Uttar Pradesh Technical University (UPTU) 2007 B.E Computer Science COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES - Question Paper

COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES

CS - 406

*V-1034*

Printed Pages : 4

(Following Paper ID and Roll No. to be filled in your Answer Book)

Roll No. | | | | | | | | | 1 | B. Tech. (SEM. IV) EXAMINATION, 2006-07 COMPUTER BASED NUMERICAL & STATISTICAL TECHNIQUES

Time : 2 Hours]    [Total Marks : 50

Note : Attempt all the questions.

1 Attempt any four of the following :    4x3=12

(a)    If u = 3 9 ⁷ - 6 9 find the percentage error in u at 9 =1, if the error in 9 is 0.05.

(b)    Compute the real root of x³ - 5x + 3 = 0 in the interval [1,2] by the Regula falsi method. Perform three iterations only.

(c)    By Newton Raphson method find the positive root of f(u) = x -2 sin x.

Choose suitable initial guess and perform three iterations.

(d)    Find the root of the equation

f(u) = x³ - 3x - 5 = 0 which lies between

2 and 3 by the Mullers method. Perform two iterations only.

(e)    Apply the quotient - difference method to obtain the approximate roots of the equation.

X³ - 7x² + lOx -2 = 0.

(f)    Define rate of convergence. Obtain rate of convergence of Newton Raphson method.

2 Attempt any four of the following :    4x3=12

(a) From the following table, find the number of students who obtained less than 45 marks by method of interpolation :

Marks : 0-30 31-40 41-50 51-60 61-70 71-80 81-100 No. of Students : 0 31 42 51 35 31 5

(b) The ordinates of the normal curve are given by the following table

x : .0 .2 .4 .6 .8 y .3989 .3910 .3683 .3332 .2897

Calculate : (i) y (.25) () y (.62)

Use Newtons method of interpolation, (c) Use Stirling formula to find y(28) given

x : 20 25 30 35 40 y 49225 48316 47236 45926 44306

(d) Applying Lagranges formula, find the interpolating polynomial f(x) for the following set of observations.

x :	0	1	4	5
y	4	3	24	39

from the following table.

x : 4 5 7 10 11 13 f(u) : 48 100 294 900 1210 2028

(f) Differentiate between interpolation and curve fitting.

3 Attempt any two parts :    7x2=14

(a) Fit a natural cubic spline to the following data:

x:	2	3	4
y-	11	49	121

Hence compute

(i) y(2.5) and y(2)

(b) Find the first and second derivative at 1.1 for the data

x :	1.00	1.2	1.4	1.6	1.8	2.00
f(u) :	0	.1280	.5440	1.2960	2.432	4.00

(c) Evaluate the integral

Simpsons rule taking four equal intervals, and hence find the value of log_e 2.

4 Attempt any two parts :    6x2=12

(a)    For a bi variate distribution n = 18,

x² =60, y² =96, X^x= 12, = 18, xy = 48

Find the equations of lines of regressions.

(b)    Fit the curve y=ax^b to the following data, using

method of least squares.

x : 1 2 3 4 5 6 y 2.98 4.26 5.21 6.1 6.8 7.5

(c) Write short notes on :    6x2=12

(i)    Quality Control Methods

(ii)    Multiple Regression Analysis.

V-1034]    4    [ 1355 ]

CS - 406

*V-1034*

Printed Pages : 4

(Following Paper ID and Roll No. to be filled in your Answer Book)

Roll No. | | | | | | | | | 1 | B. Tech. (SEM. IV) EXAMINATION, 2006-07 COMPUTER BASED NUMERICAL & STATISTICAL TECHNIQUES

Time : 2 Hours]    [Total Marks : 50

Note : Attempt all the questions.

1 Attempt any four of the following :    4x3=12

(a)    If u = 3 9 ⁷ - 6 9 find the percentage error in u at 9 =1, if the error in 9 is 0.05.

(b)    Compute the real root of x³ - 5x + 3 = 0 in the interval [1,2] by the Regula falsi method. Perform three iterations only.

(c)    By Newton Raphson method find the positive root of f(u) = x -2 sin x.

Choose suitable initial guess and perform three iterations.

(d)    Find the root of the equation

f(u) = x³ - 3x - 5 = 0 which lies between

2 and 3 by the Mullers method. Perform two iterations only.

(e)    Apply the quotient - difference method to obtain the approximate roots of the equation.

X³ - 7x² + lOx -2 = 0.

(f)    Define rate of convergence. Obtain rate of convergence of Newton Raphson method.

2 Attempt any four of the following :    4x3=12

(a) From the following table, find the number of students who obtained less than 45 marks by method of interpolation :

Marks : 0-30 31-40 41-50 51-60 61-70 71-80 81-100 No. of Students : 0 31 42 51 35 31 5

(b) The ordinates of the normal curve are given by the following table

x : .0 .2 .4 .6 .8 y .3989 .3910 .3683 .3332 .2897

Calculate : (i) y (.25) () y (.62)

Use Newtons method of interpolation, (c) Use Stirling formula to find y(28) given

x : 20 25 30 35 40 y 49225 48316 47236 45926 44306

(d) Applying Lagranges formula, find the interpolating polynomial f(x) for the following set of observations.

x :	0	1	4	5
y	4	3	24	39

from the following table.

x : 4 5 7 10 11 13 f(u) : 48 100 294 900 1210 2028

(f) Differentiate between interpolation and curve fitting.

3 Attempt any two parts :    7x2=14

(a) Fit a natural cubic spline to the following data:

x:	2	3	4
y-	11	49	121

Hence compute

(i) y(2.5) and y(2)

(b) Find the first and second derivative at 1.1 for the data

x :	1.00	1.2	1.4	1.6	1.8	2.00
f(u) :	0	.1280	.5440	1.2960	2.432	4.00

(c) Evaluate the integral

Simpsons rule taking four equal intervals, and hence find the value of log_e 2.

4 Attempt any two parts :    6x2=12

(a)    For a bi variate distribution n = 18,

x² =60, y² =96, X^x= 12, y = 18, xy = 48

Find the equations of lines of regressions.

(b)    Fit the curve y=ax^b to the following data, using

method of least squares.

x : 1 2 3 4 5 6 y 2.98 4.26 5.21 6.1 6.8 7.5

(c) Write short notes on :    6x2=12

(i)    Quality Control Methods

(ii)    Multiple Regression Analysis.

V-1034]    4    [ 1355 ]

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