ICSE Class X Math 2015 Solved Paper

Question : 23 of 52

Marks: +1, -0

a, b, c

are in continued proportion, prove that

(a+b+c)(a-b+c)=a^2+b^2+c^2 \;\text ". "\;

Solution:

To Prove :

(a+b+c)(a-b+c)=a^2+b^2+

c^2

.
Proof :

a, b, c

are in continued proportion.

∴ \;\; \;{a}/{b}=\;{b}/{c}\;=k \;\text " (let) "\;

b\;=c k

a\;=b k=(c k) k

\;=c k^2

\;\text " L.H.S. "\;\;= (c k^2+c k+c) (c k^2-c k+c)

\;=c^2 (k^2+k+1) (k^2-k+1)

\;=c^2 [ (k^2+1) ^2-(k)^2]

\;=c^2 [k^4+2 k^2+1-k^2]

\;=c^2 [k^4+k^2+1]

\;\text " R.H.S. "\;\;=c^2 k^4+c^2 k^2+c^2

\;=c^2 [k^4+k^2+1]

L.H.S. = R.H.S.

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