The value of 'a' for which the vectors
A=2i-j-4k ,
B=i-5j+ak and
C=2i+3j+4k are linearly dependent is

We now that the given vectors will be linearly dependent if

$\left(\begin{array}{ccc}2& -1& 4\\ 1& -5& a\\ 2& 3& 4\end{array}\right|=0$

solving the matrix
$\Rightarrow$ 2(-5×4 - 3×a) - (-1)(1×4 - 2×a) + 4(1×3 - 2×(-5)) = 0
$\Rightarrow$ -40-6a+4-2a+12+40 = 0
$\Rightarrow$ a = 2
So for a = 2, the given vectors will be linearly dependent.