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S Chand CLASS 10 Chapter 6 Ration and proportion Exercise 6C

  Exercise 6C

Question 1

Ans:
Given,
xa=yb=zc let xa=yb=zc=k.
x=ak,y=bk and z=ck

(i) x3a3y3b3+z3c3=xyzabc
(ak)3a3(bk)3b3+(ck)3c3
ak×bk×ckabc
a3k3a2b3k3b3+c3k3c3=k3
k3k2+k3=k3
k3=k3
Hence, proved: L.H.S=R.H.S 

(ii)  (a2x2+b2y2+c2z2a3x+b3y+c3z)32=xyzabc
(a2×a2x2+b2×b2k2+c2×c2k2a3×ak+b3×bk+c3+ck)32=ak×bk×ckabc
(a4k2+b4k2+c4k2a4k+b4k+c4k)32= abck3abc
(k2(a4+b4+c4)k(a4+b4+c4))32  =K3
(K)32=(K3)12
K32=K32. 
Hence prove : L.H.S =R.H.S

(iii)
xz+acxzac=yz+bcyzbcak×ck+a.cak×ckac=bk×ck+bcbk×ckbcack2+acack2ac=bck2+bcbck2bc
ac(k2+1)ac(k21)=bc(k2+1)bc(k21)
k2+1k21=k2+1k21
L.H.S =R.H.S 
Hence proved

(iv) ab(x+aa+y+bb+2+cc)3=27(x+a)(y+b)(2+c) ab((ak+aa+bk+bb+ck+cc)3=27(ak+a)(bk+b)(ck+c)
abc(a(k+1)a+b(k+1)b+c(k+1)c)3= 27a(k+1)b(k+1)c(k+1)
abc(k+1+k+1+k+1)3 =27abc(k+1)3
abc(3k+3)3=27abc(k+1)3
abc(3)3(k+1)3=27abc(k+1)3
27abc(k+1)3=27abc(k+1)3
L.H.S = R.H.S 
Hence proved 

(iv)(3x3+5y3+7z33a3+5b3+7c3)13
(3×a3k3+5×b3×k3+7×c3k33a3+5b3+7c3)13
(3a3k3+5b3k3+7c3k33a3+5b3+7c3)13
(k3(3a3+5b3+7b3)(3a3+5b3+7c3))13
=(k3)13
=k(3×13)
Hence proved 

Question 2

Ans:
Given,
 let ab=cd=efab=cd=ef=k
a=bk,c=dk and e=fk
(i) pa3+qc3+rc3pb3+qd3+rf3=accbdf
pb3k3+qd3k3+rj3k3pb3+qd3+rf3=
k3(pb3+qd3+rf3)(pb3+qd3+rf3)
k3=k3
Hence, prove.

(ii)˙a4+c4b4+d4=pa2+qc2pb2+qd2
b4k4+d4k4b4+d4=pb2k2+qd2k2pb2+qd2
k4(b4+d4)b4+d4=k2(pb2+qd2)pb2+qd2
$\begin{aligned}&\sqrt{k^{4}}=k^{2} \\&\sqrt{k^{2} \times  k^{2}}=k^{2}
\\&k^{2}=k^{2}\end{aligned}$
Hence, proved

(iii) 2a4b2+3a2c25c4f2b6+3b2f25f5 =a4b4
2b4k4b2+3b2k2f2k25f4k4f2b6+3b2f25f5= b4k4b4
2b6k4+3b2f2k45f5k42b6+3b2f25f5=k4
k4(2b6+3b2f25f5)2b6+3b2f25f5=k4
k4=k4.
Hence, proved.

Question 3

Ans:
Given 
a, b, c are in continued proportion 
a:b::b:c.
ab=bc.
Let ab=bc=k
b=ck,a=bk=ckk=ck2

(i) (a+b+c)(ab+c)=a2+b2+c2.
(ck2+ck+c)(ck2ck+c)=(ck2)2+(ck)2+c2
c(k2+k+1)c(k2k+1)=c2k4+c2k2+c2
c2(k4+k2+1)=c2(k4+k2+1)
Hence proved 

(ii) a2+b2b2+c2=ac
(ck2)2+(ck)2(ck)2+c2 =ck2c
c2k4+c2k2c2k2+c2=ck2c
k2=k2
Hence proved

(iii)  a3+b3+c3a2b2c2=1a3+1b3+1c3
(ck2)3+(ck)3+c3(ck2)2(ck)2c2=b3c3+a3c3+a3b3a3b3c3
c3k6+c3k3+c3c2k4c2k2c2=(ck)3c3+(ck2)3c3+(ck2)3(ck)3(ck2)3(ck)3c3
c3(x6+k3+1)c2k6c2k2c2=c3k3c3+c3k6c3+c3k6c3k3c3k6c3k3c3
c3(k6+k3+1)c6.k6=c6k3+c6k6+c6k9c3k6c3k3c3
k6+k3+1c3k61+k3+k6c3k6
1+k3+k6c3k6=1+k3+1k6c3k6
Hence proved 


(iv) (4a2+7ab+9b2):(4b2+7bc+9c2)=a:c.
4a2+7ab+9b24b2+7bc+9c2=ac.
4(ck2)2+7ck2ck+9(ck)24(ck)2+7ckc+9c2= ck2c
4c2k4+7c2k3+9c2k24c2k2+7c2k+9c2=k2
k2=k2 
Hence proved 

Question 4

Ans: Given,
a,b,c,d are in continued proportion
ab=bc=cd
let ab=bc=cd=k.
c=dk,b=ckdkk=dk2a=bk=dk2k=dk3
a, b ,c , d are in continued proportion 
ab=bc=ca=k 
c=dk,b=ckb=dk2.
a=dk,=dk2k=dk3
(bc)2+(ca)2+(db)2=(ad)2
(image to be added)

(ii) a5+b2c2+a3c2b4c+d4+b2cd2ad
(dk3)5+(dk2)2(dk)2+(dk3)5(dk)2(dk2)4dk+d4+(dk2)2dkd2=dk3d
d5k15+d2k4d2k2+d3k9d2k2d4k8dk+d4+d2k4dkd2=dk3d
d5k15+d4k6+d5k11d5k9+d4+d5k5=k3
k6=k3
k3×k3=k3
k3=k3
Hence proved 

(iii) abbc+cd=(ab+c)(bc+d)
dk3dk2dk2dk+dkd=(dk3dk2+dk)(dk2dk+d)
d2k5d2k3+d2k=dk(k2k+1)d(k2k+1)
dk2kdkk+dk=d2k(k2k+1)2
dk(k2k+1)=dk(k2k+1).
Hence proved 

(iv) 3a+5d5a+7d=3a3+5b35a3+7b3
3dk3+5d5dk3+7d=3(dk3)3+5(dk2)35(dk3)3+7(dk2)3
d(3k3+5)d(5k3+7)=3d3k9+5d3k65d3k9+7d3k6
3k3+55k3+7=3k3+55k3+7
Hence proved 

(v) a3+b3+c3b3+c3+d3=ad
(dk3)3+(dk2)3+(dk)3(dk2)3+(dk)3+d3= dk3d
d3k9+d3k6+d3k3d3k6+d3k3+d3=k3
d3k3(k6+k3+1)d2(k6+k3+1)=k3
k3=k3
Hence, proved

(vi) (to be added)

Question 5

Ans: Given,
xa=yb=zc
letxa=yb=zc=k.
x=ak,y=bk and z=ck.
axby(a+b)(xy)+bycz(b+c)(yz)+czaxc(+a)(zx)=3
aakbbk(a+b)(akbk)+bbkcck(b+c)(bkck)+c(ckaak(c+a)(ckak)=3
a2kb2k(a+b)(akbk)+b2kc2k(b+c)(bkck)+c2ka2k(c+a)(ck9k)=3
k2(a2b2)(a+b)k(ab)+ K(b2c2)(b+c))k(bc)
(a+b)(ab)(a+b)(ab)+((b+b)(bc)(b+x)(b5)+(c+a)(ca)(c+a)(ca)=3
1+1+1=3
3=3 Hence proved

Question 6

Ans: Given,
xb+ca=yc+ab=Za+bc
=x+y+zb+ca+c+ab+a+bc
=x+y+za+b+c
Hence, proved

Question 7

Ans: Given,
ab+c=bc+a=ca+b
Let ab+c=bc+c=ca+b=k
a=k(b+c),b=k(c+a),c=k(a+b)
a(bc)+b(ca)+c(ab)=0
k(b2c2)+k(c2a2)+k(a2b2)=0
K×0=0
0=0 
Hence proved 

Question 8

Ans: Given,
ax=by=cz Leb ax=by=cz=k.x=14,y=kb. and z=kc.x2yz+y22x+z2xy=bca2+cab2+abc2
(ka)2k3×kc+(kb)2kc×ka+(kc)2ka×kb=bca2+cab2+abc2
k2a2k2bc+k2b2k2ca2+k2c2k2ab=b2a2+cab2+abc2
bca2+cab2+abc2=bca2+cab2+abc2
HENCE PROVED 

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