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

#### Paper code: 135121512B.Sc. (Computer Science) (Part 2)Examination, 2015Paper No. 1.3NUMERICAL ANALYSIS

##### Time: Three Hours] [Maximum Marks: 50

Note: Attempt five questions. All questions carry equal marks. Symbols are as usual use of 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

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