LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – NOVEMBER 2012

# MT 4205 – BUSINESS MATHEMATICS

Date : 05/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

# Answer ALL the questions:                                                                                                   ( 10 X 2 = 20)

1. Find the equilibrium price and quantity for the functions  and
2. If the demand law is find the elasticity of demand in terms of x.
3. Find if .
4. Find the first order partial derivatives of .
5. Evaluate
6. Prove that .
7. If and , find .
8. If , find .
9. If  then find A and B
10. Define Linear Programming Problem.

PART B

Answer any FIVE of the following:                                                                               (5x 8=40)

1. The total cost C for output x is given by . Find (i) Cost when output is 4 units (ii) Average cost of output of 10 units (iii) Marginal cost when output is 3 units.
2. If then prove that .
3. Find the first and second order partial derivatives of .
4. Integrate with respect to x.
5. Prove that (i) , if f(x) is an even function.

(ii) , if f(x) is an odd function.

1. If  then show that .
2. Compute the inverse of the matrix .
3. Integrate with respect to x.

PART C

Answer any TWO questions:                                                              ( 2 X 20 = 40)

1. (a) If AR and MR denote the average and marginal revenue at any output, show that elasticity of demand is equal to . Verify this for the linear demand law .

(b) If the marginal revenue function for output x is given by , find the total revenue by integration. Also deduce the demand function.

1. (a) Let the cost function of a firm be given by the following equation: where C stands for cost and x for output. Calculate (i) output, at which marginal cost is minimum (ii) output, at which average cost is minimum (iii) output, at which average cost is equal to marginal cost .

(b) Evaluate .

1. (a) Find the maximum and minimum values of the function .

(b) Solve the equations  by Crammer’s rule.

1. (a) The demand and supply function under perfect competition are and Find the market price, consumer’s surplus and producer’s surplus.

(b) Food X contains 6 units of vitamin A per gram and 7 units of vitamin B per gram and costs 12  paise  per gram. Food Y contains 8 units of vitamin A per gram and 12 units of vitamin B per gram and costs 20 paise per gram. The daily minimum requirements of vitamin A and vitamin B are 100 units and 120 units respectively. Find the minimum cost of the product mix using graphical method.

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

# AB 05

B.Sc. DEGREE EXAMINATION – COMMERCE

THIRD SEMESTER – November 2008

# MT 3203 / 3200 – BUSINESS MATHEMATICS

(SHIFT – I)

Date : 13-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A

Answer ALL questions:                                                                             (10 x 2 = 20)

1. Define Profit function.
2. The total cost C for output x is given by. Find the average cost of output of 10 units.
3. Find the differential coefficient of with respect to x.
4. If find.
5. Evaluate  .
6. Integrate.
7. Find the matrix B if and.
8. If  and  then find C where 2C = A+B.
9. Resolve into partial fractions:.
10. Define a feasible solution of the linear programming problem.

SECTION B

Answer any FIVE questions:                                                                    (5 x 8 = 40)

1. If the demand law is, find the elasticity of the demand in terms of x.
2. (a) Let the production function of a firm be given by. Find the marginal productivity of labour and marginal productivity of capital. Show that .

(b) For the following pair of demand functions for two commodities X1 and X2, determine the four partial marginal demands, the nature of relationship (Complementary, Competitive or neither) between x1 and x2:     and .

1. (a) Find the maximum and minimum values of the function.

(b) Find  if.

1. If, prove that.
2. Find consumer’s surplus and producer’s surplus for the demand curve and the supply curve.
3. Evaluate.
4. Find the inverse of the matrix .
5. The manager of an oil refinery wants to decide on the optimal mix of two possible blending Processes 1 and 2 of which the inputs and outputs per production run as follows:

Input (Units)                                                     Output (Units)

Process     Crude A     Crude B                             Gasonline X        Gasonline Y

1                5                  3                                            5                           8

2                4                  5                                            4                           4

The maximum amounts available of Crudes A and B are 200 units and 150 units respectively.  Atleast 100 units of gasoline X and 80 units of gasoline Y are required. The profit per production run from processes 1 and 2 are Rs.300 and Rs.400 respectively.  Formulate the above as Linear programming problem and solve it by graphical method.

SECTION C

Answer any TWO questions:                                                                   (2 x 20 = 40)

1. (a) Let the cost function of a firm is given by the following equation:

, where C stands for cost and x for output.

Calculate    (i) Output, at which marginal cost is minimum.

(ii) Output, at which average cost is minimum.

(iii) Output, at which average cost is equal to marginal cost.

(b) If AR and MR denote the average and marginal revenue at any output, show that elasticity of demand is equal to   . Verify this for the linear demand law p = a + bx.

(10 +10)

1. (a) Find the second order partial derivative of .

(b) If y =  , show that  .

(c) If , prove that .

(6+10+4)

1. (a) Integrate .

(b) The marginal cost function of manufacturing x shoes is. The total cost of producing a pair of shoes is Rs. 12. Find the total and average cost function.

(12 + 8)

1. (a) Solve the system of the following equations using matrix method.

x +y + z = 7; x + 2y + 3z =16; x +3y +4z = 22.

(b) Prove that =.

(c) Resolve into partial fractions.

(10+6+4)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.COM. B.B.A DEGREE EXAMINATION – COMM. BUS.ADMN. & CORP.SEC.

# AB 06

THIRD SEMESTER – November 2008

# MT 3204/MT 4202/MT 3202 – BUSINESS MATHEMATICS

(SHIFT – II)

Date : 13-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART-A

• Evaluate 3A + 4B + 5I when A = A = and B =
• Define Inverse of a matrix.
• Find the determinant value of A, if A =
• Differentiate Sin (log x) w.r.t x.
• If profit function is given as , find the price at which profit is maximum.
• Evaluate .
• When total cost is given by TC = x2 + 78x +2500, find the average cost and marginal cost.
• Write any two properties of definite integrals.
• Find .
• Define objective function.

PART – B

Answer any FIVE of the following:                                                                                                           (5 X 8 = 40)

• Given the following transaction matrix, Find the gross output to meet the final

demand of 200 units of Agriculture and 800 units of Industry.

 Producing sector Agriculture Industry Final demand Agriculture 300 600 100 Industry 400 1200 400

12) If  K =   V =  and W =

Matrix K shows the stock of four types of record players R1, R2, R3 and R4 ( in columns)in three shops S1, S2 and S3 (in rows). Matrix V shows the value of the record players in hundred rupees. Matrix W gives the week’s sales. Find (a) the stock at the end of the week. (b) the order matrix to bring the stock of each of the cheaper pair of record players to 8 and the dearer pair to 5. (c) the value of the sales, (d) the value of the order.

13) Differentiate the following function (a) by substitution method

(b) by log differentiation

14) If 2x = y1/4+ y -1/4, then prove that (x2-1) .

15) Evaluate.

16) Find the consumer surplus and producer surplus under pure competition for demand

function p =  and supply function  p = , where p is price and x is

quantity.

• Resolve in to Partial fraction: .
• Solve graphically: Maximize Z = 500x + 150 y

subject to the constraints : x + y £ 60

2500x + 500y £ 50,000

x ,y ³ 0 .

PART – C

Answer any TWO of the following:                                                                                           (2 X 20 = 40)

• (a) To control a certain crop disease it is necessary to use 8 units of chemical A, 14

units of chemical B and 13 units of chemical C. One barrel of spray P contains

one unit of A, 2units of B and 3 units of C. One barrel of spray Q contains

2 units of A, 3units of B and 2 units of C. One barrel of spray R contains

one unit of A, 2units of B and 2 units of C. How many barrels of each type of

spray should be used to control the disease?                                                                                  (12)

(b) Find the rank of the matrix .                                                                                   (8)

20)  (a) The total cost function of a firm is given by C = 0.04x3 – 0.9x2 + 10x +10

Find (i) Average cost (ii) Marginal cost (iii) Slope of AC (iv) Slope of MC

(v) Value of x at which average variable cost is minimum.                                                           (10)

(b) The production function of a commodity is given by Q = 40x + 3x2 – ,

where Q is the total output and F is the unit of input.

• Find the number of units of input required to give maximum output.
• Find the maximum value of marginal product.
• Verify that when the average product is maximum, it is equal to marginal product. (10 )

21)  (a)  Evaluate                                                                                                                        (10)

• The marginal cost of production of a firm is given as MC = 5 + 0.13x and the

marginal revenue is MR = 18. Also given that C(0) = Rs.120. Compute the total profit.                                                                                                                                                                                    (10)

22)  A firm makes two types of furniture namely chairs and tables. The contribution for

each product as calculated by the accounting department is Rs. 20 per chair and Rs.

30 per table. Both products are processed on three machines M, N and O. The time

required in hours by each product and total time available in hours per week on each

machine are as follows:

 Machine Chair Table Available time M 3 3 36 N 5 2 50 O 2 6 60

How should be the manufacturer schedule his production in order to maximize

contribution?

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., B.Com., DEGREE EXAMINATION – ECO., & COMM.,

FOURTH SEMESTER – APRIL 2012

# MT 4205 / 4202 – BUSINESS MATHEMATICS

Date : 19-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Part A (Answer ALL questions)                                                                              (2 x 10 = 20)

1. The total cost C for output x is given by . Find the average cost of output of 10 units.
2. If the marginal function for output is given by, find the total revenue function by integration.
3. Find the differential coefficient of  with respect to x.
4. Find the first order partial derivatives of .
5. Evaluate.
6. Prove that.
7. If, find A2.
8. If , Can you find the inverse of the matrix of A?
9. If  then find A and
10. Define Feasible solution.

Part B (Answer any FIVE of the following)                                                           (5 x 8 = 40)

1. The total cost C for output x is given by . Find (i) Cost when output is 4 units (ii) Average cost when output is 10 units (iii) Marginal cost when output is 3 units.
2. If then prove that .
3. For the following pair of demand functions for two commodities X1 and X2, determine the four partial marginal demands, the nature of relationship (Complementary, Competitive or neither) between X1 and X2 and the four partial elasticities of demand  and .
4. Integrate with respect to x.
5. Prove that .
6. If  then show that .
7. Compute the inverse of the matrix .
8. Resolve into partial fractions: .

Part C (Answer any TWO questions)                                                                     (2 x 20 = 40)

1. (a) If AR and MR denote the average and marginal revenue at any output, show that elasticity of demand is equal to. Verify this for the linear demand law.

(b) If the marginal revenue function is , show that  is the demand law.                                                                                                          (10+10)

1. (a) If , prove that .

(b) Evaluate.                                                                              (10+10)

1. (a) Find the consumer’s surplus and producer’s surplus for the demand curve  and the supply curve .

(b) Find the maximum and minimum values of the function.

(10+10)

1. (a) Solve the equations ; ; by inverse matrix method.

(b) Solve the following LPP by graphical method:

Maximize

Subject to

.                                                                  (12+8)

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