If the corner points of the feasible region for an LPP are (0, 2) (3,…

The correct answer is Option (3) → every point on the line segment joining (0, 2) and (3, 0)

Note: The given answer is as per the official NTA Answer Key. However, more than one answer is possible: Option (3) and Option (4) are both correct, as explained below.

Values of z = 4x + 6y at the corner points:

    \(z(0, 2) = 12\)
    \(z(3, 0) = 12\)
    \(z(6, 0) = 24\)
    \(z(6, 8) = 72\)
    \(z(0, 5) = 30\)

The minimum value 12 occurs at two adjacent corners, (0, 2) and (3, 0). The edge joining them lies on the line 2x + 3y = 6, and on it

    \(z = 4x+6y\)
    \(= 2(2x+3y)\)
    \(= 2 \times 6 = 12\)

So the minimum occurs at every point of this segment, and its mid point \(\left(\frac{3}{2}, 1\right)\) is one of these points.

Option (1) (0, 2) only — Incorrect. z is also 12 at (3, 0), so the minimum is not at (0, 2) only.

Option (2) (3, 0) only — Incorrect. z is also 12 at (0, 2), so the minimum is not at (3, 0) only.

Option (3) every point on the line segment joining (0, 2) and (3, 0) — Correct. Every point of this segment gives z = 12, the least value, and this describes all such points.

Option (4) mid point of the line segment joining the points (0, 2) and (3, 0) — Should also be correct, though not the answer given by NTA. The mid point \(\left(\frac{3}{2}, 1\right)\) gives z = 6 + 6 = 12, the least value, so the minimum does occur there.

Advice for the exam: When more than one option is correct, mark the one that answers the question most completely, here every point on the line segment joining (0, 2) and (3, 0), which already includes the mid point. If NTA’s provisional answer key marks a different option, challenge it during the answer-key challenge window, giving proper reasoning. NTA does revise its key when a challenge is well justified, so a sound challenge can raise your marks.