# UPTU Previous Exam Papers

# B Tech 3rd Semester 2007

# Computer Based Numerical and Statistical Techniques

**Note : (1) Attempt all questions.**

**(2) All questions can equal marks.**

** 1. Attempt any four parts of the following :**

(a) Discuss two important computer arithmetic systems. Illustrate with examples that associatiative laws of floating point arithmetic do not hold in numerical computation, ,-^{2} .r^{4} x^{6}

(b) Derive the series : cos r – 1 + — – — . 2 ! 4 ! 6 ! Compute the number of terms required to estimate cos(rt/4) so that the result is correct to atleast two significant digits.

(e) In a triangle A ARC, a = 6 cm, c = 15 cm, Z B = 90°. Write a program in ‘C’ to find the absolute error in the computed value of A, if possible errors in a and c are 1/5% and 1/7% respectively. JJ-9967] llilll |f:ll IIIMI II 1

(d) Develop an iteration formula to find a real root of the equation x sin x + cos x = 0 Find it in the vicinity of x_{l)}=K.

(e) Use the iteration method to find a real root of the equation : 3x- _{x}/t + sin x = 0 correct to five decimal places.

(f) Use Muller’s method to obtain a root of the equation : cos x-x e^{x} =0 in the interval (0. 1).

**2 . Attempt any four parts of the following :**

(a) Prove : A-Y=-AY

(b) Estimate the missing term in the table :

.V | 0 | 1 | 2 | 4 | |

/M | 1 | “•> | 9 | 9 | 81 |

(c) Apply Stirling’s formula to find a polynomial of degree three which takes the following values of

x, .r :

x | 2 | 4 | 6 | 8 | 10 |

V | -2 | 1 | 3 | 8 | 20 |

(d) Write an algorithm of any central difference interpolation formula.

(e) Apply Langrange’s formula to find a cubic polynomial which approximates the data :

X | -1 | Z. | -> | |

y{x) | -12 | -8 | •i | 5 |

JJ-9967] | iJ| II11 111 III 111 I’ll! 2

(f) A function f(x) satisfies the conditions : /(0)= 1 , /'(0)=i, /'(1) = 0, /'(l) = 0. Use Hermite interpolation to approximate f(x) by a polynomial. Also evaluate the maximum value of f(x) in [0, l].

**3. Attempt any two parts of the following :**

(a) State the importance of numerical differentiation.

Find /'(0.6) and /”(0.6) from the following table :

,v | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 |

A*) | 1.5836 | 1.7974 | 2.0442 | 2.3275 | 2.6510 |

(b) State the need and scope of numerical integration. Use Simpson’s rule to estimate the integral.

(2 .- J e^{x} c/x with a stepsize 0.5,

(c) The area A inside the closed curve. y^{2} + .v^{2} = cos v a t ‘ ‘ o is given by A – 4j_{o} |cosx- x^{2} j ^{2} c/x where a is the positive root of the equation cos x = x~ . Compute the area with an absolute error less than 0.05.

** 4. Attempt any.two parts of the following :**

(a) Apply Runge-Kutta fourth order method to find y{0.1), v(0.2) and >'(0.3) for the initial value

problem. — = xy/r y\ y (0) = 1 . Also, find y (0.4 j using Adam’s method.

(b) Solve the initial value problem :

y’ – x + sin (rcy); v(l) = 0, 1 < x < 2 by Milne’s predictor-corrector method

(c) Discuss the stability of Euler’s method applied to the initial value problem, y’ – Xy, J'(-Vq) = j’o .

**5. Attempt any two parts of the following :**

(a) State various methods for curve-fitting. Obtain the cubic splines approximation for the function given by the following table :

X | 0 | 1 | 2 | 3 |

/(*) | ^{1} |
o | 5 | 11 |

with the end conditions Mq = 0 =

( b) State objectives of control charts. A drilling machine bores holes with a mean diameter of 0.5230 cm and a standard deviation of 0.0032 cm. Calculate the 2-sigma and 3-sigma upper and lower control limits for means of sample of 4.

(c) Define lines of regression. Find the lines of regression for the given data :

X | 50 | 100 | 150 | 200 | 250 | 300 | 350 |

y | 30 | 65 | 90 | 130 | 150 | 190 | 200 |

Also find the coefficient of correlation for the above data.