NCERT Class 12-Mathematics: Chapter –12 Linear Programming Part 2

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Question 2:

Determine the minimum value of (if any), if the feasible region for an LPP is shown in Fig.12.2.

Answer:

The feasible region (R) is unbounded. Therefore minimum of Z may or may not exist. If it exists, it will be at the corner point (Fig.12.2).

The Feasible Region (R) is Unbounded
The feasible region (R) is unbounded

Corner Point

Value of Z

Fig. 12.2-The feasible region (R) is unbounded

The Feasible Region (R) is Unbounded

Let us graph . We see that the open half plane determined by and R do not have a common point. So, the smallest value is the minimum value of Z.

Question 3:

Solve the following LPP graphically:

Maximise , subject to

Answer:

The shaded region in the Fig. 12.3 is the feasible region determined by the system of constraints and .

The feasible region OAB is bounded, so, maximum value will occur at a corner point of the feasible region.

Corner Points are and .

Evaluate Z at each of these corner point.

Evaluate Z at Each of These Corner Point
Evaluate Z at each of these corner point

Corner Point

Value of Z

Fig. 12.3-The shaded region (OAB) in the

The Shaded Region (OAB) in The

Hence, the maximum value of Z is 12 at the point

Question 4:

A manufacturing company makes two types of television sets; one is black and white and the other is colour. The company has resources to make at most 300 sets a week. It takes Rs 1800 to make a black and white set and to make a coloured set. The company can spend not more than a week to make television sets. If it makes a profit of Rs 510 per black and white set and per coloured set, how many sets of each type should be produced so that the company has maximum profit? Formulate this problem as a LPP given that the objective is to maximise the profit.

Answer:

Let x and y denote, respectively, the number of black and white sets and coloured sets made each week. Thus

Since the company can make at most sets a week, therefore,

Weekly cost (in Rs) of manufacturing the set is

and the company can spend upto . Therefore,

The total profit on x black and white sets and y colour sets is . This is the objective function.

Thus, the mathematical formulation of the problem is

Maximise

subject to the constraints:

Fig.12.4-The mathematical formulation of the problem

The Mathematical Formulation of the Problem

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