Sunday, August 2, 2020

NUMERICAL ANALYSIS - 2017 | B.SC CS 2ND YEAR | MJPRU | EXAM PAPER | My CS Tutorial

My CS Tutorial August 02, 2020

Numerical analysis | My CS Tutorial

Paper code: 13512
1512
B.Sc. (Computer Science) (Part 2)
Examination, 2017
Paper No. 1.3
NUMERICAL ANALYSIS

Time: Three Hours]

[Maximum Marks: 50

Note: Attempt five questions. All questions carry equal marks.

1. (a) Use Lagrange’s formula to find f(6) from the following table:

x
2
5
7
10
12

f(x)
18
180
448
1210
2028

(b) The population (in thousands) of a town in the year 1931, …………, 1971 are as ahead:

Year. Population

1931 15

1941 20

1951 27

1961 39

1971 52

2. (a) Use the Milne’s method to solve the equation ${y}'=x-y^{2}$ with y(0)=0 from x=0 to x=1.Find the population of the town in 1946 by applying Gauss’s backward formula.

(b) Use the Runge-Kutta method to approximate y when x=0.1 given that x=0 when y=2 and \frac{dy}{dx}=y-x.

3. (a) Find a real root of the equation $x=e^{-x}$ using the Newton-Raphson method.

(b) Find the cube root of 10 correct to three decimal places by Regula-Falsi method.

4. (a) Evaluate $\int_{0}^{6}\frac{1}{1+x^{3}}$ by Simpson’s one-third rule by dividing the interval into 6 parts.

(b) Evaluate $\int_{0}^{6}t\sin tdt$ by Trapezoidal rule.

5. (a) Solve the following equations by Gauss Elimination method:

$2x+y+z=10$

$3x+2y+3z=18$

$x+4y+9z=16$

(b) Solve by Jacobi iteration method the system of equations:

$4x+y+3z=17$

$x+5y+z=14$

$2x-y+8z=12$

6. (a) State and prove Newton’s-Gregory formula for backward interpolation.

(b) Apply Newton’s dividend difference formula to find the value of f(8) if f91)=3, f(3)=31, f(6)=223, f(10)=1011, f(11)=1343.

7. (a) Find the function $u_{x}$ in powers of x-1 given that

$u_{0}=8$ , $u_{1}=11$ , $u_{4}=68$ , $u_{5}=123$ .

(b) Write short notes on the following:

Relative Error and Absolute Error
Percentage Error and Round Off Error

8. (a) Solve the system linear of equations by the Gauss-Seidel method (4 iterations):

$2x_{1}+7x_{2}+x_{3}=19$

$4x_{1}+x_{2}+x_{3}=3$

$x_{1}+3x_{2}+12x_{3}=31$

(b) Solve the following:

(i) Prove that:

$\left ( 1+\bigtriangleup \right )\left ( 1-\bigtriangledown \right )=1$

(ii) Prove that:

$E=\left ( 1-\bigtriangledown \right )^{-1}$

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