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 (60, 0), (120, 0), (60, 30) and (40, 20). The objective function $Z = ax + by, a, b >0$ has maximum value 600 at points (120, 0) and (60, 30).
Assertion (A): Minimum value of $Z = 300$
Reason (R): $a = 5, b = 10$.
Select the correct answer from the options as given below:
Answer & explanation
Correct answer: option 1
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).
$Z = ax + by$ has maximum value at points (120, 0) and (60, 30).
So, $120a + 0 = 600 ⇒ a = 5$.
Also, $60a +30b = 600⇒ 60 × 5 + 30b = 600$
$⇒30b=300⇒ b=10$
∴ $a = 5, b = 10$
∴ Reason is true.
Now, $Z = 5x + 10y$
Value of Z at (60, 0) = 5 × 60 + 0 = 300 → Minimum
Value of Z at (120, 0) = 5 × 120 + 0 = 600
Value of Z at (60, 30) = 5 × 60 + 10 × 30 = 600
Value of Z at (40, 20) = 5 × 40 + 10 x 20 = 400
∴ Assertion is true.
Hence, both Assertion and Reason are true and Reason is the correct explanation of Assertion.