LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 02

FIRST SEMESTER – April 2009

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

 

 

 

Date & Time: 22/04/2009 / 1:00 – 4:00    Dept. No.                                                       Max. : 100 Marks

 

 

SECTION – A

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

1. Write the range and domain of
2. Check whether each of the following defines a function:

(i)  y = -3x + 8                              (ii) x² + y² = 1

1. State the principle of least square.
2. Reduce into linear form: y = a xⁿ, where a and n are constants.
3. Define  (i) general solution, (ii) particular solution of a difference equation.
4. Solve 16 y_x+2 – 8 y_x+1 + y_x = 0.
5. Find the eigenvalues of A⁵ when.
6. Find the sum and product of eigenvalues of the matrix.
7. State the Dirichlet conditions for Fourier series.
8. Find the Fourier constants a₀ and a_n for the function f(x) = x³ in (-p < 0 < p).

 

SECTION – B

Answer any FIVE questions.                                                                 (5 X 8 = 40)

1. (a)  A company sold 500 tool kits in 2000 and 20000 tool kits in 2005.  Assuming that sales are approximated by a linear function, express the company’s sales S as a function of linear time t.

(b)  A company has fixed cost of Rs. 8250 and a marginal cost of Rs. 450 for each item produced.  Express the cost C as a function of the number x of items produced and evaluate the function at x = 20 and at x = 50.

1. Using the method of least square fit a straight line to the following data.

x:               0               1              2                3               4

y:               1             1.8           3.3            4.5             6.3

 

 

1. The data in the following table will fit a formula of the type y = a + bx + cx².  Find the formula by the method of group averages

x:           87.5         84.0          77.8            63.7          66.7            36.9

y:           292          283            270            235            197            181

1. Solve:    y_n+2 – 4 y_n+1 + 4y_n = 2ⁿ + 3ⁿ + p.
2. Find the inverse of the matrix  using Cayley-Hamilton theorem.
3. Find the eigenvalues and eigenvectors of the matrix .
4. If  expand f(x) as a sine series in the interval (0, p)
5. Express f(x) = | x |, -p < x < p as a Fourier series and hence deduce that

SECTION – C

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

19. (a) Given the following total revenue R(x) = 600x – 5x² and total cost
    C(x) = 100 x + 10,500  (i) express p as a function of x, (ii) determine the maximum level of profit by finding the vertex of p(x) and (iii) find the x intercepts and draw a rough sketch of the graph.

(b)  The temperature q of a vessel of cooling water and the time t in minutes since the beginning of observation are connected by a relation of the form
q = a e^bt + c.  Their tabulated values are given below

t:             0          1         2           3          5             7          10         15        20

q:         52.2     48.8     46.0      43.5      39.7        36.5      33.0     28.7    26.0

Find the best values you can for a, b, c.

1. Solve the following equation.

• Du_x + D²u_x = cos x
• (E² – 5E + 6) y_n = 4ⁿ (n² – n + 5)

1. (a)  Expand f(x) = x (2p  – x) as a Fourier series in (0, 2p)

(b)  Obtain the Fourier series for the function f(x) = x², –p £ x £ p and from it deduce that

1. Page: 2

   Reduce the quadratic form  in to a canonical form by the method of orthogonal reduction.

 

 

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