# WBUT Question Papers Electronics Communication

# B Tech 3rd Semester 2008

## Numerical Methods And Programming

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**GROUP – A ( Multiple Choice Type Questions)**

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

i) Which of the following relations is true ?

a) E = 1 + A b) E = 1 – A

c) E = 1/A d) None of these.

ii)————————– By evaluating 2 by a numerical integration method, we can obtain an approximate value of

a) log_{e} 2 b) n

c) e d) log_{10}2.

iii) if a be the actual value and e be its estimated value, the formula for relative error is

a) a b) J-5L=JLL

c) ea d) -L^-

iv) in Trapezoidal rule, the portion of curve is replaced by

a) straight line b) circular path

c) parabolic path d) none of these.

v) The error Involved In 4^{111} order R-k method is given by

a) O (h^{2}) b) 0(h^{4})

c) O (h^{3}) d) 0(h^{5}).

vi) An n x n matrix A is said to be diagonally dominant if

a)J= 1

b) J= ii^{56} J fl

C) | a_{(t} | > X| ^{a} ij |

J= 1 < *J

d) ij J= 1

vii) Find the output of the following program main() { char a, b , a = ‘b’ ; b = a ; printf( “b = %c\n”, b ) ;

a) a

c) garbage value

b) b

d) none of these.

Vlll) Lagrange’s interpolation formula is used for

a) equispaced arguments only b) unequispaced arguments only

c) both equispaced and unequispaced arguments

d) none of these.

ix) j If /( 3 ) = 5 and /( 5 ) = 3, then the linear interpolation function f ( x) is

xa) /( x) = 8 – x c) /(x)=x^{2} If / ( x) = ^ , the divided difference [ a, b, c ] is

b)f(x) = 8 + x /(x)=x + x^{2} + 8.

a)

c)

1

a + b + c 1

a^{2}+b^{2}

b)

d)

abc

a+b-c’

If ^ = x + y and y ( 1 ) = 0, then y ( 1.1) according to Euler’s method is | h = 0-1 ]

a) 01 c) 0-5

b) 0-3 d) 0-9.

xii) Which one of the following results is correct ?

a] Ax^{n} – rxx

c) A^{n}e^{x} = e^{x}

d) A cos x = – sin x.

xiii) In the method of iteration the function ( x) must satisfy

a) | <t>’ ( x) | < 1 c) | <t> ‘( x) | = 1

b) [ <t>'(x) | > 1 d) | <fr'( x) | = 2.

xiv) The inherent error for Simpson’s ^ rd rule of integration is as (the notations have their usual meanings )

^{31} – iso-HM ^{b}) ” T5o(*o)

c) – ~j2~ f” ( x o ) d) none of these.

xv) ( A – V ) x^{2} is equal to (the notations have their usual meanings

a) h^{2} 2h^{2}

c) 2 h^{2} d) none of these.

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**GROUP -B ( Short Answer Type Questions )**

Answer any three of the following. 3×5= 15

2. From the following table And the values of / { 12 ) by Newton’s divided difference interpolation formula :

x : | 11 | 13 | 14 | 18 | 19 | 21 |

f(x): | 1342 | 2210 | 2758 | 5850 | 6878 | 9282 |

3. Solve the following system by Matrix Inversion Method : 2x + y + z = 10 3x + 2y + 3z = 18 x + 4y + 9z = 16.

4. a) Evaluate the missing terms In the following table :

x : | 0 | 1 | 2 | 3 | 4 | 5 |

fix): | 0 | — | 8 | 15 | — | 35 |

5. What is ternary operator ? Give an example.

Solve by Taylor’s series method = 2x + 3y ^{2} given y = 0 when x = 0 at x = 02.

Using Euler’s method obtain the solution of ^ = x – y. with y ( 0 ) = 1 and

h = 0-2 at x = 0-4.

6. Find the first approximation of the root lying between 0 and 1 of the equation x^{3} + 3x – 1 = Oby Newton^{:}Raphson formula.

x : | 0 | 1 | 2 | 3 | 4 |

fix): | 1 | 1 | 15 | 40 | 85 |

**GROUP -C ( Long Answer Type Questions )**

Answer any three of the following questions.

8. a) From the following table, estimate the number of students who obtained marks

between 40 and 45 :

Marks : | 30 – 40 | 40 – 50 | 50 – 60 | 60 – 70 | 70 – 80 |

No. of Students : | 31 | 42 | 51 | 35 | 31 |

10. a) b)

X : | 4 | 5 | 7 | 10 | 11 | 13 |

f(X): | 48 | 100 | 294 | 900 | 1210 | 2028 |

Find the positive real root of x^{3} = 18 using the bisection method of 4 iterations. Find the root of the equation x^{3} + x^{2} + x+ 7 = 0 using Regula Falsi method.

A curve passes through the points as given in the following table. Find the area

X | l | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

y | 0-2 | 0-7 | 1 | 1-3 | 15 | 1-7 | 1-9 | 21 | 2-3 |

Write a program in C to solve the equation x^{3}-3x-5 = 0 within ( 1, 2 ) by Bisection method correct up to 3 places of decimal.

Write a program in C using recursive function to calculate the sum of all digits of

8 + 7

any number.

11. a) Evaluate J* xe^{x} dx by using Trapezoidal rule taking n = 6.

b) Use Lagrange’s interpolation formula to find the value of /( x ) for x – 0. given the following :

x : | – 1 | – 2 | 2 | 4 |

f(x): | – 1 | – 9 | 11 | 69 |

Prove that Newton-Raphson method has a quadratic convergence. Solve the following system of equations by L-U Factorization Method :

^{X}1^{+X}2~^{X}3 ^{=} 2

2x _{t} + 3x_{2} + 5x_{3} = – 3

3x , + 2x _{2} – 3x _{3} = 6. ’

Solve the following set of equations by Gauss-Seidel method correct to 2 places of decimal :

9x – 2y + z = 50

x + 5y – 3z = 18

– 2x + 2y + 7z = 19.

Write a C program to approximate a real root of the following equation 4Bisection method.Write a C program to interpolate a given function at a specified argument by Lagrange’s interpolation formula.

Find the value of log 2 ^{1/3} from—3 dx using Simpson’s 41 + x >3 c) n = 4. R Calculate the approximate value of J sin x dx by Composite Trapezoidal Rule by using 11 ordinates. Also compare it with the actual value of the integral.