Thapar University 2006 M.C.A Numerical & Statistical Methods - Question Paper
Thapar Institute of Engineering & Technology
MCA (2ndYear)
Final Term exam
MA204 (Numerical & Statistical Methods) w
SCHOOL OF MATHEMATICS AND COMPUTER APPLICATIONS, T. I. E. T., Patiala End Semester Examination- December 12, 2006 Max. Mark: 36 Numerical and Statistical Methods (MA-204) Time: 03 Hours
Note: (a) Answer any FIVE questions by selecting any three from Group A and any two from Group B. (b)Write your tutorial group on the top of first page of answer sheet, (c) Attempt the parts of a problem at only one place.
SECTION-A
(a) Prove that the iterations generated by Bisection method always converge and its order of convergence is linear. (3. 6)
(b) Check whether the procedure to compute /(x) = VxTT - Vx at x = 12345 in six - decimal arithmetic is stable or unstable. Discuss the procedure if it is unstable. (3. 6)
(a) Show that there are several second order Runge-Kutta methods and one of them is the improved Eulers method. (3.6)
(b) The equation x3 - Ix2 + 16x -12 = 0 has a double root at x = 2. Starting with initial approximation x0 = 1, find the root correct to three decimal places using modified Newton -Raphsons method. (3.6)
(a) Let p(x) = a0 + (x - X<3[ + +(* - cn \an ) *) be a polynomial in Newton form If
C, = C2 = = c,+1, then prove that ) = J' <*,> 7 = 0, - (3.6)
(b) Given that
x : 1.0 1.5 2.0
(3.6)
logx: 0.0 0.17609 0.30103 Find Lagrange interpolating polynomial with this data. If we add one more term log 3.0= 0.47712 in the above data then find /?3(x) in such a way that P*nd r(x)explicitly. Also find
the lower bound on interpolation error at x = 2.5 for /?3(x).
dx
(a) Find the minimum number of intervals required to evaluate f- with accuracy 106, by using
1 + x
the Simpsons one-third rule and also evaluate this integral using the same method. (3.0)
(b) Using Power method, find the smallest eigenvalue and its corresponding eigenvector of the
following matrix correct to two decimal places starting with initial approximation X0 = [l \]
1 2
A- (4.2,
SECTION-B
(a) The equations of two regression lines obtained in a regression analysis are as follows: 3 X +12 Y = 19* 3y + 9 = 46. Obtain (i) the value of correlation coefficient (ii) mean values of X and Y, and (iii) the ratio of the coefficient of variability of X to that of Y. (3.5)
(b) A continuous random variable X has the probability density function: /(x) = A + Bx, 0 < x < 1. If. the mean of the distribution is 0.5, find A and B . (1.5)
(c) A coin is flipped until 3 heads in succession occur. List only those elements of the sample space space that require 6 or less tosses. Is this a discrete sample space? Explain. (2.2)
6. (a) The time it takes to repair a personal computer is random variable whose density function in hours, is
, . [1/2, 0 < x < 2 given by /(*)=-L (2.5)
[0, otherwise
The cost of the repair depends on the lime it takes and is equal to 40 + 30V* when the time isx. What is expected cost to repair a personal computer?
A
(b) Prove that the variance of normal distribution is cr . (2.0)
(c )The probability of error in transmission of binary digit over a communication channel is 0.001. Write an expression for exact probability of more than 3 errors when transmitting a block of 1000 bits. What is its approximate value? Assume independent. (2.7)
7. (a) If U is uniformly distributed on (0, l) show that a + (& - a )U is uniform on (a, b). (3.0)
(b)A manufacturer produces bolts that are specified to be between 1.19 and 1.21 inches in diameter. If its production process results in a bolts diameter being normally distributed with mean 1.20 inches a: s.d. 0.005. What percentage of bdt will not meet specification? (4.2)
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| Earning: Approval pending. |
