Question

Corners points of the feasible region for an LPP are \((1, 1)(2, 0) (3, 1)(\frac32,4)\) and \((0,5)\).Let \(z = px + 4y\),  be the objective function. If maximum of z occurs at \((\frac32,4)\) and\((3,1)\),then the value of p is : 
 

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

The Correct Option is D

To find the value of \( p \) such that the objective function \( z = px + 4y \) attains its maximum at the given points \((\frac{3}{2},4)\) and \((3,1)\), evaluate \( z \) at these points and equate the resulting expressions.

Calculate \( z \) for both points:

• At \((\frac{3}{2}, 4)\):
  \( z_1 = p \times \frac{3}{2} + 4 \times 4 = \frac{3p}{2} + 16 \)
• At \((3, 1)\):
  \( z_2 = p \times 3 + 4 \times 1 = 3p + 4 \)

Since \( z_1 = z_2 \), set the equations equal:

\(\frac{3p}{2} + 16 = 3p + 4\)

To solve for \( p \), subtract \( \frac{3p}{2} \) from both sides:

\(16 = \frac{3p}{2} + 4\)

Further simplify:

\(16 - 4 = 3p - \frac{3p}{2}\)

\(12 = \frac{3p}{2}\)

Clear fractions by multiplying the entire equation by 2:

\(24 = 3p\)

Divide by 3:

\(p = 8\)

Thus, the value of \( p \) is 8 when the maximum occurs at the given points.