LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2012

MT 1501 – GRAPHS, DIFF. EQU., MATRICES & FOURIER SERIES

 

 

 

Date : 02-05-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL the questions:                                                                                  (10 x 2 = 20 Marks)

 

1. Find the range of the following functions

(a) Let defined by f(x)  = x². (b) Let be any constant function.

2. Find the equation of the line passing through (-3,4) and (1,6).
3. Write the normal equation of y = ax+b.
4. Reduce y = ae^bx to normal form.
5. Define Difference equation with an example.
6. Solve y_x+2-4y_x=0.
7. State Cayley Hamilton theorem.
8. Find the eigen value of the matrix .
9. Find the Fourier coefficient a₀ for the function f(x) = e^x in (0,2π).
10. Define odd and even function.

PART – B

Answer any FIVE questions:                                                                                     (5 X 8 = 40 Marks) 

 

11. A company has a total cost function represented by the equation y = 2x³-3x²+12x, where y represents cost and x represent quantity.

(i) what is the equation for the Marginal cost function?

(ii)What is the equation for average cost function? What point average cost is at its minimum?

12. The total cost in Rs.of output x is given by C = Find

• Cost when output is 4 units.
• Average cost of output of 10 units.
• Marginal cost when output is 3 units.

 

13. Fit a straight line to the following data

X:	0	5	10	15	20	25
Y:	12	15	17	22	24	30

Estimate the value of  Y corresponding to X =6.

 

14. Solve y_x+2 – 5y_x+1+6y_x = x²+x+1.
15. Find the eigen vectors of the matrix A = .
16. Verify Cayley Hamilton theorem for the matrix A =
17. Expand f(x) = x (0 <x<2 π) as a Fourier series with period 2 π.

 

18. If f(x) = x  in the range (0,π)

=   0 in the range (π, 2π).  Find Fourier series of f(x) of periodicity 2 π.

 

PART – C

Answer any TWO questions:                                                                                 (2 X 20=40 Marks)

 

19. (a) Fit a second degree parabola by taking x_i as the independent variable.

X:	0	1	2	3	4
Y:	1	5	10	22	38

 

(b)  The total profit y in rupees of a drug company from the manufacture and sale of x drug bottles is

given by  . (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?         (10 +10)

20. (a) Solve y_x+2 – 7y_x+1 – 8y_x = x(x-1) 2^x.

 

(b) Solve u(x+1) – au(x) = cosnx.                                                                                       (10+10)

21. (a) Find the Fourier series of period 2 π for f(x) = x² in (o,2 π) . Deduce

(b)  Expand   in (0,2 π) as Fourier series of period 2 π.                                                 (10+10)

22. Determine the Characteristic roots and corresponding vectors for the matrix

.Hence diagonalise A.

 

 

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