Two statements are given, one labelled Assertion (A) and the other labelled Reason (R).

The corner points of the feasible region for an L.P.P. are (4, 0), (5, 0), (5, 3), (3, 5), (0, 5) and (0, 4). The objective function $Z = ax - by + 1900, a, b > 0$ has maximum value 1950 at (5,0) and minimum value 1550 at (0,5).

Assertion (A): The value of Z at point (5, 3) = 1740
Reason (R): $a = 10, b = 70$.

Select the correct answer from the options as given below:

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

The correct answer is Option (1) → Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

$Z = ax - by + 1900$ has maximum value 1950 at (5,0).

So, $5a + 1900 = 1950⇒ 5a = 50⇒ a = 10$

and Z has minimum value 1550 at (0,5).

So, $0-5b+ 1900 = 1550⇒-5b=-350 ⇒b = 70$

$∴ a = 10, b = 70$

Also, value of Z at (5, 3) = 10 × 5 - 70 × 3 + 1900 = 1740.

Hence, both Assertion and Reason are true and Reason is the correct explanation of Assertion.