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If the points $A(3,2,1), B(4, x, 5), C(4,2,-2)$ and $D(6,5,-1)$ are coplanar, then $x$ has the value
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The correct answer is:
$5$
For coplanarity
$\begin{aligned} & {\left[\begin{array}{ccc}6-3 & 5-2 & -1-1 \\ 4-3 & x-2 & 5-1 \\ 4-3 & 2-2 & -2-1\end{array}\right]=0} \\ & \Rightarrow\left[\begin{array}{ccc}3 & 3 & -2 \\ 1 & x-2 & 4 \\ 1 & 0 & -3\end{array}\right]=0 \\ & \Rightarrow 3\{-3(x-2)-0\}-3\{1 \times(-3)-1 \times 4\}-2(1 \times 0-1 \times(x-2))=0 \\ & \Rightarrow-7 x+35=0 \\ & \Rightarrow x=5\end{aligned}$
$\begin{aligned} & {\left[\begin{array}{ccc}6-3 & 5-2 & -1-1 \\ 4-3 & x-2 & 5-1 \\ 4-3 & 2-2 & -2-1\end{array}\right]=0} \\ & \Rightarrow\left[\begin{array}{ccc}3 & 3 & -2 \\ 1 & x-2 & 4 \\ 1 & 0 & -3\end{array}\right]=0 \\ & \Rightarrow 3\{-3(x-2)-0\}-3\{1 \times(-3)-1 \times 4\}-2(1 \times 0-1 \times(x-2))=0 \\ & \Rightarrow-7 x+35=0 \\ & \Rightarrow x=5\end{aligned}$
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