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 : 03-11-2006/1.00-4.00 Dept. No. Max. : 100 Marks
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 = x2 – 2x + 3.
- Reduce y = axn to the linear law.
- Solve the difference equation yx+2 – 8yx+1 + 15yx = 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.
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 – x2. 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 = x2 – 2x – 15 , (b) f(x) = .
(4 + 4 marks)
- Fit a parabola y = a + bx + cx2 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 yk+2 – 5yk+1 + 6yk = 6k.
- 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) = .
Answer ANY TWO Questions. (2 x 20 = 40)
- (a) Suppose that the price and demand for an item are related by p = 150 – 6x2, where p is the price and x is the number of items demanded. The price and supply are related by p = 10x2 + 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) yn+2 – 3yn+1 + 2yn = 0, given y1 = 0, y2 = 8, y3 = -2.
(b) u(x+2) – 4u(x) = 9x2. (8 + 12 marks)
- Expand f(x) = x2, when -< x < , in a fourier series of periodicity 2. Hence deduce that
- Diagonalize the matrix A = . Hence find A4.