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

AA 02

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

 

 

SECTION A

Answer ALL Questions.                                                                            (10 x 2 = 20)

1. 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?
2. Find the equation of the line passing through (2, 9) and (2, -9).
3. Find the domain and range of the function f(x) = .
4. Find the axis and vertex of the parabola y = x² – 2x + 3.
5. Reduce y = axⁿ to the linear law.
6. Solve the difference equation y_x+2 – 8y_x+1 + 15y_x = 0.
7. State Cayley Hamilton theorem.
8. Find the determinant value of a matrix given its eigen values are 1, 2 and 3.
9. Define periodic function. Give an example.
10. Show that = 0, when n 0.

SECTION B

Answer ANY FIVE Questions.                                                         (5 x 8 = 40)

11. 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)

 

12. 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.
13. Graph the functions (a) y = x² – 2x – 15 , (b) f(x) = .

(4 + 4 marks)

14. 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

15. Solve the difference equation y_k+2 – 5y_k+1 + 6y_k = 6^k.
16. Find the eigen values and eigen vectors of A = .
17. Using Cayley Hamilton theorem, find A^-1 if A = .
18. In (-), find the fourier series of periodicity 2for f(x) = .

SECTION C

Answer ANY TWO Questions.          (2 x 20 = 40)

19. (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)

20. Solve the following difference equations.

(a) y_n+2 – 3y_n+1 + 2y_n = 0, given y₁ = 0, y₂ = 8, y₃ = -2.

(b) u(x+2) – 4u(x) = 9x².                                                               (8 + 12 marks)

21. Expand f(x) = x², when -< x < , in a fourier series of periodicity 2. Hence deduce that

(i) .

(ii) .

(iii) .

22. Diagonalize the matrix A = . Hence find A⁴.

 

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