# Numerical Analysis 2016 – BSc Computer Science Part 2 (MJPRU)

#### Paper code: 13512 1512 B.Sc. (Computer Science) (Part 2) Examination, 2016 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) Show that:

$\sum_{k=0}^{n-1} \Delta^{2}f_{k} = \Delta f_{n}-\Delta f_{0}$

(b) Estimate the missing term in the following table:

 X F(x) 0 1 1 3 2 9 3 ? 4 81

Explain why value differs from $3^{3}$ or 27?

2. (a) Given:

$u_{0} + u_{8} = 1.9243, u_{8} +u_{7}=1.9590$
$u_{2} + u_{6} = 1.9823, u_{3} +u_{5}=1.9956$

Find $u_{n}$.

(b) Use the method of separation of symbols to prove that:

$u_{1}x + u_{2}x^{2}+u_{3}x^{3}+........=\frac{x}{1-x}u_{1}+\frac{x^{2}}{\begin{pmatrix}1-x\end{pmatrix}^{2}}\Delta u_{1}+\frac{x^{3}}{\begin{pmatrix}1-x\end{pmatrix}^{3}}\Delta^{2} u_{1}+.....$

3. (a) State and prove Newton-Gregory formula for forward interpolation.

(b)$f\begin{pmatrix}x\end{pmatrix} = \frac{1}{x^{2}}$ , find the divided differences:

f(a,b), f(a,b,c) And f(a,b,c,d) ?

1. (a) Find $\intop\nolimits_{0}^{1} \frac{dx}{1+x^{2}}$ by using Simpson’s 1/3 and 3/8 Hence obtain the approximate value of  in each case.

(b) Find first and second derivatives of the function given below at the point x=1.2:

 x y 1 0 2 1 3 5 4 6 5 8

5. (a) Show that the expirations given below are approximations to the third derivative of $y_{x}$.

1. $\Delta^{3}y_{0}+ \begin{pmatrix}x-\frac{3}{2}\end{pmatrix}\Delta ^{4} y_{0}$
2. $\Delta ^{3}y_{-1}+ \begin{pmatrix}x-\frac{1}{2}\end{pmatrix}.\frac{1}{2} \begin{pmatrix}\Delta ^{4}y_{-2}+\Delta ^{4}y_{-1}\end{pmatrix}$

(b) Define the following:

1. Inherent errors
2. Round-off errors
3. Truncation errors

6. (a) Solve the following system of equations by Gaussian elimination method:

$5x-2y+z=4$
$7x+y-5z=8$
$3x+7y+4z=10$

(b) Find the solutions of the system:

$83x+11y-4z=95$
$7x+52y+13z=104$
$3x+8y+29z=71$

7. Tabulate by Milne’s method the numerical solution of $\frac{dy}{dx}=x+y$ with $x_{0}=0, y_{0}=1$, from x=0.20 to x=0.30.

8. (a) Find the real root of the equation $x^{2}+4\sin x=0$ correct to four places of decimals by Newton-Rap son method.

(b) Show that the square root of N=AB is given by $\sqrt{N} = \frac{S}{4}+\frac{N}{5}$ where $S=A+B$.

9. (a) Determine the real root of $\tan x = x$ by iteration method.

(b) Use Runge-Kutta Method to approximate y, when x=0.1 and x=0.2 and , given that x=0, y=1 and $\frac{dy}{dx}=x+y$.

……..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.

This site uses Akismet to reduce spam. Learn how your comment data is processed.