LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 02

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

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

 

 

 

Date : 12-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A        (10 × 2 = 20)

Answer ALL the questions

 

1. What are linear functions?

 

2. Find the slope of the line x = -2y–7.

 

3. Write the normal equations of y = ax+b.

 

4. Reduce y = ae^bx to normal form.

 

5. Find the particular integral of y_n+2-4y_n+1+3y_n = 2ⁿ

 

6. Solve y_n+2 -5y_n+1 + 6y_n = 0.
7. Find the eigen values of and hence the eigen values of A²
8. Define a quadratic form.

 

9. Write down the Fourier series expansion of an even function f(x) = x in .

 

10. Find the Fourier coefficient a₀ if f(x) = x in the range 0 to π.

 

 

PART-B        (5 × 8 = 40) 

Answer any FIVE questions

 

11. (a) Graph the function f(x) = x²+4x. (4+4)

 

(b) If the cost in rupees to produce x kilograms of a milk product is given by c(x) = 500-3x+2x²,

then find

(i) Total cost for 9 kilograms.

(ii) Marginal cost.

 

12. Find the minimum average cost if the cost function is given by C(x) = 36x-10x²+2x³. Find also

the marginal cost at that point.

 

13. Using the method of least squares fit a straight line to the following data.

x	3	4	4	6	8
y	4	5	6	8	10

 

14. Solve the difference equation, y_n+2 -3y_n+1+2y_n = 5ⁿ+2ⁿ

 

15. (a) Verify Cayley – Hamilton Theorem for .

 

(b) What are the Eigen values of a triangular matrix?                                                         (6+2)

 

16. (a) Find the characteristic vectors of . (6+2)

(b) Write down the matrix form corresponding to the quadratic form x²+2y²+3z²+4xy+8yz+6zx.

 

17. Find the Fourier series expansion of e^x in the range –π to π .
18. Show that in , x(π-x) = .

 

PART-C        (2 × 10 = 20)      

Answer any TWO questions

 

19. (a) Draw the graph and find the equilibrium price (y) and quantity (x) for the demand and supply

curves given below                                                                                             (10+10)

2y = 16-x

y² = 4(x-y).

 

(b) Fit a curve of the form y = a+bx+cx² to the following data

x	0	1	2	3	4
y	1	1.8	1.3	2.5	6.3

 

20. (a) Solve y_n+2+y_n+1+y_n =  n²+n+1.                                                                               (12+8)

 

(b) Solve  y_n+2-2y_n+1+y_n  =  n²-2ⁿ

 

21. (a) Find a Fourier series expansion for the function f(x) = x² in and deduce that .

 

(b) Obtain the half range sine series for the function f(x) = cosx  .            (10+10)

 

22. Diagnolise .

 

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