JEE 2023 — Mathematics PYQ

Solution:

1. Find $k$ for infinite solutions ($\Delta =0$):

   $$\left|\begin{array}{ccc}1& amp;1& amp;k\\ 2& amp;3& amp;-1\\ 3& amp;4& amp;2\end{array}\right|=0⟹1(6+4)-1(4+3)+k(8-9)=0⟹k=3$$

2. Substitute $k=3$ in the second system:

   $$4x+5y=7$$
   $$7x+8y=10$$

3. Solve for $x,y$:

   By solving, we get $x=-2$ and $y=3$.

   Check condition: $x+y=-2+3=1$.

Correct Option: (3) unique solution satisfying $x+y=1$