Loyola College B.Sc. Mathematics April 2009 Graphs, Diff. Equ., Matrices & Fourier Series Question Paper PDF Download

      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 yx+2 – 8 yx+1 + yx = 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 an 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:    yn+2 – 4 yn+1 + 4yn = 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)

  1. (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 ebt + 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.
  • Dux + D2ux = cos x
  • (E2 – 5E + 6) yn = 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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