Exam Papers

Numerical Analysis 2019 – BSc Computer Science Part 2 (MJPRU)

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 = x2 in 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:023479
f(x):42658112466922

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 ux in powers of x-1 given that u0 = 8, u1 = 11, u4 = 68, u5 = 123.

……..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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