Applied Mathematics for Electrical Engineering — Winter 2019

Find a real root of the equation x3  x1 0by using Regula-falsi method correct to two decimal places.

State the formula for finding the qth-root and find the square root of using Newton Raphson method correct to two decimal places.

Attempt the following.

Find the positive solution of f(x)ex x by the secant method starting fromx 0,x 1.0.01 (ii) Using method of least squares, find the best fitting straight line to the given following data. x : 1 2 3 4 y : 1 3 5 6

1 If f  x   , prepare the table for finite differences and hence find x     a,b and a,b,c .

State Newton’s forward formula and use it to find the approximate value of f  1.6 , if x 1 1.4 1.8 2.2 f  x  3.49 4.82 5.96 6.5

Attempt the following.

Using quadratic Lagrange interpolation, compute ln9.2 from ln9.02.1972, ln9.52.2513, ln112.3979 (ii) State Newton’s Backward formula and use it to find the approximate value of f  7.5 , if x 3 4 5 6 7 f  x  28 65 126 217 344 513

Attempt the following.

Using the relation between the operators prove that,    1 1 1. (ii) 3 1 State Simpson’s 3 rule and hence evaluate  dx with n6.8 1 x0

1.3 Use trapezoidal rule to estimate ex2 dx using a strip of width0.2. 0.5

The velocity vof a particle at a distance sfrom a point on its linear path is given by the following data.1 Time  t  0 5 10 15 20 25 30 Speed  v  30 24 19 16 13 11 Estimate the time taken by the particle to travel the distance of 20m using1 Simpson’s rule.3

Attempt the following.

Using Euler’s method, find y  0.2  given that dy 2x  y ;y  0  1 takingh 0.1. dx y (ii) State the formula for Runge-Kutta method of fourth order and use it to calculate y  0.2  given that y'  x y,y  0  1 takingh0.1.

Define the following. 1) Favorable Events 2) Random Variable 3) Probability Density function

An urn contains 10 white and 3 black balls, while another urn contains 3 white and 5 black balls. Two balls are drawn from the first urn and put into the second urn and then a ball is drawn from the latter. What is the probability that it is a white ball?

Define the following. 1) Mutually Exclusive Events 2) Probability 3) Compound Events

State Bayes’ theorem. In a bolt factory, three machines A,Band C manufacture 25% , 35% and 40% of the total product respectively. Of these outputs 5% , 4% and 2% respectively, are defective bolts. Abolt is picked up at random and found to be defective. What are the probabilities that it was manufactured by machines A,Band C?

Attempt the following.

Aperson is known to hit the target in 3 out of 4 shots, where as another person is known to hit the target in 2 out of 3 shots. Find the probability of the target being hit at all when they both try. (ii) Out of five cars, two have tyre problems and one has brake problem and tow are in good running condition. Two cars are required for the journey. If two cars are selected among five at random and if Xdenotes the number with tyre problem, Ydenotes with brake problem then find the marginal probability function of Xand Y.2

1   Evaluate expx2 dx by using the Gaussian integration formula for n 30

Using method of least squares, find the best fitting second degree curve to the following data. x : 1 2 3 y : 6 11 18 27

Define Mean, Median and Mode for the ungrouped data.

Find the first four moments about mean x 5,10,8,13,4

Attempt the following.

In a distribution of two different groups the variances are 15 and 27, whereas the third central moments are 32.4 and 67.56 respectively. Compare the skewness of two groups. (ii) Two automatic filling machines Aand Bare used to fill mixture of cement concrete in beam. Arandom sample of beam on each machine showed following results. A 32 28 47 63 71 39 10 60 96 B 19 31 48 53 67 90 10 62 40 80 Find standard deviation of each machine and also comment on the performance of the tow machines.

The pH solution is measured eight times using the same instrument and the data obtained are as follows. 7.15 , 7.20 , 7.18 , 7.19 , 7.21 , 7.20 , 7.16 , 7.18 Calculate the mean, variance and standard deviation.

In environmental geology computer simulation was employed to estimate how far a block from a collapsing rock wall bounce down a soil slope.Based on the depth, location and angle of block soil impact marks left on the slope of the actual rock fall, the following 10 rebound lengths (meters) were estimated. Compute mean and standard deviation of the rebounds. 10.2 , 9.5 , 8.3 , 9.7 , 9.5 , 11.1 , 7.8 , 8.8 , 9.5 ,