Algebra

Linear Programming

Medium

core

An objective function $z = ax + by$ is maximum at points (15,15) and (0, 20). If $a, b \geq 0$ and $ab = 27$ , then the maximum value of the objective function is

190

180

170

160

✅ Correct Option: 2

The objective function $z = ax + by$ is maximum at two points: $(15, 15)$ and $(0, 20)$ .

When a linear objective function has maximum value at two different points, both points give the same maximum value. This occurs when the objective function line is parallel to the edge connecting these two points in the feasible region.

Since both points give the same maximum value:

$15a + 15b = 0 + 20b$

$15a = 20b - 15b$

$15a = 5b$

$3a = b$

Using the condition $ab = 27$ with $b = 3a$ :

$a \times 3a = 27$

$3a^2 = 27$

$a^2 = 9$

$a = 3$ (since $a \geq 0$ )

From $b = 3a$ :

$b = 3 \times 3$

$b = 9$

Using point $(15, 15)$ :

$z = ax + by$

$z = 3(15) + 9(15)$

$z = 45 + 135$

$z = 180$

Therefore, the maximum value of the objective function is $180$ .

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A manufacturer of electronic circuit has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits A and B. Type A requires 20 resistors, 10 transistors and 10 capacitors. Type B requires 10 resistors, 20 transistors and 30 capacitors. If the profit on type A circuit is Rs. 50 and that on type B circuit is Rs. 60, identify the constraints for this LPP, if it was assumed that x circuit B of type A and y circuits of type B was produced by the manufacturer.

A. $x + 2y \geq 15$

B. $2x + y \leq 20$

C. $x + 2y \leq 12$

D. $x, y \leq 0$

Choose the correct answer from the options given below

An objective function z=ax+byz = ax + byz=ax+by is maximum at points (15,15) and (0, 20). If a,b≥0a, b \geq 0a,b≥0 and ab=27ab = 27ab=27, then the maximum value of the objective function is

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CUET Mathematics Past Year Questions (Topic-Wise)

An objective function $z = ax + by$ is maximum at points (15,15) and (0, 20). If $a, b \geq 0$ and $ab = 27$ , then the maximum value of the objective function is