Algebra

Linear Programming

Medium

common

The corner points of the feasible region determined by $x+y \leq 8, 2 x+y \geq 8, x \geq 0, y \geq 0$ are $A(0,8), B(4,0)$ and $C(8,0)$ . If the objective function $Z=a x+$ by has its maximum value on the line segment $A B$ , then the relation between $a$ and $b$ is :

$8 a+4=b$

$a=2 b$

$\mathrm{b}=2 \mathrm{a}$

$8 b+4=a$

✅ Correct Option: 2

Let's calculate the value of $Z$ at each corner point:

At point $A(0,8)$ : $Z_A = a(0) + b(8) = 8b$

At point $B(4,0)$ : $Z_B = a(4) + b(0) = 4a$

At point $C(8,0)$ : $Z_C = a(8) + b(0) = 8a$

For a maximum to occur on segment $AB$ (not just at endpoints), the function must increase from $C$ to $B$ and from $B$ to $A$ .

For $Z$ to increase from $C$ to $B$ : $Z_B > Z_C$ which means $4a > 8a$ , so $a < 0$

For $Z$ to increase from $B$ to $A$ : $Z_A > Z_B$ which means $8b > 4a$ , so $2b > a$

For the maximum to occur exactly on segment $AB$ , the level curves of $Z$ must be parallel to $AB$ .

The slope of $AB$ is:

$\dfrac{0-8}{4-0} = \dfrac{-8}{4} = -2$

The slope of the level curves $Z=ax+by$ is $-\dfrac{a}{b}$

For these to be parallel: $-\dfrac{a}{b} = -2$

Which simplifies to: $a = 2b$

The relation between $a$ and $b$ is $a = 2b$ .

We can verify: If $a = 2b$ , then $Z_A = 8b$ and $Z_B = 4a = 4(2b) = 8b$ , confirming equal values at both points.

2025: 27 May Shift 1

The corner points of the bounded feasible region determined by the system of linear constraints are (15,0), (40,0), (4,18) and (6, 12). If objective function is Z = 30x + 20y, then the sum of the maximum and the minimum values of Z is

2026: 22 May Shift 1

The corner points of the bounded feasible region determined by $x+3y\le60, x+y\ge10, x\le y, x\ge0, y\ge0$ are A(0, 10), B(5, 5), C(15, 15) and D(0, 20). If the objective function $z=ax+by$ has its maximum value on the line segment CD, then the relation between $a$ and $b$ is:

2022: 10 Aug Shift 1

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

2023: 30 May Shift 3

The feasible region for a LPP is shown in the given figure. The maximum value of $z = 2x + 5y$ is :

2025: 29 May Shift 2

The feasible region of a linear programming problem is bounded. The corresponding objective function is Z= 3x-4y. The objective function attains

2026: 29 May Shift 2

The number of feasible solutions of the linear programming problem as

Maximum $z=15x+30y$ subject to the constraints

$3x+y \leq 12$

$x+2y \leq 10$

$x \geq 0, y \geq 0$

Solution

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