Explanation

(d)
Given system equations are
$3x - 2y + \mathcal{z} = 0$
$\lambda x - 14y + 15\mathcal{z} = 0\ and\ \ x + 2y - 3\mathcal{z} = 0$
The system of equations has infinitely many (non-trivial solutions, if $\mathrm{\Delta} = 0$.
$\Rightarrow \ \mathrm{\Delta} = \left| \begin{matrix}
3 & - 2 & 1 \\
\lambda & - 14 & 15 \\
1 & 2 & - 3
\end{matrix} \right| = 0$
$\Rightarrow 3(42 - 30) - \lambda(6 - 2) + 1( - 30 + 14) = 0$
$\Rightarrow 36 - 4\lambda - 16 = 0$
$\Rightarrow \ \lambda = 5$