Code: A-20

AMIETE – ET (OLD SCHEME)

Code: AE07 Subject: NUMERICAL ANALYSIS & COMPUTER PROGRAMMING

Time: 3 Hours Max. Marks: 100

NOTE: There are 9 Questions in all.

· Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1 Choose the correct or the best alternative in the following: (210)

a. What is the output of the following C program

#include<stdio.h>

void main( )

{

int arr[ ]={10, 20, 36, 72, 45, 36};

int *j, *k;

j = &arr[4];

k = (arr + 4);

if (j = = k)

printf("1010");

else

printf("0101");

}

(A) Error (B) 0101

(C) 1010 (D) No output

b. Consider the following program

#include<stdio.h>

void main( )

{

int x, y;

scanf("%d %d", &x, &y);

fun(x, y);

}

void fun(int a, int b)

{

a = a + b;

b = a – b;

a = a – b;

}

The above coding can be used for

(A) Addition and subtraction of two numbers.

(B) Exchanging the value of two variables

(C) Finding the Fibonacci series

(D) None of these

c. The convergence of Newton-Raphson method is

(A) linear (B) quadratic

(C) cubic (D) None of the above

d. If is the Forward Difference operator and is the shift operator, then

equal to

(A) 6x (B) 3x²

(C) 3x³ (D) None of the above

e. The value of y₆ if y₀= –8, y₁= –6, y₂= 22, y₃= 148, y₄= 492, y₅= 1222 is

(A) 2156 (B) 2554

(C) 2618 (D) None of the above

f. After Rounding of 37.46235 to four significant figures, the absolute error will be

(A) 0.00235 (B) 0.3746

(C) 6.27 x 10^–5 (D) None of the above

g. If is an eigen value of the Matrix A, then the eigen value of A^–1 is

(A) (B)

(C) (D) None of the above

h. Let and denote the lower and upper triangle matrices respectively. Then which of the following is correct.

(A) product of two lower triangular matrices is a upper triangular matrix

(B) product of two upper triangular matrices is a lower triangular matrix

(C) product of two lower triangular matrices is a lower triangular matrix

(D) All of the above

i. The approximate value of

by using mid-point rule is

(A) 0.7325 (B) 0.9589

(C) 0.6537 (D) None of the above

j. For Simpson's ^rd rule, the interpolating polynomial is of degree

(A) first (B) second

(C) third (D) fourth

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2 a. Find the root of the equation using the Secant Method correct to four decimal places. (8)

b. Find a real root of the equation correct to three decimal places using Iteration Method. (8)

Q.3 a. Solve the equations

(6)

b. Solve the system of equations by Cholesky method.

(10)

Q.4 a. The population of a town in decimal census were given in the following table. (6)

Year

: 1921

1931

1941

1951

1961

population in thousand

: 46

101

Estimate the population for the year 1955 using Newton's backward formulae.

b. Obtain the least squares polynomial approximation of degree two for on [0,1]. (10)

Q.5 a. The following values of the function , are given (8)

x	10°	20°	30°
	1.1585	1.2817	1.3660

construct the quadratic interpolating polynomial that fits the data. Hence find

b. Find the approximate value of the integral by using composite trapezoidal rule with 2,3,5,9 nodes and Romberg Integration. (8)

Q.6 a. Employ Taylor's method to obtain approximate value of y at x=0.2 for the differential equation (8)

b. Given with initial condition y = 1 at x = 0. Find y for x = 0.1 by Euler's method. (8)

Q.7 a. Assume that f(x) has a minimum in the interval where . Show that the interpolation of f(x) by a polynomial of second degree yields the approximation