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)

 

11. If the demand law is, find the elasticity of the demand in terms of x.
12. (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 .

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

(b) Find  if.

14. If, prove that.
15. Find consumer’s surplus and producer’s surplus for the demand curve and the supply curve.
16. Evaluate.
17. Find the inverse of the matrix .
18. 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)

 

19. (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)

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

(b) If y =  , show that  .

(c) If , prove that .

(6+10+4)

21. (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)

 

22. (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

  Answer ALL the following:                                                                                                           (10X2=20)

                                                                                                                       

• 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.

                                                        Purchasing sector

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 = y^1/4+ 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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