IIT-JEE 2008 · Paper 1, Mathematics question 14

Statement 1. Of the 16 choices of $(a,b,c,d)\in\{0,1\}^4$, a unique solution needs $ad-bc\ne0$:

• $ad=1$, $bc=0$: $a=d=1$ with $(b,c)\ne(1,1)$: 3 cases;
• $ad=0$, $bc=1$: $b=c=1$ with $(a,d)\ne(1,1)$: 3 cases.

Probability $\frac6{16}=\frac38$. True.

Statement 2. A homogeneous system always has the solution $x=y=0$. True.

Explanation? Statement 2 says nothing about uniqueness, so it does not explain Statement 1.

(B)