LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

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

 

Date : 10/11/2012               Dept. No.                                          Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL the questions                                                                                                    (10 X 2 = 20 Marks)

1. Let f: be defined by.  Find the  range of the function .
2. Find the equation of the line passing through (-3,4) and (1,6).
3. Write the normal equation of y = a+bx.
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 . Find  Cost when output is 4 units.

(i)   Average cost of output of 10 units.

(ii)  Marginal cost when output is 3 units.

13. Fit a straight line to the following data. Estimate the sale for 1977.

Year:	1969	1970	1971	1972	1973	1974	1975	1976
Sales(lakhs)	38	40	65	72	69	60	87	95

14. Solve y_x+2 – 4y_x = 9x².
15. Find the eigen vectors of the matrix .
16. Verify Cayley Hamilton theorem for the matrix A =
17. Expand f(x) = x (-π <x< π) as a Fourier series with period 2 π.
18. Obtain a Fourier series expansion for f(x) = – x  in the range (-π,0)

=      x   in the range [0, π).

PART – C

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

19. (a) Fit a curve of the form y = a + bx +c x² for the following table:

X:	0	1	2	3	4
Y:	1	1.8	1.3	2.5		6.3

 

(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) So lve 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)  Find the Fourier (i) Cosine series (ii) Sine series for the function f(x) =  π-x  in (0, π).                        (10+10)

 

 

22. (a) Determine the Characteristic roots and corresponding vectors for the matrix

.Hence diagonalise A.

 

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