JEE Main & Advanced
Mathematics
Vector Algebra
Question Bank
Critical Thinking
question_answer
The value of 'a' so that the volume of parallelopiped formed by \[\mathbf{i}+a\mathbf{j}+\mathbf{k},\mathbf{j}+a\,\mathbf{k}\] and \[a\,\mathbf{i}+\mathbf{k}\] becomes minimum is [IIT Screening 2003]
A)- 3
B)3
C)\[\frac{1}{\sqrt{3}}\]
D)\[\sqrt{3}\]
Correct Answer:
C
Solution :
\[V=\left| \,\begin{matrix} 1 & a & 1 \\ 0 & 1 & a \\ a & 0 & 1 \\ \end{matrix}\, \right|=1+{{a}^{3}}-a\,\,\Rightarrow \,\,\frac{dV}{da}=3{{a}^{2}}-1\] \[=3\,\left( a+\frac{1}{\sqrt{3}} \right)\,\left( a-\frac{1}{\sqrt{3}} \right)\]
\[\therefore \] Minimum at \[\frac{1}{\sqrt{3}}\].