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 𝐢 = 2 5 π‘₯ + 9 . Find the average cost of output of 10 units. 6 If the marginal function for output is given by 𝑅 π‘š = (π‘₯+2) 2 + 5 , find the total revenue 3.

4.

5.

6.

7.

8.

9.

function by integration. Find the differential coefficient of π‘₯ 2 βˆ’1 π‘₯ 2 +1 with respect to x. Find the first order partial derivatives of 𝑒 = 3π‘₯ 2 + 2π‘₯𝑦 + 4𝑦 2 . Evaluate ∫ π‘₯𝑒 π‘₯ 𝑑π‘₯ . 𝑐 Prove that ∫ 𝑓(π‘₯)𝑑π‘₯ + π‘Ž 𝑐 𝑏 π‘Ž 𝑏 . If 𝐴 = ( 4 1 2 3 ) , find

A

2 . If 𝐴 = ( 1 2 3 4 5 6 7 8 9 ) , Can you find the inverse of the matrix of A? π‘₯+1 If (π‘₯βˆ’1)(2π‘₯+1) = 𝐴 π‘₯βˆ’1 + 𝐡 2π‘₯+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

ο€½ 2 3

x

 35 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

ο€½

x x x

...

ο‚₯ then prove that

dy x dx

ο€½ 1 ο€­

y

2

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 14.

π‘₯ 3 Integrate (π‘₯ 2 +1) 3 with respect to

x

. 15.

Prove that π‘Ž ∫ 𝑓(π‘₯)𝑑π‘₯ = { 0 π‘Ž 𝑖𝑓 𝑓(π‘₯)𝑖𝑠 𝑒𝑣𝑒𝑛 . 0 𝑖𝑓 𝑓(π‘₯)𝑖𝑠 π‘œπ‘‘π‘‘ π‘₯ 1 = 4 𝑝 1 2 𝑝 2 and π‘₯ 2 = 16 𝑝 2 2 𝑝 1 . 16.

If

A

ο€½    1 0 3 2 1 ο€­ 1 1 ο€­ 1 οƒΆ οƒ· then show that

A

3 ο€­ 3

A

2 1 οƒΈ 9

I

ο€½ 0 .

1 17.

Compute the inverse of the matrix 𝐴 = ( βˆ’2 18.

π‘₯ 2 +1 Resolve into partial fractions: (π‘₯βˆ’3)(π‘₯βˆ’1) 2 . 3 0 2 βˆ’1 βˆ’4 5 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

ο€­

MR

. Verify this for the linear demand law

p

. (b) If the marginal revenue function is 𝑀𝑅 = π‘Žπ‘ (π‘₯βˆ’π‘) 2 βˆ’ 𝑐 , show that 𝑝 = π‘Ž (π‘βˆ’π‘₯) βˆ’ 𝑐 is the demand law.

(10+10)

20.

(a) If 𝑦 = (π‘₯ + √1 + π‘₯ 2 ) π‘š , prove that (1 + π‘₯ 2 )𝑦 2 + π‘₯𝑦 1 = π‘š 2 𝑦 . (b) Evaluate ∫ (3π‘₯+7) 2π‘₯ 2 +3π‘₯βˆ’2 𝑑π‘₯ .

(10+10)

21.

(a) Find the consumer’s surplus and producer’s surplus for the demand curve 𝐷(π‘₯) = 16 βˆ’ π‘₯ 2 and the supply curve 𝑆(π‘₯) = 4 + π‘₯ . (b) Find the maximum and minimum values of the function π‘₯ 4 + 2π‘₯ 3 βˆ’ 3π‘₯ 2 βˆ’ 4π‘₯ + 4 .

(10+10)

22.

(a) Solve the equations 5π‘₯ βˆ’ 6𝑦 + 4𝑧 = 15 ; 7π‘₯ + 4𝑦 βˆ’ 3𝑧 = 19 ; 2π‘₯ + 𝑦 + 6𝑧 = 46 by inverse matrix method. (b) Solve the following LPP by graphical method: Maximize 𝑍 = 2π‘₯ 1 + 5π‘₯ 2 Subject to π‘₯ 1 + π‘₯ 2 ≀ 24 3π‘₯ 1 π‘₯ 1 + π‘₯ 2 + π‘₯ 2 ≀ 21 ≀ 9 π‘₯ 1 , π‘₯ 2 β‰₯ 0 .

(12+8)

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