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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 02

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – November 2008

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

 

 

 

Date : 12-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART – A        (10 × 2 = 20)

Answer ALL the questions

 

1. What are linear functions?

 

2. Find the slope of the line x = -2y–7.

 

3. Write the normal equations of y = ax+b.

 

4. Reduce y = ae^bx to normal form.

 

5. Find the particular integral of y_n+2-4y_n+1+3y_n = 2ⁿ

 

6. Solve y_n+2 -5y_n+1 + 6y_n = 0.
7. Find the eigen values of and hence the eigen values of A²
8. Define a quadratic form.

 

9. Write down the Fourier series expansion of an even function f(x) = x in .

 

10. Find the Fourier coefficient a₀ if f(x) = x in the range 0 to π.

 

 

PART-B        (5 × 8 = 40) 

Answer any FIVE questions

 

11. (a) Graph the function f(x) = x²+4x. (4+4)

 

(b) If the cost in rupees to produce x kilograms of a milk product is given by c(x) = 500-3x+2x²,

then find

(i) Total cost for 9 kilograms.

(ii) Marginal cost.

 

12. Find the minimum average cost if the cost function is given by C(x) = 36x-10x²+2x³. Find also

the marginal cost at that point.

 

13. Using the method of least squares fit a straight line to the following data.

x	3	4	4	6	8
y	4	5	6	8	10

 

14. Solve the difference equation, y_n+2 -3y_n+1+2y_n = 5ⁿ+2ⁿ

 

15. (a) Verify Cayley – Hamilton Theorem for .

 

(b) What are the Eigen values of a triangular matrix?                                                         (6+2)

 

16. (a) Find the characteristic vectors of . (6+2)

(b) Write down the matrix form corresponding to the quadratic form x²+2y²+3z²+4xy+8yz+6zx.

 

17. Find the Fourier series expansion of e^x in the range –π to π .
18. Show that in , x(π-x) = .

 

PART-C        (2 × 10 = 20)      

Answer any TWO questions

 

19. (a) Draw the graph and find the equilibrium price (y) and quantity (x) for the demand and supply

curves given below                                                                                             (10+10)

2y = 16-x

y² = 4(x-y).

 

(b) Fit a curve of the form y = a+bx+cx² to the following data

x	0	1	2	3	4
y	1	1.8	1.3	2.5	6.3

 

20. (a) Solve y_n+2+y_n+1+y_n =  n²+n+1.                                                                               (12+8)

 

(b) Solve  y_n+2-2y_n+1+y_n  =  n²-2ⁿ

 

21. (a) Find a Fourier series expansion for the function f(x) = x² in and deduce that .

 

(b) Obtain the half range sine series for the function f(x) = cosx  .            (10+10)

 

22. Diagnolise .

 

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      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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – APRIL 2012

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

 

 

 

Date : 02-05-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL the questions:                                                                                  (10 x 2 = 20 Marks)

 

1. Find the range of the following functions

(a) Let defined by f(x)  = x². (b) Let be any constant function.

2. Find the equation of the line passing through (-3,4) and (1,6).
3. Write the normal equation of y = ax+b.
4. Reduce y = ae^bx to normal form.
5. Define Difference equation with an example.
6. Solve y_x+2-4y_x=0.
7. State Cayley Hamilton theorem.
8. Find the eigen value of the matrix .
9. Find the Fourier coefficient a₀ for the function f(x) = e^x in (0,2π).
10. Define odd and even function.

PART – B

Answer any FIVE questions:                                                                                     (5 X 8 = 40 Marks) 

 

11. A company has a total cost function represented by the equation y = 2x³-3x²+12x, where y represents cost and x represent quantity.

(i) what is the equation for the Marginal cost function?

(ii)What is the equation for average cost function? What point average cost is at its minimum?

12. The total cost in Rs.of output x is given by C = Find

• Cost when output is 4 units.
• Average cost of output of 10 units.
• Marginal cost when output is 3 units.

 

13. Fit a straight line to the following data

X:	0	5	10	15	20	25
Y:	12	15	17	22	24	30

Estimate the value of  Y corresponding to X =6.

 

14. Solve y_x+2 – 5y_x+1+6y_x = x²+x+1.
15. Find the eigen vectors of the matrix A = .
16. Verify Cayley Hamilton theorem for the matrix A =
17. Expand f(x) = x (0 <x<2 π) as a Fourier series with period 2 π.

 

18. If f(x) = x  in the range (0,π)

=   0 in the range (π, 2π).  Find Fourier series of f(x) of periodicity 2 π.

 

PART – C

Answer any TWO questions:                                                                                 (2 X 20=40 Marks)

 

19. (a) Fit a second degree parabola by taking x_i as the independent variable.

X:	0	1	2	3	4
Y:	1	5	10	22	38

 

(b)  The total profit y in rupees of a drug company from the manufacture and sale of x drug bottles is

given by  . (i) How many drug bottles must the company sell to achieve the

maximum profit? (ii) What is the profit per drug bottle when this maximum is achieved?         (10 +10)

20. (a) Solve y_x+2 – 7y_x+1 – 8y_x = x(x-1) 2^x.

 

(b) Solve u(x+1) – au(x) = cosnx.                                                                                       (10+10)

21. (a) Find the Fourier series of period 2 π for f(x) = x² in (o,2 π) . Deduce

(b)  Expand   in (0,2 π) as Fourier series of period 2 π.                                                 (10+10)

22. Determine the Characteristic roots and corresponding vectors for the matrix

.Hence diagonalise A.

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

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

 

Date : 10/11/2012               Dept. No.                                          Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

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

1. Let f: be defined by.  Find the  range of the function .
2. Find the equation of the line passing through (-3,4) and (1,6).
3. Write the normal equation of y = a+bx.
4. Reduce y = ae^bx to normal form .
5. Define Difference equation with an example.
6. Solve y_x+2 – 4y_x=0.
7. State Cayley Hamilton theorem.
8. Find the eigen value of the matrix .
9. Find the Fourier coefficient a₀ for the function f(x) = e^x in (0,2π).
10. Define odd and even function.

PART – B

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

11. A company has a total cost function represented by the equation y = 2x³-3x²+12x, where y

represents cost and x represent quantity.  (i) what is the equation for the Marginal cost function?

(ii) What is the equation for average cost function? What point average cost is at its minimum?

12. The total cost in Rs.of output x is given by . Find  Cost when output is 4 units.

(i)   Average cost of output of 10 units.

(ii)  Marginal cost when output is 3 units.

13. Fit a straight line to the following data. Estimate the sale for 1977.

Year:	1969	1970	1971	1972	1973	1974	1975	1976
Sales(lakhs)	38	40	65	72	69	60	87	95

14. Solve y_x+2 – 4y_x = 9x².
15. Find the eigen vectors of the matrix .
16. Verify Cayley Hamilton theorem for the matrix A =
17. Expand f(x) = x (-π <x< π) as a Fourier series with period 2 π.
18. Obtain a Fourier series expansion for f(x) = – x  in the range (-π,0)

=      x   in the range [0, π).

PART – C

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

19. (a) Fit a curve of the form y = a + bx +c x² for the following table:

X:	0	1	2	3	4
Y:	1	1.8	1.3	2.5		6.3

 

(b)  The total profit y in rupees of a drug company from the manufacture and sale of x drug

bottles is given by  .

(i) How many drug bottles must the company sell to achieve the maximum  profit?

(ii) What is the profit per drug bottle when this maximum is achieved ?                                                     (10+10)

20. (a) Solve y_x+2 – 7y_x+1 – 8y_x = x(x-1) 2^x.

(b) So lve u(x+1) – au(x) = cosnx.                                                                                                                                 (10+10)

21. (a) Find the Fourier series of period 2 π for  f(x) = x² in (o,2 π) . Deduce

(b)  Find the Fourier (i) Cosine series (ii) Sine series for the function f(x) =  π-x  in (0, π).                        (10+10)

 

 

22. (a) Determine the Characteristic roots and corresponding vectors for the matrix

.Hence diagonalise A.

 

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