# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 ## LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

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

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 2 −1 𝑥 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 𝑐 𝑏 𝑎 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? 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 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

x x x

...

 then prove that

dy x dx

 1 

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.

𝑥 3 Integrate (𝑥 2 +1) 3 with respect to

x

. 15.

Prove that 𝑎 ∫ 𝑓(𝑥)𝑑𝑥 = { 0 𝑎 𝑖𝑓 𝑓(𝑥)𝑖𝑠 𝑒𝑣𝑒𝑛 . 0 𝑖𝑓 𝑓(𝑥)𝑖𝑠 𝑜𝑑𝑑 𝑥 1 = 4 𝑝 1 2 𝑝 2 and 𝑥 2 = 16 𝑝 2 2 𝑝 1 . 16.

If

A

    1 0 3 2 1  1 1  1   then show that

A

3  3

A

2 1  9

I

 0 .

1 17.

Compute the inverse of the matrix 𝐴 = ( −2 18.

𝑥 2 +1 Resolve into partial fractions: (𝑥−3)(𝑥−1) 2 . 3 0 2 −1 −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

MR

. Verify this for the linear demand law

p

. (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)