# Applied Mathematics for Electrical Engineering(3130908)

BE | Semester 3
Winter - 2019|26-11-2019
Total Marks 70

## Q1 (a) Find a real root of the equation ${\mathrm{x}}^{3}-\mathrm{x}-1=0$ by using Regula - Falsi method correct to two decimal places.

Unit : Numerical Solutions |  Topic : false position method

## Q1 (b) State the formula for finding the ${\mathrm{q}}^{\mathrm{th}}$ root and find the square root of 8 using Newton Raphson method correct to two decimal places.

Unit : Numerical Solutions |  Topic : Newton-Raphson method

## Q1 (c ( i )) Find the positive solution of by the secant method starting from .

Unit : Numerical Solutions |  Topic : Secant method

## Q1 (c ( ii )) Using method of least square, find the best fitting straight line to the given following data. x 1 2 3 4 5 y 1 3 5 6 5

Unit : Curve fitting by the numerical method |  Topic : fitting of straight lines

## Q2 (a) If , prepare the table for finite differences and hence find and .

Unit : Interpolation |  Topic : Newtonâ€™s divided formula

## Q2 (b) State Newton’s forward formula and use it to find the approximate value of f(1.6), $\mathrm{x}$ $1$ $1.4$ $1.8$ $2.2$ $3.49$ $4.82$ $5.96$ $6.50$

Unit : Interpolation |  Topic : Newtonâ€™s forward interpolation formula

## Q2 (c ( i )) Using quadratic Lagrange interpolation, compute from .

Unit : Interpolation |  Topic : Lagrangeâ€™s interpolation formulae for unequal intervals

## Q2 (c ( ii )) State Newton’s backward formula and use it to find the approximate value of f(7.5), if $\mathrm{x}$ $3$ $4$ $5$ $6$ $7$ $8$ $28$ $65$ $126$ $317$ $344$ $513$

Unit : Interpolation |  Topic : Newtonâ€™s backward interpolation formula

## Q2 (c ( i )) Using the relation between the operators, prove that, .

Unit : Interpolation

## Q2 (c ( ii )) State Simpson’s $\frac{3}{8}\mathrm{th}$ rule and hence evaluate with

Unit : Numerical Integration |  Topic : Simpsonâ€™s formulae

## Q3 (a) Use Trapezoidal rule to estimate  using a strip of width $0.2$.

Unit : Numerical Integration |  Topic : Trapezoidal formula

## Q3 (b) The velocity $\mathrm{v}$ of a particle at a distance s from a point on its linear path is given by the following data:   Time   ( $\mathrm{t}$ )  $0$ $5$ $10$ $15$ $20$ $25$ $30$  Speed ( $\mathrm{v}$ ) $30$ $24$ $19$ $16$ $13$ $11$ $10$   Estimate the time taken by the particle to travel the distance of using Simpson’s ^rd rule.

Unit : Numerical Integration |  Topic : Simpsonâ€™s formulae

## Q3 (c ( i )) Using Euler’s method find , given that taking .

Unit : Numerical solution of Ordinary Differential Equations |  Topic : Euler methods

## Q3 (c ( ii )) State the formula for Runge-Kutta method of fourth order and use it to calculate , given that taking $\mathrm{h}=0.1$.

Unit : Numerical solution of Ordinary Differential Equations |  Topic : and Runge-Kutta methods

## Q3 (a) Define the following. Favorable event Random variable Probability density function

Unit : Basic Probability |  Topic : definition of probability

## Q3 (b) 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 later. What is the probability that it is a white ball?

Unit : Basic Probability |  Topic : conditional probability

## Q3 (c ( i )) In producing screws, let A mean “screw too slim” and B “screw too small”. Let $\mathrm{P}\left(\mathrm{A}\right)=0.1$ and let the conditional probability that a slim screw is also too small be $\mathrm{P}\left(\mathrm{B}/\mathrm{A}\right)=0.2$. What is the probability that the screw that we pick randomly from a lot produced will be both too slim and too short?

Unit : Basic Probability |  Topic : conditional probability

## Q3 (c ( ii )) The joint probability density function of two random variables $\mathrm{x}$ and $\mathrm{y}$ is given by . Find the marginal density function of $\mathrm{x}$ and $\mathrm{y}$.

Unit : Basic Probability |  Topic : probability density function

## Q4 (a) Define the following. Mutually exclusive events Probability Compound events

Unit : Basic Probability

## Q4 (b) State Bayes’ theorem. In a bolt factory, three machines A, B, and C manufacture 25%, 35%, and 40% of the total product respectively. Out Of these outputs 5%, 4%, and 2% respectively are defective bolts. A bolt is picked up at random and found to be defective. What are the Probabilities that it was manufactured by machines A, B, and C?

Unit : Basic Probability |  Topic : Bayes' rule

## Q4 (c ( i )) A person is known to hit the target in 3 out of 4 shots, whereas 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.

Unit : Basic Probability |  Topic : conditional probability

## Q4 (c ( ii )) Out of five cars two have tyre problems and one has break problem and two are in good running condition. Two cars are required for the journey. If two cars are selected among five at random and if X denotes the number with tyre problem, Y denotes with break problem then find the marginal probability function of X and Y.

Unit : Basic Probability

## Q4 (a) Evaluate  by Gauss integration formula with $\mathrm{n}=3$.

Unit : Numerical Integration |  Topic : Gaussian quadrature formulae

## Q4 (b) Using method of least squares, find the best fitting second degree curve to the following data. x 1 2 3 4 y 6 11 18 27

Unit : Curve fitting by the numerical method |  Topic : second degree parabola

## Q4 (c ( i )) Attempt the following. Solve the Ricatti’s equation using the Taylor’s series method for the initial condition . Where, .

Unit : Numerical solution of Ordinary Differential Equations |  Topic : Taylor method

## Q4 (c ( ii )) Find a positive root of the equation using bisection method correct to two places of decimals.

Unit : Numerical Solutions |  Topic : Bisection method

## Q5 (a) Define mean, median and mode for the ungrouped data.

Unit : Basic Statistics |  Topic : Measure of central tendency:

## Q5 (b) Find the first four moments about mean

Unit : Basic Statistics |  Topic : Moments

## Q5 (c ( i )) 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.

Unit : Basic Statistics |  Topic : skewness

## Q5 (c ( ii )) Two automatic filling machines A and B are used to fill mixture of cement, concrete and beam. A random sample of beam on each machine showed following results. $\mathrm{A}$ $\mathrm{32}$ $\mathrm{28}$ $\mathrm{47}$ $\mathrm{63}$ $\mathrm{71}$ $\mathrm{39}$ $\mathrm{10}$ $\mathrm{60}$ $\mathrm{96}$ $\mathrm{14}$ $\mathrm{B}$ $\mathrm{19}$ $\mathrm{31}$ $\mathrm{48}$ $\mathrm{53}$ $\mathrm{67}$ $\mathrm{90}$ $\mathrm{10}$ $\mathrm{62}$ $\mathrm{40}$ $\mathrm{80}$ Find standard deviation of each machine and also comment on the performance of the two machine.

Unit : Basic Statistics |  Topic : dispersion

## Q5 (a) The pH solution is measured eight times using the same instrument and data obtained are as follows: Calculate the mean, variance and standard deviation.

Unit : Basic Statistics

## Q5 (b) 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 rebounds lengths (meters) were estimated. Compute mean and standard deviation of the rebounds.

Unit : Basic Statistics

## Q5 (c ( i )) Attempt the following. Find the coefficient of quartile deviation for the following data: .

Unit : Basic Statistics

## Q5 (c ( ii )) State the formula for coefficient of skewness based on central moments and find it for the following frequency distribution. Class 50 - 55 55 - 60 60 - 65 65 - 70 70 - 75 y 8 10 15 17 8

Unit : Basic Statistics |  Topic : skewness