In a 60-minute test, a student can attempt questions of type A, each taking 3 minutes and carrying 5 marks, and questions of type B, each taking 2 minutes and carrying 4 marks. He may attempt at most 25 questions in all. If every question he attempts is answered correctly, the maximum marks he can score is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 110
Let him attempt x questions of type A and y of type B. Then
Maximise \(Z = 5x + 4y\)
subject to \(3x + 2y \le 60\), \(x + y \le 25\), \(x \ge 0\), \(y \ge 0\)
The two boundary lines meet where
\(3x + 2y - 2(x + y) = 60 - 50\)
\(x = 10, \; y = 15\)
The corner points are (0, 0), (20, 0), (10, 15) and (0, 25). Values of Z:
\(Z(0, 0) = 0\)
\(Z(20, 0) = 100\)
\(Z(10, 15) = 50 + 60 = 110\)
\(Z(0, 25) = 100\)
So the maximum is 110 marks, with 10 questions of type A and 15 of type B.
Why the other options are incorrect
Option (1) 100 — Incorrect. This is Z at (20, 0) or (0, 25), both smaller than Z at (10, 15).
Option (2) 120 — Incorrect. This needs 30 questions of type B, which breaks the limit of 25 questions.
Option (4) 125 — Incorrect. Z is at most 110 at every corner point, so 125 cannot be reached.