Algebra

Linear Programming

Medium

applied

If the objective function $Z = px + qy, p > 0, q > 0$ of a linear programming problem attains its optimal value at the points (4, 7) and (5, 5) and $pq = 50$ then

$p = 5, q = 10$

$p = 10, q = 5$

$p = 25, q = 2$

$p = \frac{40}{3}, q = \frac{15}{4}$

✅ Correct Option: 2

The objective function $Z = px + qy$ where $p > 0, q > 0$ attains its optimal value at both points $(4, 7)$ and $(5, 5)$ .

Since the optimal value occurs at two different points, both points give the same value of $Z$ .

At point $(4, 7)$ : $Z = 4p + 7q$

At point $(5, 5)$ : $Z = 5p + 5q$

Setting them equal:

$4p + 7q = 5p + 5q$

$7q - 5q = 5p - 4p$

$2q = p$

Given that $pq = 50$ , substitute $p = 2q$ :

$(2q) \times q = 50$

$2q^2 = 50$

$q^2 = 25$

$q = 5$ (taking the positive value since $q > 0$ )

Using $p = 2q$ :

$p = 2(5)$

$p = 10$

Therefore, $p = 10$ and $q = 5$ .

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(B) The graphical compliment

(D) The linear constraints

Choose the correct answer from the options given below:

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If the objective function Z=px+qy,p>0,q>0Z = px + qy, p > 0, q > 0Z=px+qy,p>0,q>0 of a linear programming problem attains its optimal value at the points (4, 7) and (5, 5) and pq=50pq = 50pq=50 then

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If the objective function $Z = px + qy, p > 0, q > 0$ of a linear programming problem attains its optimal value at the points (4, 7) and (5, 5) and $pq = 50$ then