Seat No.: ________ Enrolment No.
___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER–III (NEW) EXAMINATION – SUMMER 2021
Subject Code:3130908 Date:06/09/2021
Subject Name:Applied Mathematics for Electrical Engineering
Time:10:30 AM TO 01:00 PM Total Marks:70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
4. Simple and non-programmable scientific calculators are allowed.
Marks
Q.1 (a) Find a root of the equation x4 – x – 10 = 0 correct to three decimal places, 03
using the bisection method.
(b) By Simpson’s one-third rule, determine the area bounded by the given 04
curve and X-axis between x = 25 to x =25.6 from the data given below.
x 25 25.1 25.2 25.3 25.4 25.5 25.6
y 3.205 3.217 3.232 3.245 3.256 3.268 3.280
(c) Apply the method of least squares to determine the constants a and b such 07
that y = a ebx fits the following data:
X 0 0.5 1 1.5 2 2.5
Y 0.10 0.45 2.15 9.15 40.35 180.75
Q.2 (a) Define conditional probability. 03
A bag contains 19 tickets numbered from 1 to 19. Two tickets are drawn
successively without replacement. Find the probability that both tickets
will show even number?
(b) The following are scores of two batsmen A and B in a series of innings: 04
A: 12 115 6 73 7 19 119 36 84 29
B: 47 12 16 42 4 51 37 48 13 0
Who is the better score getter?
Who is more consistent?
(c) Discuss Newton-Rapshon method to solve non-linear equation f (x) = 0 07
numerically. Also, derive the formula to find the cube root of a positive
number N and hence compute 3 65 .
OR
(c) Discuss the fixed point iteration method. And using it find the real root of 07
x3 – 5x + 3 = 0 starting with x0 = 0.5 correct to four decimal places.
1.3
e
x2
Q.3 (a) Evaluate dx by using Simpson’s one-third rule taking h = 0.1. 03
0.5
(b) Explain the method of least squares in brief. Use it to derive normal 04
equations to fit a straight line y = ax + b.
(c) Newton’s interpolation formulas to find y at x = 0.11 and x = 0.27 from the 07
data given below.
x 0.10 0.15 0.20 0.25 0.30
y 0.1003 0.1511 0.2027 0.2553 0.3093
OR
1
Q.3 (a) Evaluate e dx by 3-point Gaussian quadrature formula.
x 2
03
0
1
(b) Define Central difference operator in terms of . 04
1
Establish the operator relations D = log(1 )
h
(c) Write Newton’s Divided difference interpolation formula for unequal 07
intervals. Determine the interpolating polynomial of degree three by using
Lagrange’s interpolation for the following data. Also find f(2)
x –1 0 1 3
f(x) 2 1 0 –1
Q.4 (a) (i) State Baye’s theorem. 03
(ii) Define Bernoulli’s trials.
(iii) Define independent events.
(b) Define probability density function. 04
If the probability density function of a random variable is given by
f ( x) k 1 x 2 , if 0 x 1
0 , elsewhere
Find the value of k and probability that X takes the value greater than 0.5
(c) What do you mean by predictor-corrector methods? State names of any 07
three predictor-corrector methods. Apply Milne’s predictor–corrector
method to obtain y(2) correct to three decimal places, if y(x) is the solution
x y where y(0) = 2, y(0.5) =2.636, y(1) = 3.595, y (1.5)=
dy 1
of
dx 2
4.968
OR
Q.4 (a) Discuss Binomial probability. The probability a man aged 60 will live to 03
be 70 is 0.65. What is the probability that out of 10 men aged 60 now, at
least 7 would live to be 70?
(b) Two cards are drawn successively with replacement from a well shuffled 04
pack of 52 cards. Find the mean and variance of the number of kings.
(c) Apply second order Runge-Kutta method to find an approximate value of 07
dy
y(0.2) given that x y 2 , y(0) = 1 and h = 0.1.
dx
Q.5 (a) State any four known methods for finding skewness. 03
Apply suitable method to compute the coefficient of skewness from the
following figures:
25, 15, 23, 40, 27, 25, 23, 25, 20
(b) Let X has the probability density function 04
1
f ( x) for 3 x 3
2 3
0 elsewhere
3
Find the actual probability P{X– } and compare it with the upper
2
bound obtained by Chebyshev’s inequality.
(c) Find kurtosis from the following data. 07
Class 0–10 10–20 20–30 30–40
interval
Frequency 1 4 3 2
OR
Q.5 (a) What do you mean by kurtosis? Illustrate the shape of three different 03
curves on the basis of value of 2 .
(b) A bag contains 6 white and 9 black balls. Four balls are drawn at a time. 04
Find the probability for the first draw to give four white balls and second
2
draw to give four black balls in each of the following case.
(i) with replacement and
(ii) without replacement
(c) Define rth moment about mean for grouped data. From the following data, 07
calculate moments about: (i) assumed mean and (ii) actual mean
Variable 0–10 10–20 20–30 30–40
Frequency 1 3 4 2
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