# WBUT Previous year Question Papers EC

## Numerical Methods And Programming B Tech Third Sem 2009-2010

Time Allotted : 3 Hours

full Marks : 70

The figures In the margin Indicate full marks.

Candidates are required to give their answers in their own words

as far as practicable.

GROUP-A ( Multiple Choice Type Questions )

1. Choose the correct alternatives for any ten of the following :                                     10 x 1 = 10

i)          If the interval of differencing is unity and f(x) – ax2 ( ‘a’ is a constant) which of the following choices is wrong ?

a)        Af (x) = a(2x +1)    b) A2/(x)~2a * c) A3 /(x) = 2 d) A4/(*) = 0.

ii)        The number of significant figures in 6,00,000 is

a)        1                   b) 7

c)         0 d) 6.

iii)       Which of the following is true ?

a)        A” xn = (n +1)!        b) A” xn = n !            /

c)                                                                 A” xn = 0  d) A”x”=n.

33502                                   [ Turn over

iv)       When Gauss elimination method is used to solve AX = B,A is transformed to a

a)        unit matrix

b)        loWer triangular matrix

c)         diagonally dominant matrix

d)        upper triangular matrix.

v)         The method of iteration formula <t> ( x ) must satisfy

a)      |<t>’ (x)|< 1        b) |VW|>1

c) (x)|-l           d) | <J>'(x) | – 2.

vi)       Regula-Falsi method is

a)        conditionally convergent

b)        linearly convergent

c)         divergent

d)        none of these.

vii)      Which of the following is true ?

a)                                                                E = 1 – A          b) E = 1 + A

c) A = 1 + E                                d) E = 1/A.

viii)     The order of h in the error expression of Trapezoidal rule is

a)                                                                6          h) 3

c) 5                                             d) 2.

ix)       The degree of precision of Simpson’s one third rule is

a)        1                                                 b) 2

c)                                                                3         d) 5.

x)        Which of the following methods is an iterative method ?

a)         Gauss Elimination method

b)        Gauss-Jordan method

c)         Gauss-Seidel method

d)        Crout’s method.

xi)       main ()

{

print(“%x”,-l«4);

}

a)      0   b) FO

c)   FFFF     d) FFFO.

xii)      main ()

{

char s[] = {‘a’,’b’,’c’,’\n’,’c’,”}; char *p, *str,*strl; p=&s; str=p;

strl=s;                                                                                ,

printfT%d”,++*p + ++*strl-32);

}

b)         122

d)        277.

xiii)    main ()

{

lnta=2,*fl, *f2; fl =f2=&a;

*£2+=*f2+=a+=2-5;

printfl”\n%d %d %d”, a, *fl, *f2);

}

a)       16 15 14   b) 16 16 16

c)       16 15 16   d) 24 24 24.

xiv)     main ()

{

printfTXnab”);

printfl”\bsi”);

printfl”\rha”);

}

What will be the output for the above code ?

a)    hai_        b) ha

c)                d) ab

GROUP – B (Short Answer Type Questions )

Answer any three of the following. 3 x5 = 15

1. a) What is the difference between interpolation and extrapolation ? Give suitable examples.       2

b)        If y ( 10 ) = 35-3, y ( 15 ) = 32-4, y ( 20 ) = 29-2, y ( 25 ) = 26 1, y ( 30 ) = 23-2 and y ( 35 ) = 20-5, find y ( 12 ) using Newton’s forward interpolation formula. 3

3. a) Use Newton’s divided difference formula to And / ( 5 ) from the following data :                                                         3

 X 0 2 3 4 7 8 fix) 4 26 58 112 466 668

b)       What do you mean by geometrical Interpretation of Simpson’s ^rd rule ?            2

4. a) Find the values of y'(x) and y”(x) at x = 11 from the

following data, using Newton’s forward interpolation formula :                                                                                                  3

 X 10 12 1-4 1-6 1-8 2-0 Y 0 0128 0-544 1-296 2-432 4

b)       What is ternary operator ? Give examples.

Find the approximate value of I-f dx/(l + x) when the

Interval is ( 0, 1 ) and h – Use trapezoidal rule. 3

2

b)       Show that A log/ ( x ) = log ( 1 + A/ ( x ) // ( x ) j, where A is the forward difference operator.                                                                                                    2

1. Solve by using Euler’s method the following differential equation for x = 1 by taking h = 0-2 :

dy/dx = xy, y = 1 when x = 0.                                                         5

1. Find the smallest positive root of the equation 3x -9x2+8-0 correct to 4 places of decimals, using Newton-Raphson                         ( Turn over

GROUP-C ( Long Answer Type Questions )

Answer any three of the following. 3×15 = 45

1. a) Solve the system of linear equations by Gauss

Elimination method :

5*! – x2 – 9 -x1+5x2-x3-4

x2 + 5x3 – – 6.                                                                             7

b)        Find the Newton-Raphson iterative formula to find the pth root of positive number N and hence find the cube- root of 17.                                                          5

. c) Evaluate the following:                                                                     3

A2 j(5x + 12)/(x2 + 5x + 6)|, taking h = 1

1. a) Write a C program to interpolate a given function as

specified argument by divided difference formula. 7

b)      Compute J*x/sin xdx, where the interval is ( 0, 1/2 ) using Simpson’s rule with h =1/4.          5

c)      Deduce trapezoidal rule for Newton-Cote’s quadrature formula.     .           3

1. a) Find the inverse of the following matrix.                            5

3         -1 1\

-15 6 -5 5 -2 2

b)       Solve the following system of equations by LU factorization method :         5

2x – 6y + 8z = 24

5x + Ay – 3z = 2

3x + y + 2z = 16

c)        Evaluate f xex dx where the interval is ( 0, -1 ) by using Trapezoidal rule taking n = 6. 5

1. a) Write a C program to solve the equation x3 – 3x – 5«0 within ( 1, 2 ) by Bisection method correct upto 3 places of decimal.                                            8

b)       Write a program in C using recursive function to calculate the GCD of any two given numbers.                                                                                                    7

1. a) Find the root of the equation 3x – cosx -1=0 that lies

between 0 and 1, correct to four places of decimal, using bisection method.         7

b)        Find the root of the equation x3 -5x-7-0, that lies between 2 and 3, correct to 4 places of decimals, using the method of false position.              7

c)         State the condition of convergence of Newton-Raphson method.      1

1. a) Solve the following system of equations, correct to four

places of decimals, by Gauss-Seidel iteration method : 8

x + y + 54 z =110 27x + 6y – z = 85 6x + 15y + 2z = 72

b)        Find the values of y ( 01 ), y ( 0-2 ) and y ( 0-3 ) using Runge-Kutta method of the fourth order, given that

dy/dx-xy + y2,y(0)-l. ‘                                                                7