The position vector of the points A, B, C and D are $3 \hat{i}-2 \hat{j}-\hat{k}, 2 \hat{i}+3 \hat{j}-4 \hat{k}, -\hat{i}+\hat{j}+2 \hat{k}$ and $4 \hat{i}+5 \hat{j}+\lambda \hat{k}$. It is known that these points are coplanar, then $\lambda$ is equal to:

$-\frac{146}{17}$

$\vec{A B}=-\hat{i}+5 \hat{j}-3 \hat{k}$

$\vec{A C}=-4 \hat{i}+3 \hat{j}+3 \hat{k}$

$\vec{A D}=\hat{i}+7 \hat{j}(\lambda+1) \hat{k}$

If vector $\vec{AB}, \vec{AC}$ and $\vec{AD}$ are coplanar, then

$\left|\begin{array}{ccc} -1 & 5 & -3 \\ -4 & 3 & 3 \\ 1 & 7 & \lambda+1 \end{array}\right|=0$

$\Rightarrow \lambda=-\frac{146}{17}$

Hence (1) is correct answer.