NCERT Class 12-Mathematics: Chapter – 12 Linear Programming Part 2 (For CBSE, ICSE, IAS, NET, NRA 2023)

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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.


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
Corner PointValue of Z
Fig. 12.2-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


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
Corner PointValue of Z
Fig. 12.3-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 ₹ 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 ₹ 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.


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 ₹) 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


subject to the constraints:

Fig. 12.4-The Mathematical Formulation of the Problem