Saturday, August 1, 2020

NUMERICAL ANALYSIS - 2018 | B.SC CS 2ND YEAR | MJPRU | EXAM PAPER | My CS Tutorial

My CS Tutorial August 01, 2020

Numerical analysis | My CS Tutorial

Paper code: 13512
1512
B.Sc. (Computer Science) (Part 2)
Examination, 2018
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 of calculator is allowed.

1. (a) Evaluate :

$\Delta ^{2} \cos 2x$

(b) If :

$u_{0} + u_{8} = 1.9243$

$u_{1} + u_{7} = 1.9590$

$u_{2} + u_{6} = 1.9823$

$u_{3} + u_{5} = 1.9956$

find $u_{4}$ .

2. (a) Given :

$\sum_{1}^{10}u_{x}=500426$ $\sum_{4}^{10}u_{x}=329240$ $\sum_{7}^{10}u_{x}=175212$

and $u_{10}=40365$ . Find $u_{1}$ .

(b) Find y, when x = 8 for :

x	0	5	10	15	20	25
y	7	11	14	18	24	32

3. (a) If $f(x) = \frac{1}{x^{2}}$ , find the divided differences $f(a,b)$ , $f(a,b,c)$ and $f(a,b,c,d)$ .

(b) Given lof 654 = 2.8156, log 658 = 2.8182, log 659 = 2.8189 and log 661 = 2.8202. Find log 656.

4. Solve the following system by iteration method :

$27x + 6y - z = 85$

$6x+15y+2z=72$

$x+y+54z=110$

5. Solve the system :

$5x-2y+z=4$

$7x+y-5z=8$

$3x+7y+4z=10$

by :

(i) Gauss’s elemination method

(ii) Gauss’s Jordan method

6. (a) Evaluate :

$\int_{0}^{6}\frac{1}{1+x^{2}}dx$

by using Simpson’s 3/8 rule.

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

$\int_{1}^{3}U_{x}dx = 2U_{2}+\frac{1}{12}(U_{0}-2U_{2}+U_{4})$

and hence find :

$\int_{-1/2}^{1/2} e^{-\frac{x^{2}}{10}}dx$

7. (a) Evaluate :

$\int_{0.2}^{1.4}(\sin x - \log_{e}x + e^{x})dx$

by Weddle’s rule.

(b) Using Euler’s modification method to compute y for x = 0.05. Given that :

$\frac{dy}{dx}=x+y$

with the initial condition x0 = 0 and y0 = 1.

8. Solve initial value problem :

$\frac{dy}{dx} = 1 + xy^{2}$ , $y(0)=1$

for x = 0.4, 0.5 by using Milne’s method when it is given :

x	0.1	0.2	0.3
y	1.105	1.223	1.355

9. Solve :

${y}'' = (x^{2} + y^{2})(1+y^{2})$

for x = 0.5 and x = 1.0 by using Runge-Kutta method, with x =0, y = 1, y’ = 0.

10. (a) Find $\sqrt{12}$ to five places of decimal by Newton’s-Raphson method.

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

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