Consider the system of equations $ax + by = 0; cx + dy = 0$, where $a, b, c, d \in \{0, 1\}$ STATEMENT I : The probability that the system of equations has a unique solution is $\frac{3}{8}$. and STATEMENT II : The probability that the system of equations has a solution is 1.

$\text{Given: } ax+by=0;\quad cx+dy=0$

$a,b,c,d \in \{0,1\} \ \Rightarrow\ 2^4 = 16 \text{ equally likely matrices}$

$\text{homogeneous system (always has } x=y=0)$

$\text{Check Statement I: } P(\text{unique solution}) \overset{?}{=} \tfrac{3}{8}$

$\text{Check Statement II: } P(\text{has a solution}) \overset{?}{=} 1$

$\text{homogeneous} \Rightarrow (x,y)=(0,0) \text{ always solves it}$

$\Rightarrow P(\text{has a solution}) = 1 \ \Rightarrow\ \text{Statement II is TRUE}$

Unique solution iff Δ = a b c d = ad-bc ≠ 0

$a,b,c,d \in \{0,1\} \Rightarrow ad,\ bc \in \{0,1\} \Rightarrow \Delta \in \{-1,0,1\}$

$\Delta \neq 0 \iff \text{exactly one of } ad,\ bc \text{ equals } 1$

$ad=1,\ bc=0:\ a=d=1,\ (b,c)\neq(1,1) \Rightarrow 3 \text{ matrices}$

$ad=0,\ bc=1:\ b=c=1,\ (a,d)\neq(1,1) \Rightarrow 3 \text{ matrices}$

$\text{Nonzero } \Delta: 3+3 = 6$

$P(\text{unique}) = \dfrac{6}{16} = \dfrac{3}{8} \ \Rightarrow\ \text{Statement I is TRUE}$

$\text{Both statements are TRUE}$

$\text{II is about existence (trivial solution), not the cause of uniqueness}$

$\Rightarrow \text{Option } (2)$