LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

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

FOURTH SEMESTER – APRIL 2012

Date : 19-04-2012

Time : 1:00 - 4:00

MT 4205 / 4202 - BUSINESS MATHEMATICS

Dept. No. Max. : 100 Marks

Part A (Answer ALL questions)

1.

2.

(2 x 10 = 20)

The total cost C for output x is given by 𝐶 =

2

5 𝑥 + 9 . Find the average cost of output of 10 units.

6

If the marginal function for output is given by 𝑅 𝑚

=

(𝑥+2) 2

+ 5 , find the total revenue

3.

4.

5.

6.

7.

8.

9.

function by integration.

Find the differential coefficient of 𝑥 𝑥

2 −1

2 +1

with respect to x.

Find the first order partial derivatives of 𝑢 = 3𝑥 2 + 2𝑥𝑦 + 4𝑦 2

.

Evaluate ∫ 𝑥𝑒 𝑥 𝑑𝑥 . 𝑐

Prove that ∫ 𝑓(𝑥)𝑑𝑥 + 𝑎

If 𝐴 = (

4 1

2 3

) , find A

2

.

1 2 3 𝑐 𝑏 𝑎 𝑏

∫ 𝑓(𝑥)𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 .

If 𝐴 = ( 4 5 6 ) , Can you find the inverse of the matrix of A?

7 8 9 𝑥+1

If

(𝑥−1)(2𝑥+1)

=

𝐴 𝑥−1

+

𝐵

2𝑥+1

then find A and B.

10.

Define Feasible solution.

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

11.

The total cost C for output x is given by C

2 35 x

3

2

. Find (i) Cost when output is 4 units (ii)

Average cost when output is 10 units (iii) Marginal cost when output is 3 units.

12.

If y

 x x x ...

then prove that x dy dx y

2

1

 y log x

.

13.

For the following pair of demand functions for two commodities X

1

and X

2

, determine the four partial marginal demands, the nature of relationship (Complementary, Competitive or neither) between X

1

and X

2

and the four partial elasticities of demand 𝑥

14.

Integrate

(𝑥 𝑥

3

2 +1) 3

with respect to x .

1

= 𝑝

1

2

4 𝑝

2

and 𝑥

2

=

16 𝑝

2

2 𝑝

1

.

15.

Prove that 𝑎

∫ 𝑓(𝑥)𝑑𝑥 = {

𝑖𝑓 𝑓(𝑥)𝑖𝑠 𝑒𝑣𝑒𝑛

.

0 𝑖𝑓 𝑓(𝑥)𝑖𝑠 𝑜𝑑𝑑

16.

If A

1 2 1

0 1

1

3

1 1

then show that A 3 

3 A 2  

9 I

0 .

17.

Compute the inverse of the matrix 𝐴 = (

1 0 −4

−2 2 5 ) .

18.

𝑥 2 +1

Resolve into partial fractions:

(𝑥−3)(𝑥−1) 2

.

3 −1 2

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

19.

(a) If AR and MR denote the average and marginal revenue at any output, show that elasticity of demand is equal to

AR

AR

MR

. Verify this for the linear demand law p a bx .

(b) If the marginal revenue function is 𝑀𝑅 = 𝑎𝑏

(𝑥−𝑏) 2

− 𝑐 , show that 𝑝 = 𝑎

(𝑏−𝑥)

− 𝑐 is the demand law.

(10+10)

20.

(a) If 𝑦 = (𝑥 + √1 + 𝑥 2 ) 𝑚

, prove that (1 + 𝑥 2 )𝑦

2

+ 𝑥𝑦

1

= 𝑚 2 𝑦 .

(b) Evaluate ∫

(3𝑥+7)

2𝑥 2 +3𝑥−2 𝑑𝑥 . (10+10)

21.

(a) Find the consumer’s surplus and producer’s surplus for the demand curve 𝐷(𝑥) =

16 − 𝑥 2

and the supply curve 𝑆(𝑥) = 4 + 𝑥 .

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

4

+ 2𝑥

3

− 3𝑥

2

− 4𝑥 + 4 .

(10+10)

22.

(a) Solve the equations 5𝑥 − 6𝑦 + 4𝑧 = 15 ; 7𝑥 + 4𝑦 − 3𝑧 = 19 ; 2𝑥 + 𝑦 + 6𝑧 = 46 by inverse matrix method.

(b) Solve the following LPP by graphical method:

Maximize 𝑍 = 2𝑥

1

+ 5𝑥

2

Subject to 𝑥

1

+ 𝑥

2

≤ 24

3𝑥

1

𝑥

1

+ 𝑥

2

+ 𝑥

2

≤ 21

≤ 9

𝑥

1

, 𝑥

2

≥ 0 . (12+8)

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