Let S be the of all column matrices b 1 b 2 b 3 such that b 1 ,   b 2 ,   b 3 ∈ ℝ and…

We find $D=0$ & since no pair of planes are parallel, so there are an infinite number of solutions. So, the infinite solutions shall lie on a common line of intersection of these planes. Hence, we can write any plane as  a linear combination of other two planes: 

$\alpha P_1-\lambda P_2=P_3$
$\Rightarrow$ $P_1+7P_2=13P_3$ (by comparing coefficients of $x, y, z$)
$\Rightarrow$ $b_1+7b_2=13b_3$
(a) $D\ne 0\Rightarrow$ unique solution for any $b_1, b_2, b_3$
(b) $D=0$ but $P_1+7P_2\ne 13P_3$
(c) $D=0$ Also as they are parallel planes, they will have at least one solution only if they are all coincident, so that $b_2=-2b_1, b_3=-b_1$
So each of $b_1, b_2, b_3$ that satisfy $b_1+7b_2=13b_3$, they don't always need to satisfy $b_2=-2b_1, b_3=-b_1$. Hence option is wrong.

(d) $D\ne 0 \Rightarrow$ unique solution for any $b_1, b_2, b_3$