No. of Printed Pages: 7 IBCS-054I
BACHELOR OF COMPUTER APPLICATIONS (Revised)
Term-End Examination
June,

BCS-054 COMPUTER ORIENTED NUMERICAL TECHNIQUES
Time hours Maximum Marks: 100
Note: Simple (but not scientific) calculator is allowed. Question no. 1 is compulsory. Attempt any three questions from the next four questions.

1. Explain the concepts of chopping, rounding, each with a suitable example. 3

Using 8-decimal digit floating-point representation (with four digits for mantissa, two for exponent and one each for sign of exponent and mantissa),
represent the following numbers in normalized floating point form (use chopping, if required) 3

89543

-89.766

0.0007345

For two floating point numbers

x1 0.7108 x 10^5 and x2 0.8701 x find x1 x2 . 2

Find the product of the two numbers given in question no. above. 3

Write the following system of equations in matrix form: 2


3x+ 4y=-17

Show one iteration of solving the following system of linear equations using any iterative method. You may assume x y 0 as the initial estimate 3

4x+ 7y=-1l

Find an interval in which the following equation has a root:

x2 9x 20 0 2

Write the formula used in Secant method for finding the root of an equation. 2

Write the three expressions which are obtained by applying each of the following operators to for some h 3

E

d

V

Write each of V and 0 in terms of E. 2

State the following two formulae for interpolation: 3

Newton's backward difference formula

Bessel's formula

Construct a difference table for the following data: 2

x 4 5 6 7
f(x) 13 22 33 46

From the Newton's backward difference formula asked in part above, derive the formula for finding the derivative of a function at x x0.

State Simpson's 1/3 rule for finding the value of Integral(f(x)) dx ,the limits are a to b .

Explain each of the following concepts with a suitable example: 4

Order of a differential equation

Initial Value Problem

Degree of a differential equation

Non-linear"differential equation

2. For each of the three numbers of Q.No. find relative error in its
normalized floating point representation. 6

Using Maclaurin's series expansion, find the value of at x by taking the first three terms and find truncation error. 4

Attempt to solve the following system of linear equations using the Gauss elimination method:

3x1 2x2 x3

2x1 x2 x3

6x1 2x2 4x3

Does the solution exist? If yes, how many? 5

Starting with Xu perform two iterations to find an approximate root of the equation x^3 -4x using Newton-Raphson method. 5

3. Solve the following system of linear equations using Gaussian elimination method with partial pivoting condensation:

3x2 4x3= 2

4x1 2x2 18

3x1 4x2 5x3 11 Compute upto two decimals only. 12

Give the formula for next approximation of values of x1, x2 and x3 using Gauss-Jacobi iterative method for solving the following system of linear equations: 4

a11x1 a12x2 a13x3 b1

a21x1 a22x2 a23x3 b2

a3lx1 a32x2 a33x3 b3

Discuss the relative merits and demerits of direct methods over iterative methods for solving a given system of linear equations. 4

4. Construct a difference table for the following data and mark the forward differences by underlying the numbers:

x 1 2 3 4 5 6 7 8
y 7 13 18 25 35 48 62 78

Derive the operators d and V in terms of E. 5

Find Newton's backward difference form of interpolating polynomial for the following data:

x 3 5 7 9 11 13
16 36 64 100 144 196

Hence evaluate f(12). 7

5. Attempt any two parts of and given below:

Find the approximate value of

I Integral(dx/(2+3x) using Simpson's 1/3 rule (three points). The limits are 0 to 1 10


The values of y sqrt(x) are given below for x =1.5(0.5)3.5.

x 1.5 2.0 2.5 3.0 3.5
1.2247 1.4142 1.5811 1.7320 1.8708

Find, and y" at x 1.75 using FD formula. 10·

Solve the following IVP using Euler's method

given

Find the solution on with h =0.2. 10