Exercise 6C
Question 1
Ans:
Given,
xa=yb=zc let xa=yb=zc=k.
∴x=ak,y=bk and z=ck
(i) x3a3−y3b3+z3c3=xyzabc
(ak)3a3 - (bk)3b3+(ck)3c3
ak×bk×ckabc
a3k3a2−b3k3b3+c3k3c3=k3
k3−k2+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+acxz−ac=yz+bcyz−bcak×ck+a.cak×ck−ac=bk×ck+bcbk×ck−bcack2+acack2−ac=bck2+bcbck2−bc
ac(k2+1)ac(k2−1)=bc(k2+1)bc(k2−1)
k2+1k2−1=k2+1k2−1
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+3a2c2−5c4f2b6+3b2f2−5f5 =a4b4
2b4k4b2+3b2k2f2k2−5f4k4f2b6+3b2f2−5f5= b4k4b4
2b6k4+3b2f2k4−5f5k42b6+3b2f2−5f5=k4
k4(2b6+3b2f2−5f5)2b6+3b2f2−5f5=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=ck−k=ck2
(i) (a+b+c)(a−b+c)=a2+b2+c2.
(ck2+ck+c)(ck2−ck+c)=(ck2)2+(ck)2+c2
c(k2+k+1)c(k2−k+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)2⋅c2=b3⋅c3+a3⋅c3+a3−b3a3b3c3
c3k6+c3k3+c3c2k4⋅c2k2⋅c2=(ck)3⋅c3+(ck2)3−c3+(ck2)3−(ck)3(ck2)3−(ck)3−c3
c3(x6+k3+1)c2k6⋅c2k2−c2=c3k3⋅c3+c3k6⋅c3+c3k6⋅c3k3c3k6⋅c3k3⋅c3
c3(k6+k3+1)c6.k6=c6−k3+c6⋅k6+c6k9c3k6⋅c3k3⋅c3
k6+k3+1c3k6 = 1+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+7ck2⋅ck+9(ck)24⋅(ck)2+7ck⋅c+9⋅c2= 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=ck−dk⋅k=dk2a=bk=dk2k=dk3
a, b ,c , d are in continued proportion
ab=bc=ca=k
c=dk,b=ck⇒b=dk2.
a=dk,=dk2k=dk3
(b−c)2+(c−a)2+(d−b)2=(a−d)2
(image to be added)
(ii) √a5+b2c2+a3c2b4c+d4+b2cd2 = ad
√(dk3)5+(dk2)2(dk)2+(dk3)5−(dk)2(dk2)4⋅dk+d4+(dk2)2⋅dk⋅d2=dk3d
√d5k15+d2k4⋅d2k2+d3k9⋅d2k2d4k8⋅dk+d4+d2⋅k4⋅dk⋅d2=dk3d
√d5k15+d4⋅k6+d5k11d5k9+d4+d5k5=k3
√k6=k3
√k3×k3=k3
k3=k3
Hence proved
(iii) √ab−√bc+√cd=√(a−b+c)(b−c+d)
√dk3⋅dk2−√dk2⋅dk+√dk⋅d=√(dk3−dk2+dk)(dk2−dk+d)
√d2k5−√d2k3+√d2k=√dk(k2−k+1)d(k2−k+1)
dk2√k−dk√k+d√k=√d2k(k2−k+1)2
d√k(k2−k+1)=d√k(k2−k+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.
ax−by(a+b)(x−y)+by−cz(b+c)(y−z)+cz−axc(+a)(z−x)=3
a−ak−b⋅bk(a+b)(ak−bk)+bbk−c⋅ck(b+c)(bk−ck)+c(ck−a⋅ak(c+a)(ck−ak)=3
a2k−b2k(a+b)(ak−bk)+b2k−c2k(b+c)(bk−ck)+c2k−a2k(c+a)(ck−9k)=3
k2(a2−b2)(a+b)k(a−b)+ K(b2−c2)(b+c))k(b−c)
(a+b)(a−b)(a+b)(a−b)+((b+b)(b−c)(b+x)⋅(b−5)+(c+a)(c−a)(c+a)(−c−a)=3
1+1+1=3
3=3 Hence proved
Question 6
Ans: Given,
xb+c−a=yc+a−b=Za+b−c
=x+y+zb+c−a+c+a−b+a+b−c
=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(b−c)+b(c−a)+c(a−b)=0
k(b2−c2)+k(c2−a2)+k(a2−b2)=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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