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) (9y5)(9y5)(9y5)2=(9y)22(9y)5+(+5)2((ab)2=a22ab+b2)=81y290y+25

(iii) (4x+11y)(4x11y)

=(4x)2(11y)2((a+b)(ab)=a2b2)

=16x2121y2


(v) (2a+5b)(2a+5b)

(2a+5b)2=(2a)2+22a5b+(5b)2((a+b)2=a2+2ab+b2)=4a2+20ab+25b2


(vi) (p22+2q2)(p222q2)

(p22)2(2q2)2((a+b)(ab)=a2b2

p444q4

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=(1003)2=(100)221003+32((ab)2=a22ab+b2]=10000600+9=9409

(iii)
 1052=(100+5)2=1002+2.1005+52((a+b)2=a2+2ab+b2)=10000+1000+25=11025

(iv) 
9972=(10003)2=(1000)2210003+32((ab)2=a22ab+b2)=10000006000+99972=994009

(v) 
6.12=(6+0.1)2=62+26(0.1)+0.12((a+b)γ=a2+2ab+b2)=36+1.2+0.016.12=31.21

(vi) 
496×504=(5004)(500+4)=(500)242((a+b)(ab)=a2b2)=25000016496×504=249984


(vii)
 20.5×19.5=(20+0.5)(200.5)=2020.52(19+b)(ab)=a2b2)=4000.2520.5×19.5=399.75

(viii) 
9.62=(100.4)2=1022.10(0.4)+(0.4)2((ab)2=a22ab+b2)=1008+0.169.62=92.16



Question 3

Sol :
(i) 
(pq+5r)2=(Pq)2+2pq5r+(5r)2((a+b)2=a2+b2+2ab)=p2qr+10pqr+2522


(ii) 
(52a35b)2=(52a)2252a35b+(35b)2((ab)2=a22ab+b2)=254a23ab+925b2


(iii) 
(2a+3b)2=(2a)2+22a3b+(3b)2((a+b)2=a2+2ab+b2)=2a2+26ab+3b2


(iv) 
(2x3y3y2x)2=(2x3y)222x3y3y2x+(3y2x)2 ((a+b)2=a2+b2+2ab)

=4x29y22+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)(3x5)=(3x)v+(4+(5))(3x)+4x5

((x+a)(x+b)=x2+(a+b)x+ab)

=9x23x20


(iii) (p2+2q)(p23q)=(p2)2+(2q+(3q))p2+2qx3q
               
((x+a)(x+b)=x2+(a+b)x+ab)

=p42q26q2

=p4p2q6q2


(iv) (abc+3)(abc5)=(abc)2+(3+(5))abc+3x5

((x+a)(x+b)=x2+(a+b)x+ab)

=(abc)22abc15
 

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)(1007)

=(100)2+(7+(7))100+7x7
((x+a)(x+b)=x2+(a+b)x+ab)

=10000+0.100=49
=9951

Question 6 

Sol :
(i) 532472=(53+47)(5347) (a2b2=(a+b)(ab))
=(100)(6)
=600

(ii) (2.05)2(0.95)2=(2.05+0.95)(2.050.95)
=3×0.1
=0.3

Question 7

Sol :
i. (2x+5y)2+(2x5y)2

(2x)2+(5y)2+22x5y+(2x)2+(5y)222x5y
(a+b)2=a2+b2+2ab

2(2x)2+2(5y)2

2[4x1+25y2]

8x2+50y2
                                


(ii) (72a52b)2(52az2b)2

=(72a)2+(52b)2272a52b[(52a)n+(72b)2252a72b]

=494a2+254b227252b254a2494b2+252a72b

=(494254)a2+(254494)b2

=244a2244b2

=6(a2b2)



(iii) (p2qr2)2+2p2q22

(p2)22p2q22+(q22)2+2p2q2r   ((a+b)2=a2+2ab+b2

p42p2q2r+q4r2+2p2q2r

p4+q4r2

Question 8

Sol :
L H S
(i) (4x+7y)2(4x7y)n

(4x)2+(7y)2+24x7y[(4x)2+(7y)22.4x7y]

((ab)2=a2+b22ab)

(4x)2+(7y)2+2.4x7y(4x)2(7y)2+24x7y

4.4x7y

112 x y= R.H.S


(ii) (37p76q)2+pq

(37p)2+(76q)2237p16q+pq

((ab)2=a2+b22ab)

949p2+4936q2pq+pq

949p2+4936q2=RHSL.HS=RHS


(iii) LHs=(pq)(p+q)+(pr)(q+r)+(rp)(r+p)

=p2q2+q2r2+r2p2

(+(a+b)(ab)=a2b2]

=0=RHS

LHS=RHS


Question 9

Sol :
Given : (x+1x)=2

Squaring on both sides
(i) (x+1x)2=22

x2+2x1x+(1x)2=4((a+b)2=a2+b2+2ab)

x2+2+1x2=4

x2+1x2=42

x2+1x2=2


(ii) Again squaring on both sides

(x2+1x2)2=22

(x2)n+2x21x2+(1x2)2=4

x4+2+1x4=4

x4+1x4=42

x4+1x4=2n

Question 10

Sol :
(i) x1x=7

Squaring on both sides

(x1x)2=72

x22x1x+(1x)2=49

((ab)2=a22ab+b2)

x22+1x2=49

x2+1x2=49+2

x2+1x2=51



(ii) x2+1x2=51

squaring on both sides

(x2+1x2)2=512

(x2)2+2x21x2+(1x2)2=2601

x4+2+1x4=2601

x4+1x4=26012

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+2x1x=25(a2+b2+2ab=(a+b)2)

(x+1x)2=25

x+1x=5



(ii)  x+1x2=252

Subtract '2' on both sides

x2+1x22=232

(x)2+(1x)n2x1x=21

(x1x)2=21(a2+b22ab=(ab)2)

x1x=21

x1x=33

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 ab=6,a2+b2=42
a-b=6

Squaring on both Sides

(ab)2=62

a2+b22ab=36

422ab=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 

(ab)2=a2+b22ab=412×4=418=33.(ab)2=33ab=33

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