LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
SUPPLEMENTARY EXAMINATION – JUNE 2009
MT 1501 - GRAPHS, DIFF. EQU., MATRICES & FOURIER SERIES
Date & Time: 25/06/2009 / 10:00 - 1:00 Dept. No.
Max. : 100 Marks
Part A (10 x 2 = 20)
Answer All questions.
1. Find the inverse of the function 1 – 2-x
2. Determine whether the following are functions from A to B where A = { 1,2,3,4,5 }
and B = { a,b,c,d,e}.
(i)
f = { (1,a), (2,b), (3,c), (5,e) }
(ii)
g = {(1,e), (2,c), (3,a), (4,b), (5,d)}
3. Write the normal equations of y = ax + b
4. Define residual.
5. Solve 16ux+2 – 8 ux+1 + ux = 0
6. Find the complementary function of yx+2 – 6yx+1 +8yx = 4x
7. Define the characteristic equation of a matrix.
a h g
8. Find the eigen values of 0 b 0
0 0 c
9. Find the Fourier coefficient a 0 for the function f (x) = ex in (0, 2  )
10. Define even and odd functions and give an example of each.
Part B (5 x 8 = 40)
Answer any Five questions.
11. The cost in dollars to produce x kilograms of chocolate candy is given by
C(x) = 3.5x + 800. Find each of the following:
(i) The fixed cost
(ii) The total cost for 12 kilograms
(iii) The marginal cost per kilogram
(iv) The marginal cost of the 40th kilogram
12. (a) Graph the function f (x) = x2 + 8x + 18 by completing the square.
(b) The total profit y in rupees of a drug company from the manufacture and sale of x
x2
drug bottle is given by y = + 2x – 80
400
(i) How many drug bottles must the company sell to achieve the maximum profit ?
(ii) What is the profit per drug bottle when this maximum is achieved ?
13. Reduce the following into linear law.
(a) y = axn
(b) y = aebx
1
14. Using the method of least squares fit a straight line to the following data.
X
Y
0
7
5
11
10
16
15
20
20
26
15. Solve the difference equation y k+2 – 5 y k+1 +6 y k = 6 k
1
16. Verify Cayley Hamilton theorem for the matrix
5
1
3
2
6
 2 1  3
2 3
3 2
18. Obtain the Fourier expansion for f(x) = x –  in the interval ( -  ,  )
17. Find the eigen vectors of the matrix
Part C (2 x 20 = 40)
Answer any Two questions.
19.(a) The following table gives the levels of prices in certain years.
Year 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985
Price 88
87
81
78
74
79
85
84
90
92
100
level
Fit a curve of the form y = a + bx + cx2
(b) Suppose that the price and demand for an item are related by p = 150 – 6 x2, the
demand function where p is the price and x is the number of items demanded (in
hundreds).The price and supply are related by p = 10 x2 + 2, the supply function where
x is the supply of the item ( in hundreds). Draw the graph and find the equilibrium
demand and equilibrium price.
(12 + 8 )
20. Solve the difference equations:
n
(a) y n+2 –16yn = cos
2
(b) y n+2 + y n+1 – 56 y n = 2 n ( n 2 – 3)
(12 +8)
21. (a) Obtain the half range cosine series for f(x) = x in (0,  ) and deduce that the sum
1
1
1
2
of the series 2 + 2 + 2 + … =
1
3
5
8
(b) Find a Fourier series expansion for the function f (x) = -  in -  < x < 0
= x in 0 < x < 
(10 + 10)
2 2 3
22. Diagonalize the matrix 1 1
1
1
3
1
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