13. Ifa/b=c/d=e/f, prove that each ratio is equal to
(2a^3 + 5c^3 + 7e^3/2b^3 + 5d^3 + 7f^3)^1/3
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If a/b=c/d=e/f, prove that each ratio is
(i)
3b
2
−5d
2
+7f
2
3a
2
−5c
2
+7e
2
(ii)[
2b
3
+5d
3
+7f
3
2a
3
+5c
3
+7e
3
]
3
1
Answer
It is given that
a/b=c/d=e/f=k
So we get
a=k,c=dk,e=fk
(i)
3b
2
−5d
2
+7f
2
3a
2
−5c
2
+7e
2
Substituting the values
=
3b
2
−5d
2
−7f
2
3b
2
k
2
−5d
2
k
2
+7f
2
k
2
Taking k as common
=k
3b
2
−5d
2
+7f
2
3b
2
−5d
2
+7f
2
=k
Therefore, it is proved.
(ii)[
2b
3
+5d
3
+7f
3
2a
3
+5c
3
+7e
3
]
3
1
Substituting the values
=[
2b
3
+5d
3
+7f
3
2b
3
k
3
+5d
3
k
3
+7f
3
k
3
]
3
1
Taking k as common
=k[
2b
3
+5d
3
+7f
3
2b
3
+5d
3
+7f
3
]
3
1
=k
Therefore, it is proved.
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