The number of real values of a for which the vectors $\hat i +2\hat j+\hat k, a\hat i + \hat j + 2\hat k$ and $\hat i + 2\hat j + a\hat k$ are coplanar, is

2

For given vectors to be coplanar, we must have

$\begin{vmatrix}1 &2& 1\\a &1& 2\\1 &2& a\end{vmatrix}=0$

$⇒(a-4)-2(a^2-2)+(2a −1) = 0$

$⇒2a^2-3a+1=0⇒ (2a-1) (a-1)=0⇒a=1,\frac{1}{2}$

Hence, there are two values of a.