If the minimum value of the objective function $Z = ax + by$ of an LPP occurs at two points (3, 5) and (5, 3), then

$a = b$

The correct answer is Option (3) → $a = b$

Given that the minimum value of $Z = ax + by$ occurs at two points $(3,5)$ and $(5,3)$.

If an objective function attains the same minimum value at two distinct points, then it has the same value at every point on the line segment joining them.

Hence, for points $(3,5)$ and $(5,3)$,

$Z_1 = a(3) + b(5)$ and $Z_2 = a(5) + b(3)$

Since both are equal:

$3a + 5b = 5a + 3b$

$\Rightarrow 2b = 2a$

$\Rightarrow a = b$

Therefore, $a = b$