# Document 15510660 ```LOYOLA COLLEGE (AUTONOMOUS), CHENNAI  600 034.
M.A. DEGREE EXAMINATION  ECONOMICS
SECOND SEMESTER  APRIL 2003
MT 2900 / M 875  MATHEMATICAL METHODS  II
28.04.2003
1.00  4.00
Max : 100 Marks
I. a) Evaluate
b) Evaluate
 xe d x.
 x e dx
x2
2
(3)
x
(OR)
Marginal cost y  as a function of units produced x is given by y   1.064  0.005 x Find
the total and average cost functions if fixed cost is 16.3.
(7)
C) i) The quantity demanded and the corresponding price, under pure competition are
 x2 
determined by the demand and supply functions y  36  x 2 and y  6   
 4 
respectively. Determine the corresponding consumers’ surplus and producers’
surplus.
(10)
ii) If investment flow is given by I t   15t 4 and the initial capital stock at t = 0 is 30,
find the function representing capital, K
(5)
(OR)
iii) Find the profitmaximizing output and the total profit at that point if the marginal
revenue and marginal cost functions are given by
MR  25  5 x  2 x 2
(10)
MC  15  2 x  x 2
xd x
iv) Evaluate  2
(5)
x 1
1
dy y
(3)

dx x
dy
b) Solve 1  x 2
 xy0
dx
(OR)
2
Solve x  x  y 2 d x  y e y  2 x y d y  0
(7)
c) i) The change in price, y, with change in quantity demanded, x, of a particular
d y  2 x y  24 x 
commodity is given by
. Find the relationship between price

dx
x 2  16
and quantity demanded if the price is 7.5 when the quantity demanded is 4.
(10)
ii) Discuss Domar Macro Model.
(5)
(OR)
iii) Obtain the general solution of the difference equation y x  2  6 y x 1  8 y x  9 and a
particular solution if y 0  10, y1  25
(10)
1
iv) Solve the equation 2 y x 1  4 y x  3 and find the particular solution if y 0 
(5)
2
II. a) Solve


 2 3
 4 6 0
 B  
 find A B.
III. a) If A  
 4 0
 5 2 1
(3)
10 3 

b) Find the characteristic roots of the matrix A  
 3 2
(OR)
Solve the equation 3 y x 1  3 y x  7 and find the particular solution if y 0  3 .
(7)
 0  2  3


c) i) By Gaussian Elimination method, find the inverse of the matrix  1
3
3 
  1  2  2


(10)
 1 2


 4 6  1 3
 1 1
 and B  
ii) If A  
Prove that A B  B A .
(5)
1 6
 0  1 2 1


 2 3


(OR)
iii)
Solve
3x  x 2  x3  2
x1  2 x 2  x 3  9
(10)
4 x1  3x 2  2 x3  1
0
 3

iv) If A  
  4  1
0
 
 3 5  7  C    1


B  
  Prove that ABC   C B A
6
0 1 8 
 3 
 x1 
 
 
 B X 
IV. a) If B    2  and X   x 2  find
.
X
 1 
x 
 
 3
 x1 
 3 1  2
 


  X AX 
 2 AX
b) If X =  x 2  and A =  1 0 3  then prove that
X
x 
 2 3 2 
 3


(OR)
b1  b2 
 x1 
 2 b1
 


z 
  X BZ 
 BZ
If X   x 2  B   b2  2b3
3 b2  Z   1  Prove that

X
z
 2
x 
 b b
4 b32 
 3
 1 3
c) Determine the maxima and minima of f x, y, z   e  x  y 2  z 2  2 y  x z
(OR)
Determine the maximum of the function
f x, y, z    x 2  2 y 2  z 2  x y  z if x  y  z  35
2
+ + + + +
(5)
(3)
(7)
(15)
(15)
```