ALCCS

 

 

Code: CS41                                                                      Subject: NUMERICAL COMPUTING

Flowchart: Alternate Process: AUGUST 2009Time: 3 Hours                                                                                                     Max. Marks: 100

 

NOTE:

·      Question 1 is compulsory and carries 28 marks. Answer any FOUR questions from the rest.  Marks are indicated against each question.

·      Parts of a question should be answered at the same place.

·      All calculations should be up to three places of decimals.

 

 

  Q.1     a.  Find the number of terms of the exponential series such that their sum gives the value of ex correct to six decimal places at x = 1.

 

             b.  Find a real root of the equation , by the method of False Position using the two iterations.

 

             c.  Factorize the matrix  using LU decomposition.

 

             d.  Evaluate .

                 

             e.  Using Givens Method, reduce the following matrix to the tri-diagonal form: .

 

             f.   Determine f(x) as a polynomial in x for the  following data:

                 

x

- 4

   - 1

   0

  2

  5

f(x)

1245

    33

   5

   9

1335

 

             g.  Estimate approximately the distance covered in 20 minutes using Simpson’s  rule. The velocity v (km/min.) of a moped which starts from rest, is given at fixed intervals of time t(min) as follows:        (7  4)

 

                 

t

2

4

6

8

10

12

14

16

18

20

v

10

18

25

29

32

20

11

5

2

0

 

  Q.2     a.  Find a real root of   using Iteration Method.                                              

 

             b.  If a, b, c, d are the arguments of , show that .          (9+9)            

 

  Q.3     a.  Solve by Gauss-Seidel method, the following system of equations:

                                                                                                                         

 

             b.  Solve the following system of equations by Crout’s method:

                                                                                                    (9+9)

 

  Q.4     a.  Determine the largest eigenvalue and its corresponding eigenvector of the matrix

                                                                                                                           

 

             b.  Using inverse interpolation, find the real root of the equation  which is close to 1.2.     (9+9)

 

  Q.5     a.  The following are data from the steam table:

                 

            tempC0(t)

140

150

160

170

180

Pressure kgf/cm2 (P)

3.685

4.854

6.302

8.076

10.225

                  Using Newton’s divided difference interpolation formula, find the pressure of steam for temperature 1420 and 1750.                                                                                                                                

 

             b.  Assuming that the following values of (x, y) and y(x) a polynomial of degree four given, compute the two missing values.                                                                                                          (9+9)

x

2

4

6

8

10

12

14

y

2

3

5

8

9

----

---

                 

 

 

Q.6      a.         From the given data, find the maximum value of y:                                                                                 

x

-1

1

2

3

y

-21

15

12

3

 

 

       

             b.  A curve is drawn to pass through the following points:

                 

x

1

  1.5

2

2.5

3

3.5

4

y

2

2.4

2.7

2.8

3

2.6

2.1

                  Estimate the area  bounded by the curve, x-axis and lines x = 1, x = 4. Also find the volume of solid generated by revolving this area using Simpson’s 3/8 rule.                                          (9+9)

 

  Q.7     a.  Using Runge-Kutta method  of fourth order, solve for y(0.2), y(0.4) given that  .    

 

             b.  Using Taylor’s series method, find the values of y(0.1) and y(0.2) where .     (12+6)