The GATE Grind

GATE 2023 ME – Question 14

Operations Research · Linear programming and the simplex method · 1 mark · Multiple choice

Which one of the options given represents the feasible region of the linear programming model:
Maximize $45X_1 + 60X_2$
$X_1 \le 45$
$X_2 \le 50$
$10X_1 + 10X_2 \ge 600$
$25X_1 + 5X_2 \le 750$

a sketch (not to scale) in the $X_1$-$X_2$ plane with a vertical line, a horizontal line, a flatter line sloping down to the right, and a steeper line sloping down to the right. Four regions are marked: P (a large hatched region on the lower left), Q (a small triangle just below the horizontal line, between the two sloping lines), R (below the flatter line and to the right of the steeper line) and S (a hatched region to the right of the steeper line and below the horizontal line).
  1. Region P
  2. Region Q
  3. Region R
  4. Region S

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Show answer and explanation

Correct answer: (B) Region Q

Explanation

The constraint $10X_1 + 10X_2 \ge 600$ means $X_1 + X_2 \ge 60$, so the feasible points are on or above the flatter line. The constraint $25X_1 + 5X_2 \le 750$ keeps points below or on the steeper line. Together with $X_2 \le 50$ (below the horizontal line) and $X_1 \le 45$, the points that are above the flatter line, below the steeper line and below the horizontal line form the small triangle Q. Region P is below the flatter line, and regions R and S are on the wrong side of one of the lines.