A) \[{{\cos }^{-1}}\left( \frac{1}{3} \right)\]
B) \[{{\cos }^{-1}}\left( \frac{1}{2} \right)\]
C) \[{{\cos }^{-1}}\left( \frac{1}{4} \right)\]
D) \[(a-b).(b-c)\times (c-a)\]
Correct Answer: A
Solution :
Let \[\overrightarrow{a}=3\hat{i}-2\hat{j}-\hat{k},\] \[\overrightarrow{b}=2\hat{i}+3\hat{j}-4\hat{k},\]\[\overrightarrow{c}=-\hat{i}+\hat{j}+2\hat{k}\]and \[\overrightarrow{d}=4\hat{i}+5\hat{j}+\lambda \hat{k}\]are coplanar so, \[[d\,b\,c]+[d\,c\,a]+[d\,a\,b]=[a\,b\,c]\] \[\Rightarrow \]\[\left| \begin{matrix} 4 & 5 & \lambda \\ 2 & 3 & -4 \\ 2 & 3 & -4 \\ \end{matrix} \right|+\left| \begin{matrix} 4 & 5 & \lambda \\ -1 & 1 & 2 \\ 3 & -2 & -1 \\ \end{matrix} \right|\] \[\left| \begin{matrix} 4 & 5 & \lambda \\ 3 & -2 & -1 \\ 2 & 3 & -4 \\ \end{matrix} \right|=\left| \begin{matrix} 3 & -2 & -1 \\ 2 & 3 & -4 \\ -1 & 1 & 2 \\ \end{matrix} \right|\] \[\Rightarrow \] \[40+5\lambda +37-\lambda +94+13\lambda =25\] \[\Rightarrow \] \[\lambda =-\frac{146}{17}\]
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