Sunday, July 5, 2020

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

My CS Tutorial July 05, 2020 EXAM PAPER

NUMERICAL ANALYSIS B.SC CS 2ND YEAR MJPRU EXAM PAPER

Numerical analysis | My CS Tutorial

Paper code: 13512
1512
B.Sc. (Computer Science) (Part 2)
Examination, 2019
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) By means of Lagrange’s formula, prove that:

$y_{0} = \frac{1}{2}\left ( y_{1} - y_{-1} \right ) - \frac{1}{3} \left [ \frac{1}{2} \left ( y_{3} - y_{1} \right ) - \frac{1}{2} \left ( y_{-1} - y_{-3} \right ) \right ]$

(b) Use Newton’s divided difference formula to find f(6) if f(3) = 24, f(5) = 120, f(8) = 504, f(9) = 720 and f(12) = 1716.

2. (a) Solve for y by Euler’s Method, up to second approximation, the differential equation

$\frac{dy}{dx} = 2 - \left ( \frac{y}{x} \right )$ , where y = 2 when x =1 (take h = 0.5).

(b) Find y(0.2) by Runge-Kutta method, given that:

$\frac{dy}{dx} = 3x + \frac{1}{2}y$ , y (0) = 1, taking h = 0.1

3. (a) Solve by Newton-Raphson’s method, to find real root of cos x = x²in three significant figures.

(b) Find real cube root of 18 by Regula-Falsi method.

4. (a) Evaluate

$\int_{0}^{6} \frac{dx}{1+x^{2}}$ by Weddle’s rule.

(b) Compute

$\int_{0.2}^{1.4} \left ( \sin x -\log_{e} x + e^{x} \right )dx$ by Simpson’s 3/8 rule.

5. (a)Solve the following system of equations by Jacobi iteration method.

$3x + 4y + 15z = 54.8$

$x + 12y + 3z = 39.66$

$10x + y - 2z = 7.74$

(b) using Gauss-Seidel iteration method, solve the system of equations :

$2x + y + z = 4$

$x + 2y + z = 4$

$x + y + 2z = 4$

6. (a) Find y₂₅, using Newton-Gregory’s formula. Given that:

y₂₀= 24, y₂₄= 32, y₂₈= 35, y₃₂= 40

(b) Given the following data, find f(x) as a polynomial in powers of (x-5).

x:	0	2	3	4	7	9
f(x):	4	26	58	112	466	922

7. (a) Use Runge-Kutta method to find y when x = 1.2 in steps of 0.1 given that

$\frac{dy}{dx} = x^{2} + y^{2}$ and y(1) =1.5

(b) Suppose 1.414 is used as an approximation to

$\sqrt{2}$ . Find the bounds on the absolute and relative errors.

8. Prove that :

$\triangle ^{n} 0^{n+1} = \frac{n(n+1)}{2} \triangle ^{n}0^{n}$

(b) Find the function u_x in powers of x^-1 given that u₀= 8, u₁= 11, u₄= 68, u₅= 123.

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