# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

## LOYOLA COLLEGE (AUTONOMOUS), CHENNAI β 600 034

### Dept. No. Max. : 100 Marks

1.

2.

(2 x 10 = 20)

The total cost C for output x is given by πΆ = 2 5 π₯ + 9 . Find the average cost of output of 10 units. 6 If the marginal function for output is given by π π = (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 β1 π₯ 2 +1 with respect to x. Find the first order partial derivatives of π’ = 3π₯ 2 + 2π₯π¦ + 4π¦ 2 . Evaluate β« π₯π π₯ ππ₯ . π Prove that β« π(π₯)ππ₯ + π π π π π . If π΄ = ( 4 1 2 3 ) , find

A

2 . If π΄ = ( 1 2 3 4 5 6 7 8 9 ) , Can you find the inverse of the matrix of A? π₯+1 If (π₯β1)(2π₯+1) = π΄ π₯β1 + π΅ 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

ο« 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 β« π(π₯)ππ₯ = { 0 π ππ π(π₯)ππ  ππ£ππ . 0 ππ π(π₯)ππ  πππ π₯ 1 = 4 π 1 2 π 2 and π₯ 2 = 16 π 2 2 π 1 . 16.

If

A

ο½ ο¦ ο§ ο§ 1 0 3 2 1 ο­ 1 1 ο­ 1 οΆ ο· then show that

A

3 ο­ 3

A

2 1 οΈ 9

I

ο½ 0 .

1 17.

Compute the inverse of the matrix π΄ = ( β2 18.

π₯ 2 +1 Resolve into partial fractions: (π₯β3)(π₯β1) 2 . 3 0 2 β1 β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 ππ = ππ (π₯βπ) 2 β π , show that π = π (πβπ₯) β π is the demand law.

(10+10)

20.

(a) If π¦ = (π₯ + β1 + π₯ 2 ) π , prove that (1 + π₯ 2 )π¦ 2 + π₯π¦ 1 = π 2 π¦ . (b) Evaluate β« (3π₯+7) 2π₯ 2 +3π₯β2 ππ₯ .

(10+10)

21.

(a) Find the consumerβs surplus and producerβs surplus for the demand curve π·(π₯) = 16 β π₯ 2 and the supply curve π(π₯) = 4 + π₯ . (b) Find the maximum and minimum values of the function π₯ 4 + 2π₯ 3 β 3π₯ 2 β 4π₯ + 4 .

(10+10)

22.

(a) Solve the equations 5π₯ β 6π¦ + 4π§ = 15 ; 7π₯ + 4π¦ β 3π§ = 19 ; 2π₯ + π¦ + 6π§ = 46 by inverse matrix method. (b) Solve the following LPP by graphical method: Maximize π = 2π₯ 1 + 5π₯ 2 Subject to π₯ 1 + π₯ 2 β€ 24 3π₯ 1 π₯ 1 + π₯ 2 + π₯ 2 β€ 21 β€ 9 π₯ 1 , π₯ 2 β₯ 0 .

(12+8)