Question

For a given Linear Programming problem, the objective function is \( z = 3x + 2y \). Subject to constraints are: \( 4x + 3y \leq 60,\ x \geq 3,\ y \leq 2x,\ y \geq 0 \). \(P\) is one of the corner points of the feasible region. Then the coordinate of \(P\) is

• A. \( (3,6) \)
• B. \( (0,20) \)
• C. \( (0,0) \)
• D. \( (12,6) \)

Hint:

Corner points in LPP are found by solving boundary equations pairwise and checking feasibility.

Answer 1

The Correct Option is A

Step 1: Write the constraints.
\[ 4x + 3y \leq 60,\quad x \geq 3,\quad y \leq 2x,\quad y \geq 0. \]

Step 2: Convert inequalities into boundary lines.
\[ 4x + 3y = 60,\quad x = 3,\quad y = 2x,\quad y = 0. \]

Step 3: Find intersection of \(x=3\) and \(y=2x\).
\[ y = 2(3) = 6. \]
So, point is:
\[ (3,6). \]

Step 4: Verify constraint \(4x+3y \leq 60\).
\[ 4(3) + 3(6) = 12 + 18 = 30 \leq 60. \]
So, it satisfies all constraints.

Step 5: Check other options quickly.
\((0,20)\): violates \(x \geq 3\)
\((0,0)\): violates \(x \geq 3\)
\((12,6)\): violates \(y \leq 2x\)? check: \(6 \leq 24\) OK but check first constraint: \(48+18=66>60\) invalid

Step 6: Confirm feasible point.
Thus, \( (3,6) \) lies in feasible region.

Step 7: Final conclusion.
Hence, correct answer is:
\[ \boxed{(3,6)}. \]