Consider the feasible region determined by the constraints \(y \le 2x\), \(x + y \le 6\), \(x \ge 0\), \(y \ge 0\).
(A) The origin is a corner point of the feasible region
(B) The point (1, 3) lies in the feasible region
(C) The feasible region has exactly 3 corner points
(D) The maximum value of \(Z = x + 2y\) on this region is 12
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (C) only
The line y = 2x passes through the origin. The region lies below it, below the line x + y = 6, and above the x-axis.
The lines y = 2x and x + y = 6 meet where
\(x + 2x = 6\)
\(x = 2,\ y = 4\)
The region is the triangle with corner points (0, 0), (6, 0) and (2, 4).
Values of Z = x + 2y at the corners:
\(Z(0, 0) = 0\)
\(Z(6, 0) = 6\)
\(Z(2, 4) = 2 + 8 = 10\)
So the region is a triangle with corners (0, 0), (6, 0), (2, 4), and the maximum of Z is 10.
Checking the statements
(A) The origin is a corner point of the feasible region — Correct. The lines y = 2x and y = 0 meet at (0, 0).
(B) The point (1, 3) lies in the feasible region — Incorrect. Here y = 3 but 2x = 2, so \(y \le 2x\) fails.
(C) The feasible region has exactly 3 corner points — Correct. They are (0, 0), (6, 0) and (2, 4).
(D) The maximum value of \(Z = x + 2y\) on this region is 12 — Incorrect. The largest corner value is Z(2, 4) = 10; the value 12 is at (0, 6), which breaks \(y \le 2x\).