# Numerical Analysis 2018 – BSc Computer Science Part 2 (MJPRU)

#### Paper code: 135121512B.Sc. (Computer Science) (Part 2)Examination, 2018Paper 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) 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 y 0 5 10 15 20 25 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 y 0.1 0.2 0.3 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) Copute the real root of $x \log_{10} x -1.2 = 0$ correct to five decimal places by Regula-Falsi method.

……..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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