Algebra

Linear Programming

Medium

core

The corner points of the bounded feasible region determined by the system of linear constraints are (0, 0), (5, 0), (3, 4) and (0, 5). Let $Z = px + qy$ where $p, q > 0$ . Condition on p and q so that the maximum of Z occurs at both (5, 0) and (3, 4) is

$3q = p$

$q = p$

$2q = p$

$2p = q$

✅ Correct Option: 3

The corner points of the bounded feasible region are (0, 0), (5, 0), (3, 4) and (0, 5). For the objective function $Z = px + qy$ where $p, q > 0$ , the maximum occurs at two corner points when both points give equal Z values.

For maximum at both (5, 0) and (3, 4), these points must yield the same value of Z.

At point (5, 0):

$Z = p(5) + q(0)$

$Z = 5p$

At point (3, 4):

$Z = p(3) + q(4)$

$Z = 3p + 4q$

For maximum to occur at both points:

$5p = 3p + 4q$

$5p - 3p = 4q$

$2p = 4q$

$p = 2q$

This can also be written as $2q = p$ .

In linear programming, when the maximum occurs at two corner points, the objective function line is parallel to the edge connecting those points. The condition $2q = p$ ensures this property.

Therefore, the condition is $2q = p$ .

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(B) $x \geq 0$ , $y \geq 0$ , $2x + y \leq 2$

(D) $x + 2y \geq 8$ , $x - y \geq 1$

Choose the correct answer from the options given below:

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Which one of the following set of constraints depicted the shaded region given below?

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2023: 15 June Shift 2

If corner points of a feasible region are (0, 0), (2, 0), $\left(\frac{20}{19}, \frac{45}{19}\right)$ and (0, 3), then

(A) Maximum value of $z = 5x + 3y$ is 10

(B) Minimum value of $z = 5x + 3y$ is 0

(D) Maximum value of $z = 5x + 3y$ is 10 and minimum value is 0

Choose the correct answer from the options given below :

The corner points of the bounded feasible region determined by the system of linear constraints are (0, 0), (5, 0), (3, 4) and (0, 5). Let Z=px+qyZ = px + qyZ=px+qy where p,q>0p, q > 0p,q>0. Condition on p and q so that the maximum of Z occurs at both (5, 0) and (3, 4) is

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CUET Mathematics Past Year Questions (Topic-Wise)

The corner points of the bounded feasible region determined by the system of linear constraints are (0, 0), (5, 0), (3, 4) and (0, 5). Let $Z = px + qy$ where $p, q > 0$ . Condition on p and q so that the maximum of Z occurs at both (5, 0) and (3, 4) is