Applied Mathematics for Electrical Engineering — Winter 2020

Find a root of the equation x4 −x−10 = 0 using Bisection method. Perform only four iterations.

Use Newton’s divided difference formula to find f(x) from the following data: x 3 7 9 11 y 168 120 72 48 Hence evaluate y for x = 6.

( i ) Use Trapezoidal rule to evaluate ∫ 1 x2dx considering five subintervals.0 ( ii ) Apply Runge-Kutta fourth order method to find an approximate value of y04 when x = 0.2 given that dy 2x = y− , y(0) = 1, ℎ = 0.2. dx y

Find the mean, median and standard deviation for the following data: 48,43,65,57,31,60,37,48,59,78.

If the probability density of a random variable is given by k(1−x2), for 0 < x < 1 f(x) = { , 0, elsewℎere find k. Also find the probabilities that a random variable having this probability density will take on a value ( a ) between 0.1 and 0.2 ( b ) greater than 0.5.

( i ) Find a root of the equation xex −cos x = 0 in the interval (0,1) using Newton-Raphson Method correct up to 𝜀 < 1 %. Take x = 0.5. a 0 ( ii ) Find a real root of the equation x3 +x2 −100 = 0 correct to two decimal places using Fixed Point Iteration method.

Use Newton’s backward interpolation formula to find the value of f(175) from the following table: x 140 150 160 170 180 f(x) 3685 4845 6302 8076 10225

If y(1) = −3,y(3) = 9,y(4) = 30,y(6) = 132, find the Lagrange’s interpolation polynomial that takes the same values as y at the given point.

The following show the gain in reading speed of 8 students in a speed-reading program, and the number of weeks they have been in the program: No. of weeks 3 5 2 8 6 9 3 Speed gain 86 118 49 193 164 232 73 109 Fit a straight line by the method of least squares.

The population (in thousands) of a town is given below. Estimate the population for the year 1975 using interpolation. Year 1971 1981 1991 2001 2011 Population 46 66 81 93 101

In usual notations, prove the following identities:1

1+𝜇2𝛿2 = (1+ 𝛿2) (ii) 𝜇𝛿 = ∆E−1 + ∆.222

Fit a parabola y = a+bx+cx2 to the following data: x 1 2 3 4 y 9.7468 24.4451 47.9318 78.4660 164.4186

Find the value of y(0.4) from the following differential equation with the given initial condition by Euler’s method: dy = log(x+y), y(0) = 2, ℎ = 0.1. dx

Evaluate ∫ 4 (x2 +2x) dx by using Gauss’ quadrature formula with n = 3.2

( i ) An assembly plant receives its voltage regulators from three different suppliers, 60 % from supplier B , 30 % from supplier B , and 10 % from12 supplier B .If 95 % of the voltage regulators from B , 80 % of those from31 B , and 65 % of those from Bperform according to specifications, what23 is the probability that any one voltage regulator received by the plant will perform according to specifications? Also, find the probability that a particular voltage regulator, known to perform according to specifications, came from supplier B .3 ( ii ) Find the missing frequencies f and f if the mean of the following12 frequency distribution of 100 families (f) is 30.4: x 0−10 10−20 20−30 30−40 40−50 50−60 f 10 f 25 30 f12

Find, by Taylor’s series method, the value of y at x = 0.1 to five places of decimals from dy = x2y−1, y(0) = 1. dx

Evaluate ∫ 1.4 (2+xlog x−cosx)dx with ℎ = 0.2 by Simpson’s one-third rule 0.2 and Simpson’s three-eighth rule.

( i ) The probability that an integrated circuit chip will have defective etching is 0.12, the probability that it will have a crack defect is 0.29, and the probability that it has both defects is 0.07. What is the probability that a newly manufactured chip will have neither defect? ( ii ) Astandard cell whose voltage is known to be 1.10 volts was used to test the accuracy of two volt meters Aand B. Ten independent readings of the voltage of the cells were taken with the two volt meters as per the following data. Which of these two is more reliable? A 1.11 1.15 1.14 1.10 1.09 1.11 1.12 1.15 1.13 1.14 B 1.12 1.06 1.02 1.08 1.11 1.05 1.56 1.03 1.04 1.06

Find the mode for the following frequency distribution: Class 0−6 6−12 12−18 18−24 24−30 f 20 30 25 16

Calculate the coefficient of skewness based on the Method of Moments from the following data: Class 0−4 5−9 10−14 15−19 20−24 Frequency 7 12 15 10

( i ) For a random variable X, if E(3X−5) = 16 and E(X2) = 58, find the standard deviation of X. ( ii ) If the events Aand Bare independent, then show that the events Aand B′ are also independent.2

Calculate the mean and standard deviation from the following data: Value 90-99 80-89 70-79 60-69 50-59 40-49 30-39 Frequency 2 12 22 20 14 4 1

Find the mean deviation from median for the following data: Marks 0−10 10−20 20−30 30−40 40−50 Students 8 11 15 9

( i ) Three students A,Band Care running in a race. Aand Bhave the same probability of winning and each is twice as likely to win as C. Find the probability that Bor Cwins. ( ii ) The quantities of milk (in liters) produced by a dairy farm on ten consecutive days are shown below: 218.2,199.7,207.3,185.4,213.7,184.7,179.5,194.4,224.3,203.5. Evaluate the mean and the first four central moments of the milk yield data (in litres) of dairy farm.