If $\vec a, \vec b, \vec c$ are non-coplanar vectors and λ is a real number, then the vectors $\vec a +2\vec b+3\vec c,λ\vec b+4\vec c$ and $(2λ-1)\vec c$ are non-coplanar for

all except two values of λ

Let $\vec α =\vec a+2\vec b+3\vec c, \vec β-λ\vec b+4\vec c$ and $\vec γ = (2λ-1) \vec c$.

Then,

$[\vec α\,\vec β\,\vec γ]=\begin{vmatrix}1&2&3\\0&γ&4\\0&0&(2λ-1)\end{vmatrix}[\vec a\,\vec b\,\vec c]$

$⇒[\vec α\,\vec β\,\vec γ]=λ(2λ-1)[\vec a\,\vec b\,\vec c]$

$⇒[\vec α\,\vec β\,\vec γ]=0$, if $λ=0,\frac{1}{2}$    $[∵[\vec a\,\vec b\,\vec c]≠0]$

Hence, $\vec α,\vec β,\vec γ$ are non-coplanar for all values of λ except two values 0 and $\frac{1}{2}$.