Loyola College Numerical Methods Question Papers Download

Loyola College B.Sc. Mathematics April 2009 Numerical Methods Question Paper PDF Download

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        LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 38

SIXTH SEMESTER – April 2009

MT 6605 – NUMERICAL METHODS

 

 

 

Date & Time: 23/04/2009 / 9:00 – 12:00 Dept. No.                                                     Max. : 100 Marks

 

 

 

PART – A (10 ´ 2 = 20)

Answer ALL questions.

 

  1. What do you mean by partial pivoting?
  2. State Cramer’s rule.
  3. What is the order of convergence in Newton-Raphson method?
  4. Explain bisection method.
  5. What is meant by interpolation?
  6. Write the error polynomial in the Newton’s forward interpolation formula?
  7. Write the Stirling’s central difference interpolation formula.
  8. Write any two advantages of central difference interpolation formula.
  9. What is the order of error in the Trapezoidal rule?
  10. Write the formula for third order Range-Kutta method.

 

PART – B (5 ´ 8 = 40)

Answer any FIVE questions.

  1. Solve the following system of equations by Gauss elimination method

28x + 4y – z = 32,       x + 3y + 10z = 24    and     2x + 17y +4z = 35.

  1. Solve for a positive root of x3 – 4x + 1 = 0 by Regula Falsi method.
  2. Write a C program to for Lagrange’s interpolation formula.
  3. Obtain Newton’s forward interpolation formula for equal intervals.
  4. Find the first two derivatives of at x = 50 and x = 56 given the table below:

x                        :              50            51              52          53           54           55             56

:           3.6840     3.7084      3.7325    3.7563   3.7798     3.8030      3.8259

 

  1. Use Laplace-Everett’s formula to obtain f(1.15) given that f(1) = 1, f(1.1) =1.049,  f(1.2) = 1.096, f(1.3) = 1.14.
  2. Evaluate by        (i) Trapezoidal rule      (ii) Simpson’s 1/3  rule   and

(iii) Simpson’s 3/8 rule.

  1. Solve in the range 0 £ x £2 using (i) Euler’s method  (ii) improved Euler’s method

 

 

 

 

 

PART – C (2 ´ 20 = 40)

Answer any TWO questions.

  1. (a) Solve by Gauss-Seidel Method, the following system of equations.

                                    28x + 4y – z = 32,      

                                    x + 3y + 10z = 24,     

                        and      2x + 17y + 4z = 35

(b) Find the real positive root of 3x – cos x – 1 = 0 by Newton-Raphson method correct to 6 decimal places.

  1. (a) From the following table find f(x) and hence f(6) using Newton’s divided difference formula.

x          :           1             2            7             8

                        f(x)       :           1             5            5             4

  • The following table gives the value of density of saturated water for various temperatures of saturated stream.

Tempo C ( = T)            :           100      150      200      250      300

Density hg/m3 (= d)     :           958      917      865      799      712

      Find by interpolation, the densities when the temperatures are 130oC and 275oC respectively.

 

  1. (a) Using Gauss’s forward interpolation formula, find the value of log 337.5 from the following table.

x                 :              310           320            330            340          350          360

       yx = log x        :           2.4914       2.5052       2.5185      2.5315      2.5441   2.5563

  • Using Bessel’s formula, find the derivative of f(x) at x = 3.5 from the following table.

x          :           3.47     3.48     3.49     3.50     3.51     3.52     3.53

f(x)       :           0.193   0.195   0.198   0.201   0.203   0.206   0.208

  1. (a) Using Range-Kutta method of fourth order, solve for y(0.1) and y(0.2) given that                 y¢ = xy + y2, y(0) =1.

(b)  Develop a C program to implement Simpson’s 3/8 rule.

 

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Loyola College B.Sc. Mathematics April 2011 Numerical Methods Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – APRIL 2011

MT 6605 – NUMERICAL METHODS

 

 

 

Date : 09-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer ALL questions.                                                                                                 (10 ´ 2 = 20)

  1. What is the condition of convergence for solving a system of linear equations by iteration procedure?
  2. What do you mean by partial pivoting?
  3. Explain the method of successive approximation.
  4. What is the order of convergence in regula falsi method?
  5. Write a short note on interpolation.
  6. Write the Gregory-Newton backward interpolation formula.
  7. State the relation between Bessel’s and Laplace-Everett’s formulae.
  8. Write Bessel’s central difference interpolation formula.
  9. What is the order of error in Simpson’s 1/3
  10. Using Euler’s method, Solve y¢ = x + y, given y(0) = 1 for x = 0.2

 

PART – B

Answer any FIVE questions.                                                                                      (5 ´ 8 = 40)

  1. Using Gauss elimination method, solve the system

                  10x + y + z = 12,        2x + 10y + z = 13,      2x + 2y + 10z = 14

  1. Find an approximate root of x log10 x – 1.2 = 0 by regula falsi method.
  2. Find a real root of the equation cos x = 3x – 1 correct to 3 decimal places.
  3. Find a polynomial which takes the following values and hence compute yx at x = 2, 12

x:         1          3          5          7          9          11

yx:        3          14        19        21        23        28

  1. Obtain Newton’s divided difference formula for unequal intervals.
  2. The population of a certain town (as obtained from census data) is shown in the following table. Find the rate of growth of the population in the year 1981.

Year:                           1951                1961                1971                1981                1991

Population:                 19.96               36.65               58.81               77.21               94.61

(in thousands)

  1. Evaluate using (i) Simpson’s 1/3 rule and (ii) Simpson’s 3/8
  2. Using Modified Euler method, find y(0.1), y(0.2) given

PART – C

Answer any TWO questions.                                                                                   (2 ´ 20 = 40)

  1. (a) Solve by Gauss-Seidel method, the following system of equations

            10x – 5y – 2z = 3,                   4x – 10y + 3z = –3,                 x + 6y + 10z = –3

(b)  Find the positive root of f(x) = 2x2 – 3x – 6 = 0 by Newton-Raphson method correct to 3 decimal places.                                                                                                                                  ( 12 + 8)

  1. (a) Using Lagrange’s formula of interpolation find y(9.5) given

x:         7          8          9          10

y:         3          1          1          9

(b)  The population of as town is a follows

Year     x:                                 1941       1951       1961          1971       1981     1991

Population in lakhs y:               20           24           29              36           46         51

(10 + 10)

  1. The following table gives the values of the probability integral for certain values of x.  Find the values of this integral when x = 0.5437 using (i) Stirling’s formula (ii) Bessel’s formula and (iii) Laplace-Everett’s formula.

x:                           0.51                 0.52                 0.53                0.54                  0.55

y = f(x):           0.5292437       0.5378987       0.5464641       0.5549392       0.5633233

x:                           0.56                 0.57

y = f(x):           0.5716157       0.5798158

  1. (a) Develop a C-program to implement Trapezoidal rule.

(b)  Using Runge-Kutta method of fourth order, solve given y(0) = 1 at
x = 0.2, 0.4                                                                                                               (8 + 12)

 

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Loyola College B.Sc. Mathematics April 2012 Numerical Methods Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – MATHEMATICSSIXTH SEMESTER – APRIL 2012MT 6605 – NUMERICAL METHODS
Date : 20-04-2012 Dept. No.         Max. : 100 Marks                 Time : 1:00 – 4:00
PART – A Answer ALL questions: (10 X 2 = 20 marks)
1. Explain Cramer’s rule.2. Distinguish between Gauss Elimination and Gauss Seidel methods.3. State the condition for convergence in Newton Raphson method.4. What do you mean by transcendental equation?5. Write Newton forward interpolation formula.6. Write any two properties of divided differences.7. Write Stirling’s formula using central difference notation.8. Write the derivatives using Newton’s forward difference formula.9. Define numerical integration.10. Write Simpson’s 1/3rd and 3/8th rule.
PART – B
Answer any FIVE questions: (5 X 8 = 40 marks)
11. Solve by Gauss elimination method: 12. Find the real root of  correct to three significant figures using Regula falsi method.13. Write a C program to  interpolate using Newton’s forward interpolation formula.14. Use Lagrange’s formula to find the value of y at x = 6 from the following data: x = 3, 7, 9, 10 and the corresponding value of y = 168, 120, 72, 63.15. Using the following data, find f’(5) by Newton’s divided difference formula:             : 0 2 3 4 7 9                   : 4 26 58 112 466 92216. Derive Laplace Everett’s formula.17. Apply Simpson’s rule to evaluate  to two decimal places by dividing the range into 4 equal parts.18. Solve  with the initial condition x = 0, y = 0 using Euler’s modified formula.

PART – C
Answer any TWO questions:                (2 X 20 = 40 marks)
19. (a)  Solve by Gauss Seidel method:
(b)  Find by Newton’s method the root of the equation  , which is approximately 2, correct        to three places of decimals.
20. (a) Given   : 0 1 2 5             : 2 3 12 14    find the cubic function of x using Newton’s          divided difference formula.
(b) Using Newton’s formula find the value of f(1.5) from the following data: :    0    1    2    3   4        : 858.3 869.6 880.9 892.3 903.6
21. (a) Use Stirling’s formula to find f(1.63) given  : 1.50    1.60      1.70        1.80       1.90       : 17.609   20.412   23.045   25.527    27.875            (b)Given  X: 0 4 8 12Y: 143 158 177 199 calculate  y5 by Bessel’s formula.
22. (a) Apply the fourth order Runge–Kutta method, to find an approximate value of   when   = 0.2      given that            (b) Write a C program to find the value of  using Simson’s 1/3 rule.

 

 

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