NIMCET 2008 Question Paper with solutions

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Question : 23 of 120

Marks: +1, -0

The value of

λ

for which the volume of parallelopiped formed by the vectors

i+λ j+k, j+λ k

and

λ i+k

is minimum is given by

-3
3
${1}/{√{3}}$
$-√{3}$

Solution:

Volume of parallelepiped will be

V=|\table 1 , λ , 1 ; 0 , 1 , λ ; λ , 0 , 1|=1+λ(λ^{2}-1) ⇒ V=λ^{3}-λ+1

For minimum value

\;{d V}/{d λ}=3 λ^{2}-1=0

⇒ λ=1/√3

{d^2V}/{dλ^2}=6λ

At

λ=1/√3

⇒{d^2V}/{dλ^2}=6/√3 > 0

(min)

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