y - z) (x + y + z) [x2 + (y + z)2]
(iv) x4 - (x - z)4 = (x2)2 - [(x - z)2]2
= [x2 -(x - z)2][x2 + (x - z)2] [ ∵ a2 - b2 = (a - b)(a +b)]
= [x - (x - z)][x + (x - z)] [x2 + (x - z)2] [∵ a2 - b2 = (a - b)(a +b)]
= [x - x + z] [x + x - z] [x2 + x2 - 2xz + z2] [∵ (a -b)2 = a2 -2ab + b2]
= z(2x - z) (2x2 - 2xz + z2)
(v) a4 - 2a2b2 + b4 = (a2)2 - 2a2b2 + (b2)2
= (a2 - b2)2 [∵ a2 - b2 = (a - b)(a +b)]
= [(a - b)(a + b)]2 [∵ a2 - b2 = (a - b)(a +b)]
= (a -b)2 (a + b)2 [ (xy)m = xmym]
5. Factorize the following expressions:
(i) p2 + 6p + 8
(ii) q2 - 10q + 21
(iii) p2 + 6p - 16
Answer
(i) p2 + 6p + 8 = p2 + ( 4 + 2)p + 4 × 2
= p2 + 4p + 2p + 4 ×2
= p(p + 4) + 2 ( p + 4)
= (p + 4)(p + 2)
(ii) q2 - 10q + 21 = q2 - ( 7 + 3)q + 7 × 3
= q2 - 7q - 3q + 7 × 3
= q(q - 7) - 3(q - 7)
= (q - 7)( q - 3)
(iii) p2 + 6p - 16
= p2 + (8 - 2)p - 8×2
= p2 + 8p - 2p - 8×2
= p(p + 8) - 2(p + 8)
= ( p + 8)(p -2)
Page No. 227
Exercise 14.3
1. Carry out the following divisions:
(i) 2x4 ÷ 56x
(ii) -36y3 ÷ 9y2
(iii) 66pq2r3 ÷ 11 qr2
(iv) 34x3y3x3 ÷ 51xy2z3
(v) 12a8b8 ÷ (-6a6b4)
Answer
(i) 2x4 ÷ 56x
= 28x4/56x
= 28/56 × x4/x
= 1/2 x3 [xm ÷ xn = xm-n]
(ii) -36y3 ÷ 9y2 = -36y3/9y2
= -36/9 × y3/y2
= -4y [xm ÷ xn = xm-n]
(iii) 66pq2r3 ÷ 11qr2
= 66pq2r3/11qr2
= 66/11 × pq2r3/qr2
= 6pqr [xm ÷ xn = xm-n]
(iv) 34x3y3z3 ÷ 51xy2z3
= 34x3y3z3/51xy2z3
= 34/51 ×x3y3z3/xy2z3
= 2/3x2y [xm ÷ xn = xm-n]
(v) 12a8b8 ÷ (- 6a6b4)
= 12a8b8/- 6a6b4
= 12/-6 × a8b8/a6b4
= -2a2b4 [xm ÷ xn = xm-n]
2. Divide the given polynomial by the given monomial:
(i) (5x2 - 6x) ÷ 3x
(ii) (3y8 - 4y6 + 5y4) ÷ y4
(iii) 8(x3y2z2 + x2y3z2 + x2y2z3) ÷ 4x2y2z2
(iv) (x3 + 2x2 + 3x) ÷2x
(v) (p3q6 - p6q3) ÷ p3q3
Answer
(i) (5x2 - 6x) ÷3x
= (5x2 - 6x)/3x
= 5x2/3x - 6x/3x = (5/3)x - 2 = 1/3 (5x - 6)
(ii) (3y8 - 4y6 + 5y4) ÷ y4
= (3y8 - 4y6 + 5y4)/ y4
= 3y8/y4 - 4y6/y4 + 5y4/y4 = 3y4 - 4y2 + 5
(iii) 8(x3y2z2 + x2y3z2 + x2y2z3) ÷ 4x2y2z2
= {8(x3y2z2 + x2y3z2 + x2y2z3)}/4 x2y2z2
= 8 x3y2z2/4 x2y2z2 + 8 x2y3z2/4x2y2z2 + 8 x2y2z3/4x2y2z2
= 2x + 2y + 2z
= 2(x + y + z)
(iv) (x3 + 2x2 + 3x) ÷ 2x
= (x3 + 2x2 + 3x)/2x
= x3/2x + 2x2/2x + 3x/2x = x2/2 + 2x/2 + 3/2
= 1/2( x2 + 2x + 3)
(v) (p3q6 - p6q3) ÷ p3q3
= (p3q6 - p6q3)/p3q3
= p3q6/p3q3 - p6q3/p3q3 = q3 - p3
3. Work out the following divisions:
(i) (10x - 25) ÷ 5
(ii) (10x - 25) ÷ (2x - 5)
(iii) 10y (6y + 21) ÷ 5(2y + 7)
(iv) 9x2y2(3z - 24) ÷ 27xy(z - 8)
(v) 96abc(3a - 12)(5b - 30) ÷ 144(a -4)(b - 6)
Answer
(i) (10x - 25) ÷ 5
= (10x - 25)/5
= {5(2x - 5)}/5
= 2x -5
(ii) (10x - 25) ÷ (2x - 5)
= (10x - 25)/(2x - 5)
= {5(2x - 5)/(2x - 5)
= 5
(iii) 10y(6y + 21) ÷ 5(2y + 7)
= {10y(6y + 21)}/5(2y + 7)
= {2×5×y× 3(2y + 7)}/5(2y + 7)
= 2×y×3
= 6y
(iv) 9x2y2(3z - 24) ÷ 27xy(z - 8)
= {9x2y2(3z - 24)}/27xy(z - 8)
= 9/27 × {xy × xy × 3(z - 8)}/xy(z - 8)
= xy
(v) 96abc(3a - 12)(5b - 30) ÷ 144(a- 4)(b - 6)
= {96abc(3a - 12)(5b - 30)}/144(a - 4)(b - 6)
= {12×4×2×abc× 3(a-4) × 5(b-6)}/{12×4×3 (a - 4)(b - 6)
= 10abc
4. Divide as directed:
(i) 5(2x + 1)(3x + 5) ÷ (2x + 1)
(ii) 26xy(x + 5)(y - 4) ÷ 13x(y - 4)
(iii) 52pqr(p + q)(q + r)(r + p) ÷ 104pq(q + r)(r + p)
(iv) 20(y + 4)(y2 + 5y + 3) ÷ 5(y + 4)
(v) x(x + 1)(x + 2)(x + 3) ÷ x(x + 1)
Answer
(i) 5(2x + 1)(3x + 5) ÷ (2x + 1)
= {5(2x + 1)(3x +5)}/(2x + 1)
= 5(3x + 5)
(ii) 26xy( x + 5)(y - 4) ÷ 13x(y - 4)
26xy( x + 5)(y -4) ÷ 13x(y - 4)
= {26xy(x + 5)(y - 4)}/13x(y - 4)
= {13×2×xy(x + 5)(y - 4)}/13x(y - 4)
= 2y(x + 5)
(iii) 52pqr( p + q)(q + r)( r + p) ÷ 104pq(q + r)(r + p)
= {52pqr(p + q)(q + r)( r + p)}/{52 × 2 × pq(q + r)(r + p)}
= (1/2)r (p + q)
(iv) 20( y + 4)(y2 + 5y + 3) ÷ 5(y + 4)
= {20(y + 4)(y2 + 5y + 3)}/5(y + 4)
= 4(y2 + 5y + 3)
(v) x( x + 1)(x + 2)(x + 3) ÷ x(x + 1)
= {x(x + 1)(x + 2)(x + 3)}/x(x + 1)
= (x + 2)(x + 3)
5. Factorize the expressions and divide them as directed:
(i) (y2 + 7y + 10) ÷ (y + 5)
(ii) (m2 - 14m - 32) ÷ (m + 2)
(iii) (5p2 - 25p + 20) ÷ (p - 1)
(iv) 4yz(z2 + 6z - 16) ÷ 2y( z + 8)
(v) 5pq(p2 - q2) ÷ 2p(p + q)
(vi) 12xy(9x2 - 16y2) ÷ 4xy(3x + 4y)
(vii) 39y3(50y2 - 98) ÷ 26y2(5y + 7)
Answer
(i) (y2 + 7y + 10) ÷ (y + 5)
= (y2 + 7y + 10)/(y + 5)
= {y2 + ( 2 + 5)y + 2 × 5}/(y +5)
= (y2 + 2y + 5y + 2 × 5)/(y + 5)
= {(y + 2)(y + 5)}/(y + 5) [∵ x2 + (a+b)x + ab = (x +a)(x+b)]
= y + 2
(ii) (m2 - 14m + 32) ÷ (m + 2)
= (m2 - 14m + 32)/(m +2)
= { m2 + (-16 + 2)m + (-16) × 2}/(m + 2)
= {(m - 16)(m + 2)}/(m +2) [∵ x2 + (a+b)x + ab = (x +a)(x+b)]
= (m - 16)
(iii) (5p2 - 25p + 20) ÷ (p -1)
= (5p2 - 25p + 20)/(p -1)
= (5p2 - 20p -5p + 20)/(p -1)
= {5p(p - 4) -5 (p - 4)}/(p -1)
= {(5p - 5)(p - 4)}/(p -1) = {5(p -1)(p -4)}/(p - 1)
= 5 (p - 4)
(iv) 4yz (z2 + 6z - 16) ÷ 2y(z + 8)
= {4yz(z2 + 6z - 16)}/2y(z + 8)
= [4yz{z2 + (8 - 2)z + 8 × (-2)}]/2y(z + 8)
= {4yz(z - 2)(z + 8)}/2y(z + 8) [∵ x2 + (a+b)x + ab = (x +a)(x+b)]
= 2z ( z -2)
(v) 5pq(p2 - q2) ÷ 2p( p + q)
= {5pq(p2 - q2)}/2p(p + q)
= {5pq(p - q)(p + q)}/2p( p + q) [∵ a2 - b2 = (a - b)(a + b)]
= (5/2)q (p - q)
(vi) 12xy(9x2 - 16y2) ÷ 4xy(3x + 4y)
= {12xy (9x2 - 16y2)}/4xy(3x + 4y)
= {12xy[(3x)2 - (4y)2]}/4xy(3x + 4y)
= {12xy(3x - 4y)(3x + 4y)}/4xy(3x + 4y) [∵ a2 - b2 = (a - b)(a + b)]
= 3(3x - 4y)
(vii) 39y3(50y2 - 98) ÷ 26y2(5y + 7)
= {39y3(50y2 - 98)}/26y2(5y + 7)
= {39y3 × 2(25y2 - 49)}/26y2(5y + 7)
= {39y2 × 2[(5y)2 - (7)2]}/26y2(5y + 7)
= {39y2 × 2(5y - 7)(5y + 7)}/26y2(5y + 7) [∵ a2 - b2 = (a - b)(a + b)]
= 3y(5y - 7)
Page No. 228
Exercise 14.4
1. Find and correct the errors in the following mathematical statements:
4(x-5) = 4x-5
Answer
L.H.S. = 4(x-5) = 4x- 20 ≠R.H.S.
Hence, the correct mathematical statement is 4(x-5) = 4x- 20.
2. x(3x+2) = 3x2+ 2
Answer
L.H.S. = x(3x+2) = 3x2+ 2 ≠ R.H.S.
Hence, the correct mathematical statement is x(3x+2) = 3x2+ 2
3. 2x + 3y = 5xy
Answer
L.H.S. = 2x + 3y ≠ R.H.S.
Hence, the correct mathematical statement is 2x+ 3y = 2x+ 3y
4. x+ 2x +3x = 5x
Answer
L.H.S. = x+ 2x + 3x = 6x ≠R.H.S.
Hence, the correct mathematical statement is x+ 2x + 3x = 6x.
5. 5y + 2y+ y-7y = 0
Answer
L.H.S. = 5y + 2y+ y - 7y = 8y-7y = y ≠ R.H.S.
Hence, the correct mathematical statement is 5y+ 2y+y- 7y = 4
6. 3x+2x = 5 x2
Answer
L.H.S. = 3x+ 2x = 5x ≠ R.H.S.
Hence the correct mathematical statement is 3x+ 2x = 5x
7. (2x)2+ 4(2x) + 7 = 2x2+ 8x+ 7
Answer
L.H.S. = (2x)2 + 4(2x) + 7 = 4x2 + 8x+ 7 ≠ R.H.S.
Hence, the correct mathematical statement is (2x)2 + 4(2x) + 7 = 4x2 + 8x+ 7
8. (2x)2+ 5x = 4x+ 5x = 9x
Answer
L.H.S. = (2x)2 + 5x = 4x2+ 5x ≠ R.H.S.
Hence the correct mathematical statement is (2x)2 + 5x = 4x2+ 5x.
9. (3x + 2)2= 3x2 + 6x + 4
Answer
L.H.S. = (3x + 2)2 = 3x2 + 2 × 3x × 2+ (2)2
= 9x2 + 12x + 4 ≠ RHS
Hence, the correct mathematical statementsis (3x + 2)2 = 9x2 + 12X + 4 × 3x
10. Substituting x = -3 in:
(a) x2 + 5X + 4 gives (-3)2 + 5(-3) + 4 = 9+ 2+4 = 15
(b) x2 - 5X + 4 gives (-3)2 - 5(-3) + 4 = 9 - 15 + 4 = -2
(c) x2 + 5X gives (-3)2 + 5(-3) = -9 - 15 = -24
Answer
(a) L.H.S. = x2 + 5x + 4
Putting x = -3 in given expression,
= (-3)2 + 5(-3) + 4 = 9 - 15 + 4 = -2 R.H.S.
Hence, x2 + 5x + 4 gives (-3)2 + 5(-3) + 4 = 9 - 15 + 4 = -2
(b) L.H.S. = x2 - 5X + 4
Putting x = -3 cin given expression,
= (-3)2 - 5(-3) + 4 = 9 + 15 + 4 = 28 ≠ R.H.S.
Hence x2 – 5x + 4 gives (-3)2 - 5(-3) + 4 = 9 + 15 + 4 = 28
(c) L.H.S. = x2 + 5X
Putting x= -3 in given expression,
= (-3)2 + 5(-3) = 9 - 15 = -6 ≠ R.H.S.
Hence, x2 + 5X gives (-3)2 + 5(-3) = 9 - 15 = -6
11. (y-3)2= y2 - 9
Answer
L.H.S. = (y-3)2 = y2 - 2 × y × 3 +(3)2 [ (a-b)2 = a2 - 2ab + b2]
= y2 - 6y + 9 ≠ R.H.S.
Hence, the correct statement is (y-3)2 = y2 - 6y + 9
12. (z+5)2 = z2 + 25
Answer
L.H.S. = (z+5)2 = z2 + 2 × z×5+ (5)2
= z2 + 10z +25 [ (a-b)2 = a2 - 2ab + b2]
Hence, the correct statement is (z+5)2 = z2 + 10z + 25
13. (2a +3b)(a-b) = 2a2- 3b2
Answer
L.H.S. = (2a + 3b)(a-b) = 2a(a-b) + 3b(a-b)
= 2a2 - 2ab + 3ab - 3b2
= 2a2 + ab - 3b2≠ R.H.S.
Hence, the correct statement is (2a +3b)(a-b) = 2a2 + ab - 3b2
14. (a + 4) (a + 2) = a2+ 8
Answer
L.H.S. = (a+4)(a+2) =a(a+2) + 4(a+2)
= a2 + 2a + 4a + 8 = a2 + 6a + 8 ≠ R.H.S.
Hence, the correct statement is (a+4)(a+2) = a2+6a+ 8
15. (a-4)(a-2) = a2- 8
Answer
L.H.S. = (a-4)(a-2) = a(a-2)-4(a-2)
= a2 - 2a -4a+8 = a2- 6a + 8 ≠ R.H.S
Hence, the correct statement is (a-4)(a-2) = a2- 6a + 8
16. 3x2/3x2= 0
Answer
L.H.S. = 3x2/3x2 =1/1 = 1 ≠ R.H.S.
Hence, the correct statement is 3x2/3x2 =1
17. 3x2 + 1 / 3x2 = 1+ 1 = 2
Answer
L.H.S. = 3x2 + 1 / 3x2 = 3x2/ 3x2 + 1/3x2
= 1 + 1 / 3x2 R.H.S.
Hence, the correct statement is 3x2 + 1 / 3x2 = 1 + 1/3x2
18. 3x/(3x+2) = 1/2
Answer
L.H.S. = 3x/(3x+2) ≠ R.H.S.
Hence, the correct statement is 3x/(3x+2) = 3x/(3x+2)
19. 3/(4x+3) = 1/4x
Answer
L.H.S. = 3/(4x+3) ≠ R.H.S.
Hence, the correct statement is 3/(4x+3) = 3/(4x+3)
20. (4x+5)/4x = 5
Answer
L.H.S. = (4x+5)/4x = 4x/4x + 5/4x = 1 + 5/4x ≠R.H.S.
Hence the correct statement is (4x+ 5)/4x = 1 + 5/4x
21. (7x+5)/5 = 7x
Answer
L.H.S. = (7x+5)/5 = 7x/5 + 5/5 = 7x/5 + 1 ≠ R.H.S.
Hence, the correct statement is (7x+5)/5 = 7x/5 +1
Go to Chapters list NCERT Solutions of Class 8 Maths