LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOV 2006
MT 1501 – GRAPHS, DIFF. EQU., MATRICES & FOURIER SERIES
Date & Time : 03112006/1.004.00 Dept. No. Max. : 100 Marks
SECTION A
Answer ALL Questions. (10 x 2 = 20)
 A firm producing poultry feeds finds that the total cost C(x) of producing x units is given by C(x) = 20x + 100. Management plans to charge $24 per unit for the feed. How many units must be sold for the firm to break even?
 Find the equation of the line passing through (2, 9) and (2, 9).
 Find the domain and range of the function f(x) = .
 Find the axis and vertex of the parabola y = x^{2} – 2x + 3.
 Reduce y = ax^{n} to the linear law.
 Solve the difference equation y_{x+2} – 8y_{x+1} + 15y_{x} = 0.
 State Cayley Hamilton theorem.
 Find the determinant value of a matrix given its eigen values are 1, 2 and 3.
 Define periodic function. Give an example.
 Show that = 0, when n 0.
SECTION B
Answer ANY FIVE Questions. (5 x 8 = 40)
 The marginal cost for raising a certain type of fruit fly for a laboratory study is $12 per unit of fruit fly, while the cost to produce 100 units is $1500.
(a) Find the cost function C(x), given that it is linear.
(b) Find the average cost per unit to produce 50 units and 500 units.(4 + 4 marks)
 The profit P(x) from the sales of x units of pies is given by P(x) = 120x – x^{2}. How many units of pies should be sold in order to maximize profit? What is the maximum profit? Draw the graph.
 Graph the functions (a) y = x^{2} – 2x – 15 , (b) f(x) = .
(4 + 4 marks)
 Fit a parabola y = a + bx + cx^{2} using method of group averages for the following data.
x 0 2 4 6 8 10
y 1 3 13 31 57 91
 Solve the difference equation y_{k+2} – 5y_{k+1} + 6y_{k} = 6^{k}.
 Find the eigen values and eigen vectors of A = .
 Using Cayley Hamilton theorem, find A^{1} if A = .
 In (), find the fourier series of periodicity 2for f(x) = .
SECTION C
Answer ANY TWO Questions. (2 x 20 = 40)
 (a) Suppose that the price and demand for an item are related by p = 150 – 6x^{2}, where p is the price and x is the number of items demanded. The price and supply are related by p = 10x^{2} + 2x, where x is the supply of the item. Find the equilibrium demand and equilibrium price.
(b) Fit a straight line by the method of least squares for the following data.
x 0 5 10 15 20 25
y 12 15 17 22 24 30 (10 + 10 marks)
 Solve the following difference equations.
(a) y_{n+2} – 3y_{n+1} + 2y_{n} = 0, given y_{1} = 0, y_{2} = 8, y_{3} = 2.
(b) u(x+2) – 4u(x) = 9x^{2}. (8 + 12 marks)
 Expand f(x) = x^{2}, when < x < , in a fourier series of periodicity 2. Hence deduce that
(i) .
(ii) .
(iii) .
 Diagonalize the matrix A = . Hence find A^{4}.