Numerical Analysis 2015 – BSc Computer Science Part 2 (MJPRU)

Paper code: 13512
1512
B.Sc. (Computer Science) (Part 2)
Examination, 2015
Paper No. 1.3
NUMERICAL ANALYSIS

Time: Three Hours] [Maximum Marks: 50

 

Note: Attempt five questions. All questions carry equal marks. Symbols are as usual use or calculator is allowed.

1. (a) Find:

 \Delta ^{2} \begin{pmatrix}\frac{5x+12}{x^{2}+5x+6}\end{pmatrix}

(b) Find the relation between may be expressible in one terms in the factorial notation.

2. (a) Show that:

 \Delta x^{m}-\frac{1}{2}\Delta ^{2}x^{m}+\frac{1.3}{2.4}\Delta ^{3}x^{m}-\frac{1.3.5}{2.4.6}\Delta^{4}x^{m}+.......... to m-terms = \begin{pmatrix}x+\frac{1}{2}\end{pmatrix}^{m}-\begin{pmatrix}x-\frac{1}{2}\end{pmatrix}^{m}

(b) Given:

 \sin 45^{\circ}=0.7071, \sin 50^{\circ}=0.7660,
 \sin 55^{\circ}=0.8192, \sin 60^{\circ}=0.8660,

Find \sin 55^{\circ} by using any method of interpolation.

3. (a) Show that the n^{th} divided difference can be expressed as the quotient of two determinants each of order n+1.

(b) By means of Lagrange’s formula, prove that:

 y_{1} = y_{3}-0.3 \begin{pmatrix}y_{5}-y_{-3}\end{pmatrix} +0.2\begin{pmatrix}y_{-3}-y_{-5}\end{pmatrix}

Approximately.

4. (a) Solve the following system of equations by Jacobi’s method:

10x+2y+z=9

2x+20y-2z=-44

-2x+3y+10z=22

(b) Solve the system:

 x+\frac{1}{2}y+\frac{1}{3}z=1

 \frac{1}{2}x+\frac{1}{3}y+\frac{1}{4}z=0

 \frac{1}{3}x+\frac{1}{4}y+\frac{1}{5}z=0

By Gauss’ elimination method.

5. (a) Solve the following system by Gauss-Seidel iteration method:

 27x+6y-z=85

 6x+15y+2z=72

 x+y+54z=101

(b) What do you know about Cote’s formula for interpolation?

6. (a) Evaluate the value of the integral:

By using

  1. Simpson’s 3/8 rule
  2. Weddle’s rule

(b) If U_{x}=a+bx+cx^{2}, prove that:

 \intop\nolimits_{1}^{3}U_{x}dx=2U_{2}+\frac{1}{12} \begin{pmatrix}U_{0}-2U_{2}+U_{4}\end{pmatrix}

And hence find an approximate value for:

 \intop\nolimits_{-\frac{1}{2}}^{\frac{1}{2}}exp\begin{pmatrix}-\frac{x^{2}}{10}\end{pmatrix}dx

7. (a) Obtain the Newton-Rapson extended formula:

x_{1}=x{0}-\frac{f\left ( x_{0} \right )}{f\left (x_{0} \right )}-\frac{1}{2}\frac{\left \{ f\left ( x_{0} \right ) \right \}^{2}f''\left ( x_{0} \right )}{\left \{ f\left ( x_{0} \right ) \right \}^{3}}

For the root of the equation.

(b) Compute the real root of x{\log_{10}}x-1.2=0 correct to five decimal places by Regula-Falsi method.

8. (a) Show that the rate of convergence of Newton’s method is quadratic.

(b) Solve the equation \log_{e}x=\cos x to five decimal by nay suitable method.

9. Solve y^{n}=\left(x^{2}+y^{2} \right )\left(1+y^{2} \right ) for using Runge-Kutta method. If x=0, y=1, y'=0.

10. State and prove Newton’s Interpolation formula for unequal arguments.

……..End……..

Lokesh Kumar

Being EASTER SCIENCE's founder, Lokesh Kumar wants to share his knowledge and ideas. His motive is "We assist you to choose the best", He believes in different thinking.

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