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

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) 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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