LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FOURTH SEMESTER – NOVEMBER 2012
MT 4205 - BUSINESS MATHEMATICS
Date : 05/11/2012
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
PART A
Answer ALL the questions:
( 10 X 2 = 20)
1. Find the equilibrium price and quantity for the functions 𝑄𝑑 = 2 − 0.02𝑃 and 𝑄𝑠 = 0.2 + 0.07𝑃
2. If the demand law is 𝑝 =
10
(𝑥+1)2
find the elasticity of demand in terms of x.
𝑑𝑦
3. Find 𝑑𝑥 if 𝑦 = 𝑥 2 + 𝑦 2 − 2𝑥.
4. Find the first order partial derivatives of 𝑢 = 3𝑥 2 + 2𝑥𝑦 + 4𝑦 2 .
5. Evaluate 
 3x
c
6. Prove that
7. If 𝐴 = (1
3
8. If 𝐴 = (4
2

a
1
 4 x 2  3 x  8  dx
b
b
c
a
f ( x)dx   f ( x)dx   f ( x)dx
2
1
) and 𝐵 = (
4
2
1
2
), find 𝐴 .
3
.
0
), find 𝐴 + 𝐵.
−3
x 1
A
B


 x  1 2 x  1 x  1 2 x  1
9. If
then find A and B
10. Define Linear Programming Problem.
PART B
Answer any FIVE of the following:
(5x 8=40)
2
35
x  . Find (i) Cost when output is 4 units
3
2
Average cost of output of 10 units (iii) Marginal cost when output is 3 units.
dy
log x

12. If x y  e x  y then prove that
.
dx 1  log x 2
11. The total cost C for output x is given by C 
13. Find the first and second order partial derivatives of log  x 2  y 2  .
14. Integrate x 2 e x with respect to x.
a
15. Prove that (i)

a
a
f ( x) dx  2  f ( x) dx , if f(x) is an even function.
0
a
(ii)

f ( x) dx  0 , if f(x) is an odd function.
a
1 2 1 


16. If A  0 1 1 then show that A3  3 A2  A  9 I  0 .
 3 1 1 


(ii)
 1 0 4 
17. Compute the inverse of the matrix A   2 2 5  .
 3 1 2 


x5
18. Integrate
with respect to x.
2
 x  1 x  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
AR
demand is equal to
. Verify this for the linear demand law p  a  bx .
AR  MR
6
 5 , find the total
(b) If the marginal revenue function for output x is given by Rm 
2
 x  2
revenue by integration. Also deduce the demand function.
1
20. (a) Let the cost function of a firm be given by the following equation: C  300 x  10 x 2  x 3
3
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 .
 3x  7  dx .
(b) Evaluate  2
2 x  3x  2
21. (a) Find the maximum and minimum values of the function x 4  2 x3  3x 2  4 x  4 .
(b) Solve the equations x  2 y  3z  4, 2 x  y  3z  5,  x  y  2 z  3 by Crammer’s rule.
22. (a) The demand and supply function under perfect competition are y  16  x 2 and y  2 x 2  4
respectively. Find the market price, consumer’s surplus and producer’s surplus.
(b) Food X contains 6 units of vitamin A per gram and 7 units of vitamin B per gram and costs 12
paise per gram. Food Y contains 8 units of vitamin A per gram and 12 units of vitamin B per
gram and costs 20 paise per gram. The daily minimum requirements of vitamin A and vitamin B
are 100 units and 120 units respectively. Find the minimum cost of the product mix using
graphical method.
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