ML AGGARWAL CLASS 8 CHAPTER 10 Algebraic Expressions and Identities Exercise 10.5
Exercise 10.5
Question 1
Sol :
(i) (3 x+5)(3 x+5)
(3x+5)2=(3x)2+2(3x).5+52(∴(a+b)2=a2+2ab+b2)=9x2+30x+25
(ii) (9y−5)(9y−5)(9y−5)2=(9y)2−2(9y)5+(+5)2(∵(a−b)2=a2−2ab+b2)=81y2−90y+25
(iii) (4x+11y)(4x−11y)
=(4x)2−(11y)2(∵(a+b)(a−b)=a2−b2)
=16x2−121y2
(v) (2a+5b)(2a+5b)
(2a+5b)2=(2a)2+2⋅2a⋅5b+(5b)2(∵⋅(a+b)2=a2+2ab+b2)=4a2+20ab+25b2
(vi) (p22+2q2)(p22−2q2)
(p22)2−(2q2)2(∵(a+b)(a−b)=a2−b2
p44−4q4
Question 2
Sol :
(i) 812=(80+1)2=802+2.80.1+12(∵(a+b)2=a2+2ab+b2)=6400+160+1812=6561
(ii)
972=(100−3)2=(100)2−2⋅100⋅3+32(∵(a−b)2=a2−2ab+b2]=10000−600+9=9409
(iii)
1052=(100+5)2=1002+2.100⋅5+52(⋯(a+b)2=a2+2ab+b2)=10000+1000+25=11025
(iv)
9972=(1000−3)2=(1000)2−2⋅1000⋅3+32(∵(a−b)2=a2−2ab+b2)=1000000−6000+99972=994009
(v)
6.12=(6+0.1)2=62+2⋅6⋅(0.1)+0.12(∵(a+b)γ=a2+2ab+b2)=36+1.2+0.016.12=31.21
(vi)
496×504=(500−4)(500+4)=(500)2−42(∵⋅(a+b)(a−b)=a2−b2)=250000−16496×504=249984
(vii)
20.5×19.5=(20+0.5)(20−0.5)=202−0.52(19+b)(a−b)=a2−b2)=400−0.2520.5×19.5=399.75
(viii)
9.62=(10−0.4)2=102−2.10⋅(0.4)+(0.4)2(∵⋅(a−b)2=a2−2ab+b2)=100−8+0.169.62=92.16
Question 3
Sol :
(i) (pq+5r)2=(Pq)2+2⋅pq⋅5r+(5r)2(−(a+b)2=a2+b2+2ab)=p2qr+10pqr+2522
(ii)
(52a−35⋅b)2=(52⋅a)2−2⋅52⋅a⋅35⋅b+(35b)2(∵(a−b)2=a2−2ab+b2)=254a2−3ab+925b2
(iii)
(√2⋅a+√3⋅b)2=(√2⋅a)2+2√2⋅a⋅√3⋅b+(√3⋅b)2(∴(a+b)2=a2+2ab+b2)=2a2+2√6ab+3b2
(iv)
(2x3y−3y2x)2=(2x3y)2−2⋅2x3y⋅3y2x+(3y2x)2 (∴(a+b)2=a2+b2+2ab)
=4x29y2−2+9y24x2
Question 4
Sol :
(i)
(x+7)(x+3)=x2+(7+3)x+7×3(:(x+a)(x+b)=x2+(a+b)x+ab)=x2+10x+21
(ii) (3x+4)(3x−5)=(3x)v+(4+(−5))(3x)+4x−5
(∵(x+a)(x+b)=x2+(a+b)x+ab)
=9x2−3x−20
(iii) (p2+2q)(p2−3q)=(p2)2+(2q+(−3q))p2+2qx−3q
=p4−2q2−6q2
=p4−p2q−6q2
(iv) (abc+3)(abc−5)=(abc)2+(3+(−5))⋅abc+3x−5
(∵(x+a)(x+b)=x2+(a+b)⋅x+ab)
=(abc)2−2abc−15
Question 5
Sol :
(i) 203×204=(200+3)(200+4)
=(200)2+(3+4)200+3×4(∵(x+a)(x+b)=x2+(a+b)+x+ab
=40000+1400+12
=41412
(ii) 8.2×8.7=(8+0.2)(8+0.7)
=82+(0.2+0.7)8+0.2×0.7
=64+7.2+0.14
=71.34
(iii) 107×93=(100+7)(100−7)
=(100)2+(7+(−7))⋅100+7x−7
(∵(x+a)(x+b)=x2+(a+b)⋅x+ab)=10000+0.100=49
=9951
Question 6
Sol :
(i) 532−472=(53+47)(53−47) (∴a2−b2=(a+b)(a−b))
=(100)(6)
=600
(ii) (2.05)2−(0.95)2=(2.05+0.95)(2.05−0.95)
=3×0.1
=0.3
Question 7
Sol :
i. (2x+5y)2+(2x−5y)2
(2x)2+(5y)2+2⋅2x⋅5y+(2x)2+(5y)2−2⋅2x⋅5y
∵(a+b)2=a2+b2+2ab
2(2x)2+2(5y)2
2⋅[4x−1+25y2]
8x2+50y2
(ii) (72a−52b)2−(52a−z2b)2
=(72a)2+(52b)2−2⋅72a⋅52⋅b−[(52a)n+(72b)2−2⋅52⋅a⋅72b]
=494a2+254b2−2⋅72⋅52⋅b−254a2−494b2+2⋅52a⋅72⋅b
=(494−254)a2+(254−494)b2
=244⋅a2−244⋅b2
=6(a2−b2)
(iii) (p2−qr2⋅)2+2p2q22
(p2)2−2⋅p2⋅q2⋅2+(q2⋅2)2+2p2q2⋅r (∴(a+b)2=a2+2ab+b2
p4−2p2q2⋅r+q4⋅r2+2p2q2⋅r
p4+q4⋅r2
Question 8
Sol :
L H S
(i) (4x+7y)2−(4x−7y)n
(4x)2+(7y)2+2⋅4x⋅7y−[(4x)2+(7y)2−2.4x−7y]
(∵(a−b)2=a2+b2−2ab)
(4x)2+(7y)2+2.4x⋅7y−(4x)2−(7y)2+2⋅4x⋅7y
4.4x⋅7y
112 x y= R.H.S
(ii) (37p−76q)2+pq
(37p)2+(76q)2−2⋅37p⋅16⋅q+pq
(∵(a−b)2=a2+b2−2ab)
949p2+4936q2−pq+pq
949⋅p2+4936q2=R⋅H⋅S∴L.H⋅S=R⋅H⋅S
(iii) L⋅H⋅s=(p−q)(p+q)+(p−r)(q+r)+(r−p)(r+p)
=p2−q2+q2−r2+r2−p2
(∵+(a+b)(a−b)=a2−b2]
=0=RHS
∴LHS=RHS
Question 9
Sol :
Given : (x+1x)=2
Squaring on both sides
(i) (x+1x)2=22
x2+2⋅x⋅1x+(1x)2=4(∵(a+b)2=a2+b2+2ab)
x2+2+1x2=4
x2+1x2=4−2
x2+1x2=2
(ii) Again squaring on both sides
(x2+1x2)2=22
(x2)n+2⋅x2−1x2+(1x2)2=4
x4+2+1x4=4
x4+1x4=4−2
x4+1x4=2n
Question 10
Sol :
(i) x−1x=7
Squaring on both sides
(x−1x)2=72
x2−2⋅x⋅1x+(1x)2=49
(∵(a−b)2=a2−2ab+b2)
x2−2+1x2=49
x2+1x2=49+2
x2+1x2=51
(ii) x2+1x2=51
squaring on both sides
(x2+1x2)2=512
(x2)2+2⋅x2⋅1x2+(1x2)2=2601
x4+2+1x4=2601
x4+1x4=2601−2
x4+1x4=2599
Question 11
Sol :
x2+1x2=23
(i) x2+1x2=23
Adding 2 on both sides
x2+1x2+2=23+2
(x)2+(1x)2+2⋅x⋅1x=25(∵a2+b2+2ab=(a+b)2)
(x+1x)2=25
x+1x=5
(ii) x+1x2=252
Subtract '2' on both sides
x2+1x2−2=23−2
(x)2+(1x)n−2⋅x⋅1x=21
(x−1x)2=21(∵a2+b2−2ab=(a−b)2)
x−1x=√21
x−1x=3√3
Question 12
Sol :
given a+b=9, a b=10
Squaring on both sides
(a+b)2=92
a2+b2+2ab=81
a2+b2+2×10=81(∵ given ab=10)
a2+b2+20=81
a2+b2=61
Question 13
Sol :
given a−b=6,a2+b2=42
a-b=6
Squaring on both Sides
(a−b)2=62
a2+b2−2ab=36
42−2ab=36(∵a2+b2=42)
42-36=2 a b
2 a b=6
a b=3
Question 14
Sol :
given a2+b2=41,ab=4
(i) Consider
(a+b)2=a2+b2+2ab(a+b)2=41+2×u=41+8(a+b)2=49a+b=7
(ii) Consider
(a−b)2=a2+b2−2ab=41−2×4=41−8=33.(a−b)2=33a−b=√33
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