question_answer

If the volume of parallel piped 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:                                                                           [JEE Main Held on 12-4-2019 Morning]

A) \[\sqrt{3}\]                              

B) \[-\frac{1}{\sqrt{3}}\]

C) \[\frac{1}{\sqrt{3}}\] 

D) \[-\sqrt{3}\]

Correct Answer: C

Solution :

Volume of parallelepiped\[=\left| \left| \begin{matrix}    1 & \lambda  & 1  \\    0 & 1 & \lambda   \\    \lambda  & 0 & 1  \\ \end{matrix} \right| \right|\] \[f(x)=|{{\lambda }^{3}}-\lambda +1|\] Its graph as follows where \[\alpha \approx -1.32\] \[\because \]Question is asking minimum value of volume of parallelopiped & corresponding value of \[\lambda \]; the minimum value is zero, Q cubic always has atleast 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. In JEE (Screening) 2003 same Question was asked and answer was given to be none of these, where the options were: [a]\[-3\]                                                [b]\[3\] [c] \[\frac{1}{\sqrt{3}}\]         [d] none of these