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