JEE MAIN - Mathematics (2019 - 11th January Morning Slot - No. 11)
Let $$\overrightarrow a = \widehat i + 2\widehat j + 4\widehat k,$$ $$\overrightarrow b = \widehat i + \lambda \widehat j + 4\widehat k$$ and $$\overrightarrow c = 2\widehat i + 4\widehat j + \left( {{\lambda ^2} - 1} \right)\widehat k$$ be coplanar vectors. Then the non-zero vector $$\overrightarrow a \times \overrightarrow c $$ is :
$$ - 10\widehat i - 5\widehat j$$
$$ - 10\widehat i + 5\widehat j$$
$$ - 14\widehat i + 5\widehat j$$
$$ - 14\widehat i - 5\widehat j$$
Explanation
$$\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] = 0$$
$$ \Rightarrow \left| {\matrix{ 1 & 2 & 4 \cr 1 & \lambda & 4 \cr 2 & 4 & {{\lambda ^2} - 1} \cr } } \right| = 0$$
$$ \Rightarrow {\lambda ^3} - 2{\lambda ^2} - 9\lambda + 18 = 0$$
$$ \Rightarrow {\lambda ^2}\left( {\lambda - 2} \right) - 9\left( {\lambda - 2} \right) = 0$$
$$ \Rightarrow \left( {\lambda - 3} \right)\left( {\lambda + 3} \right)\left( {\lambda - 2} \right) = 0$$
$$ \Rightarrow \lambda = 2,3, - 3$$
So, $$\lambda $$ = 2 (as $$\overrightarrow a $$ is parallel to $$\overrightarrow c $$ for $$\lambda $$ = $$ \pm $$3)
Hence $$\overrightarrow a \times \overrightarrow c = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 1 & 2 & 4 \cr 2 & 4 & 3 \cr } } \right|$$
$$ = - 10\widehat i + 5\widehat j$$
$$ \Rightarrow \left| {\matrix{ 1 & 2 & 4 \cr 1 & \lambda & 4 \cr 2 & 4 & {{\lambda ^2} - 1} \cr } } \right| = 0$$
$$ \Rightarrow {\lambda ^3} - 2{\lambda ^2} - 9\lambda + 18 = 0$$
$$ \Rightarrow {\lambda ^2}\left( {\lambda - 2} \right) - 9\left( {\lambda - 2} \right) = 0$$
$$ \Rightarrow \left( {\lambda - 3} \right)\left( {\lambda + 3} \right)\left( {\lambda - 2} \right) = 0$$
$$ \Rightarrow \lambda = 2,3, - 3$$
So, $$\lambda $$ = 2 (as $$\overrightarrow a $$ is parallel to $$\overrightarrow c $$ for $$\lambda $$ = $$ \pm $$3)
Hence $$\overrightarrow a \times \overrightarrow c = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 1 & 2 & 4 \cr 2 & 4 & 3 \cr } } \right|$$
$$ = - 10\widehat i + 5\widehat j$$
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