The number of distinct real values of λ for which the vectors $\vec a=λ^3\hat i+\hat k,\vec b=\hat i-λ^3\hat j$ and $\vec c=\hat i(2λ-\sin λ)\hat j-λ\hat k$ are coplanar is

1

For $\vec a,\vec b,\vec c$ to be coplanar, we must have

$[\vec a\,\vec b\,\vec c]=0$

$⇒\begin{vmatrix}1&0&1\\1&-λ^3&0\\1&2λ-\sin λ&-λ\end{vmatrix}=0$

$⇒λ^7+λ^3+2λ=\sin λ$

This is true for $λ = 0$.

For non-zero values of λ it gives

$λ^6+λ^2+2=\frac{\sin λ}{λ}$   ...(i)

We know that $\frac{\sin x}{x}<1$ for all $x≠ 0$. Therefore, LHS of (i) is greater than 2 and RHS is less than 1. So, (i) is not true for any non-zero λ.

Hence, there is only one value of λ.