HSC Commerce (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

State whether the following statement is True or False:

Of all the points of feasible region, the optimal value is obtained at the boundary of the feasible region

State whether the following statement is True or False:

The point (6, 4) does not belong to the feasible region bounded by 8x + 5y ≤ 60, 4x + 5y ≤ 40, 0 ≤ x, y

State whether the following statement is True or False:

The graphical solution set of the inequations 0 ≤ y, x ≥ 0 lies in second quadrant

A set of values of variables satisfying all the constraints of LPP is known as ______

The feasible region represented by the inequations x ≥ 0, y ≤ 0 lies in ______ quadrant.

If the feasible region is bounded by the inequations 2x + 3y ≤ 12, 2x + y ≤ 8, 0 ≤ x, 0 ≤ y, then point (5, 4) is a ______ of the feasible region

A company manufactures 2 types of goods P and Q that requires copper and brass. Each unit of type P requires 2 grams of brass and 1 gram of copper while one unit of type Q requires 1 gram of brass and 2 grams of copper. The company has only 90 grams of brass and 80 grams of copper. Each unit of types P and Q brings profit of ₹ 400 and ₹ 500 respectively. Find the number of units of each type the company should produce to maximize its profit

A dealer deals in two products X and Y. He has ₹ 1,00,000/- to invest and space to store 80 pieces. Product X costs ₹ 2500/- and product Y costs ₹ 1000/- per unit. He can sell the items X and Y at respective profits of ₹ 300 and ₹ 90. Construct the LPP and find the number of units of each product to be purchased to maximize its profit

A company manufactures two types of ladies dresses C and D. The raw material and labour available per day is given in the table.

P is the profit, if P = 50x + 100y, solve this LPP to find x and y to get the maximum profit

Smita is a diet conscious house wife, wishes to ensure certain minimum intake of vitamins A, B and C for the family. The minimum daily needs of vitamins A, B, and C for the family are 30, 20, and 16 units respectively. For the supply of the minimum vitamin requirements Smita relies on 2 types of foods F₁ and F₂. F₁ provides 7, 5 and 2 units of A, B, C vitamins per 10 grams and F₂ provides 2, 4 and 8 units of A, B and C vitamins per 10 grams. F₁ costs ₹ 3 and F₂ costs ₹ 2 per 10 grams. How many grams of each F₁ and F₂ should buy every day to keep her food bill minimum

A chemist has a compound to be made using 3 basic elements X, Y, Z so that it has at least 10 litres of X, 12 litres of Y and 20 litres of Z. He makes this compound by mixing two compounds (I) and (II). Each unit compound (I) had 4 litres of X, 3 litres of Y. Each unit compound (II) had 1 litre of X, 2 litres of Y and 4 litres of Z. The unit costs of compounds (I) and (II) are ₹ 400 and ₹ 600 respectively. Find the number of units of each compound to be produced so as to minimize the cost

A wholesale dealer deals in two kinds of mixtures A and B of nuts. Each kg of mixture A contains 60 grams of almonds, 30 grams of cashew and 30 grams of hazel nuts. Each kg of mixture B contains 30 grams of almonds, 60 grams of cashew and 180 grams of hazel nuts. A dealer is contemplating to use mixtures A and B to make a bag which will contain at least 240 grams of almonds, 300 grams of cashew and 540 grams of hazel nuts. Mixture A costs ₹ 8 and B costs ₹ 12 per kg. How many kgs of each mixture should he use to minimize the cost of the kgs

Maximize Z = 2x + 3y subject to constraints

x + 4y ≤ 8, 3x + 2y ≤ 14, x ≥ 0, y ≥ 0.

Maximize Z = 5x + 10y subject to constraints

x + 2y ≤ 10, 3x + y ≤ 12, x ≥ 0, y ≥ 0

Maximize Z = 400x + 500y subject to constraints

x + 2y ≤ 80, 2x + y ≤ 90, x ≥ 0, y ≥ 0

Minimize Z = 24x + 40y subject to constraints

6x + 8y ≥ 96, 7x + 12y ≥ 168, x ≥ 0, y ≥ 0

Minimize Z = x + 4y subject to constraints

x + 3y ≥ 3, 2x + y ≥ 2, x ≥ 0, y ≥ 0

Minimize Z = 2x + 3y subject to constraints

x + y ≥ 6, 2x + y ≥ 7, x + 4y ≥ 8, x ≥ 0, y ≥ 0

Amartya wants to invest ₹ 45,000 in Indira Vikas Patra (IVP) and in Public Provident fund (PPF). He wants to invest at least ₹ 10,000 in PPF and at least ₹ 5000 in IVP. If the rate of interest on PPF is 8% per annum and that on IVP is 7% per annum. Formulate the above problem as LPP to determine maximum yearly income.

Solution: Let x be the amount (in ₹) invested in IVP and y be the amount (in ₹) invested in PPF.

x ≥ 0, y ≥ 0

As per the given condition, x + y ______ 45000

He wants to invest at least ₹ 10,000 in PPF.

∴ y ______ 10000

Amartya wants to invest at least ₹ 5000 in IVP.

∴ x ______ 5000

Total interest (Z) = ______

The formulated LPP is

Maximize Z = ______ subject to 

______

Solve the following LPP graphically:

Maximize Z = 9x + 13y subject to constraints

2x + 3y ≤ 18, 2x + y ≤ 10, x ≥ 0, y ≥ 0

Solution: Convert the constraints into equations and find the intercept made by each one of it.

The feasible region is OAPC, where O(0, 0), A(0, 6),

P( ___, ___ ), C(5, 0)

The optimal solution is in the following table:

∴ Z is maximum at __( ___, ___ ) with the value ___.