Match each set of constraints in List-I with the number of corner points of its feasible region in List-II (in each case, \(x \ge 0\) and \(y \ge 0\) also hold).
| List-I | List-II | ||
|---|---|---|---|
| Constraints | Number of corner points | ||
| (A) | \(x + y \le 4\) | (I) | 4 |
| (B) | \(x + y \ge 3\), \(x \ge 1\) | (II) | 2 |
| (C) | \(x + y \le 6\), \(x \le 4\) | (III) | 5 |
| (D) | \(x + 2y \le 10\), \(x + y \le 6\), \(x \le 5\) | (IV) | 3 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(IV), (B)-(II), (C)-(I), (D)-(III)
A corner point is a point of the region where two boundary lines meet.
(A) \(x + y \le 4\)
The region is the triangle with corners (0, 0), (4, 0) and (0, 4). This matches (IV) 3.
(B) \(x + y \ge 3\), \(x \ge 1\)
The line \(x = 1\) meets \(x + y = 3\) at (1, 2), and \(x + y = 3\) meets the x-axis at (3, 0). The region is unbounded with only these two corners. This matches (II) 2.
(C) \(x + y \le 6\), \(x \le 4\)
The line \(x = 4\) meets \(x + y = 6\) at (4, 2). The corners are (0, 0), (4, 0), (4, 2) and (0, 6). This matches (I) 4.
(D) \(x + 2y \le 10\), \(x + y \le 6\), \(x \le 5\)
\(x + 2y = 10\) and \(x + y = 6\) meet at (2, 4)
\(x + y = 6\) and \(x = 5\) meet at (5, 1)
The corners are (0, 0), (5, 0), (5, 1), (2, 4) and (0, 5). This matches (III) 5.