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Find the value of a so that the volume o...

Find the value of `a` so that the volume of the parallelepiped formed by vectors ` hat i+a hat j+k , hat j+a hat ka n da hat i+ hat k` becomes minimum.

A

-3

B

3

C

`1//sqrt(3)`

D

`sqrt(3)`

Text Solution

Verified by Experts

The correct Answer is:
C
We know that volume of parallelopiped whose edges
are `vec(a) , vec(b) , vec(c ) = [vec(a) ,vec(b) ,vec( c)].`
`:. |{:(1,,alpha,,1),(0,,1,,alpha),(alpha,,0,,1):}|=1 + alpha^(3) - alpha`
Let ` f(a) = a^(3) -a +1`
`rArr f' (a) = 3a^(2) -1`
`rArr f''(a) =6a`
For maximum or minimum pu f' (a) =0
`rArr a= +- (1)/(sqrt(5)) ` which shows f(a) is minimum at `a= (1)/(sqrt(3))` and maximum at `a=-(1)/(sqrt(3))`
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