If the volume of parallelepiped formed by the vectors i + j + k, … - Vidyadip

MCQ

If the volume of parallelepiped formed by the vectors $\hat i + \lambda \hat j + \hat k$, $\hat j + \lambda \hat k$ and $\lambda \hat i + \hat k$ is minimum, then $\lambda $ is equal to

A
$\sqrt 3 $
B
$\frac{1}{{\sqrt 3 }}$
C
$-\frac{1}{{\sqrt 3 }}$
✓
None of these

✓

Answer

Correct option: D.

None of these

d
Volume of paralleopiped $ = \left\| {\begin{array}{*{20}{l}}
1&\lambda &1\\
0&1&\lambda \\
\lambda &0&1
\end{array}} \right\|$

$f(\lambda)=\left|\lambda^{3}-\lambda+1\right|$

Its graphs as follows

where $\lambda=-1.32$

For minimum value of volume of paralelopiped and corresponding value of $\lambda$; the minimum value is zero, $\because$ cubic always has at least one real root.

Hence answer to the question must be root of cubic $\lambda^{3}-\lambda+1=0 .$ None of the options satisfies the cubic.

Hence Question must be Bonus.

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