Applied Mathematics for Electrical Engineering (3130908)

AMEE-3130908

Winter-2019

BE | Semester 3

Winter - 2019|26-11-2019

Total Marks 70

Q1 (a) Find a real root of the equation $x^{3} - x - 1 = 0$ 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 $q^{th}$ 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 $f (x) = e^{- x} - x$ by the secant method starting from $x_{0} = 0, x_{1} = 1$ .

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 $f (x) = \frac{1}{x}$ , prepare the table for finite differences and hence find $[a, b]$ and $[a, b, c]$ .

Unit : Interpolation | Topic : Newton’s divided formula

Q2 (b) State Newton’s forward formula and use it to find the approximate value of f(1.6),

$x$ $1$ $1.4$ $1.8$ $2.2$

$f (x)$ $3.49$ $4.82$ $5.96$ $6.50$

Unit : Interpolation | Topic : Newton’s forward interpolation formula

Q2 (c ( i )) Using quadratic Lagrange interpolation, compute $\ln 9.2$ from $\ln 9.0 = 2.1972, \ln 9.5 = 2.2513, \ln 11 = 2.3979$ .

Unit : Interpolation | Topic : Lagrange’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

$x$ $3$ $4$ $5$ $6$ $7$ $8$

$f (x)$ $28$ $65$ $126$ $317$ $344$ $513$

Unit : Interpolation | Topic : Newton’s backward interpolation formula

OR

Q2 (c ( i )) Using the relation between the operators, prove that, $(1 + ∆) (1 - \nabla) = 1$ .

Unit : Interpolation

Q2 (c ( ii )) State Simpson’s $\frac{3}{8} th$ rule and hence evaluate $\int_{0}^{3} \frac{1}{1 + x} d x$ with $n = 6 .$

Unit : Numerical Integration | Topic : Simpson’s formulae

Q3 (a) Use Trapezoidal rule to estimate $\int_{0.5}^{1.3} e^{x^{2}} d x$ using a strip of width $0.2$ .

Unit : Numerical Integration | Topic : Trapezoidal formula

Q3 (b)
The velocity $v$ of a particle at a distance s from a point on its linear path is given by the following data:

Time ( $t$ ) $0$ $5$ $10$ $15$ $20$ $25$ $30$

Speed ( $v$ ) $30$ $24$ $19$ $16$ $13$ $11$ $10$

Estimate the time taken by the particle to travel the distance of $20 m$ using Simpson’s $\frac{1}{3}$ ^rd rule.

Unit : Numerical Integration | Topic : Simpson’s formulae

Q3 (c ( i )) Using Euler’s method find $y (0.2)$ , given that $\frac{dy}{dx} = y - \frac{2 x}{y}; y (0) = 1$ taking $h = 0.1$ .

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 $y (0.2)$ , given that $y^{'} = x + y, y (0) = 1$ taking $h = 0.1$ .

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

OR

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 $P (A) = 0.1$ and let the conditional probability that a slim screw is also too small be $P (B / A) = 0.2$ . 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 $x$ and $y$ is given by $f (x, y) = \{\begin{cases} k (x + 2 y); & 0 < x < 1, 0 < y < 2 \\ 0; & elsewhere \end{cases}$ . Find the marginal density function of $x$ and $y$ .

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

OR

Q4 (a)
Evaluate $\int_{0}^{1} \exp (- x^{2}) d x$ by Gauss integration formula with $n = 3$ .

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 $y^{'} = x^{2} + y^{2}$ using the Taylor’s series method for the initial condition $y (0) = 0$ . Where, $0 \leq x \leq 0.2$ .

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

Q4 (c ( ii ))
Find a positive root of the equation $x - \cos x = 0$ 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 $x = 5, 10, 8, 13, 4 .$

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.

$A$ $32$ $28$ $47$ $63$ $71$ $39$ $10$ $60$ $96$ $14$

$B$ $19$ $31$ $48$ $53$ $67$ $90$ $10$ $62$ $40$ $80$

Find standard deviation of each machine and also comment on the performance of the two machine.

Unit : Basic Statistics | Topic : dispersion

OR

Q5 (a)
The pH solution is measured eight times using the same instrument and 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.

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.
$10.2, 9.5, 8.3, 9.7, 9.5, 11.1, 7.8, 8.8, 9.5, 10 .$

Unit : Basic Statistics

Q5 (c ( i ))
Attempt the following.

Find the coefficient of quartile deviation for the following data: $6, 8, 10, 4, 20, 18, 16, 14, 12, 10 .$ .

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

Time ( $t$ )	$0$	$5$	$10$	$15$	$20$	$25$	$30$
Speed ( $v$ )	$30$	$24$	$19$	$16$	$13$	$11$	$10$

$A$	$32$	$28$	$47$	$63$	$71$	$39$	$10$	$60$	$96$	$14$
$B$	$19$	$31$	$48$	$53$	$67$	$90$	$10$	$62$	$40$	$80$

Applied Mathematics for Electrical Engineering (3130908)

Q1 (a) Find a real root of the equation x3-x-1=0 by using Regula - Falsi method correct to two decimal places.

Q1 (b) State the formula for finding the qth root and find the square root of 8 using Newton Raphson method correct to two decimal places.

Q1 (c ( i )) Find the positive solution of f(x) = e-x - x by the secant method starting from x0 = 0, x1 = 1.

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

Q2 (a) If f ( x ) = 1 x , prepare the table for finite differences and hence find [ a , b ] and [ a , b , c ] .

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

Q2 (c ( i )) Using quadratic Lagrange interpolation, compute ln 9.2 from ln 9.0 = 2.1972 , ln 9.5 = 2.2513 , ln 11 = 2.3979.

Q2 (c ( 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 8 f x 28 65 126 317 344 513

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Q2 (c ( i )) Using the relation between the operators, prove that, ( 1 + ∆ ) ( 1 - ∇ ) = 1.

Q2 (c ( ii )) State Simpson’s 38th rule and hence evaluate ∫0311 + x dx with n = 6.

Q3 (a) Use Trapezoidal rule to estimate ∫0.51.3ex2 dx using a strip of width 0.2.

Q3 (b) The velocity v of a particle at a distance s from a point on its linear path is given by the following data: Time ( t ) 0 5 10 15 20 25 30 Speed ( v ) 30 24 19 16 13 11 10 Estimate the time taken by the particle to travel the distance of 20 m using Simpson’s 1 3 ^rd rule.

Q3 (c ( i )) Using Euler’s method find y ( 0.2 ) , given that dy dx = y - 2x y ; y ( 0 ) = 1 taking h = 0.1.

Q3 (c ( 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 taking h=0.1.

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

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?

Q3 (c ( ii )) The joint probability density function of two random variables x and y is given by f(x,y) = k(x+2y) ;0<x<1 , 0<y<20 ;elsewhere. Find the marginal density function of x and y.

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

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.

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Q4 (a) Evaluate ∫01exp(-x2) dx by Gauss integration formula with n=3.

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

Q4 (c ( i )) Attempt the following. Solve the Ricatti’s equation y' = x2 + y2 using the Taylor’s series method for the initial condition y(0) = 0. Where, 0 ≤ x ≤ 0.2.

Q4 (c ( ii )) Find a positive root of the equation x - cos x = 0 using bisection method correct to two places of decimals.

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

Q5 (b) Find the first four moments about mean x = 5, 10, 8, 13, 4.

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.

(adsbygoogle = window.adsbygoogle || []).push({}); OR

Q5 (a) The pH solution is measured eight times using the same instrument and 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.

Q5 (c ( i )) Attempt the following. Find the coefficient of quartile deviation for the following data: 6, 8, 10, 4, 20, 18, 16, 14, 12, 10 ..

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

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

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

Q1 (c ( i )) Find the positive solution of $f (x) = e^{- x} - x$ by the secant method starting from $x_{0} = 0, x_{1} = 1$ .

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

Q2 (a) If $f (x) = \frac{1}{x}$ , prepare the table for finite differences and hence find $[a, b]$ and $[a, b, c]$ .

Q2 (b) State Newton’s forward formula and use it to find the approximate value of f(1.6),

$x$ $1$ $1.4$ $1.8$ $2.2$

$f (x)$ $3.49$ $4.82$ $5.96$ $6.50$

Q2 (c ( i )) Using quadratic Lagrange interpolation, compute $\ln 9.2$ from $\ln 9.0 = 2.1972, \ln 9.5 = 2.2513, \ln 11 = 2.3979$ .

Q2 (c ( ii )) State Newton’s backward formula and use it to find the approximate value of f(7.5), if

$x$ $3$ $4$ $5$ $6$ $7$ $8$

$f (x)$ $28$ $65$ $126$ $317$ $344$ $513$

OR

Q2 (c ( i )) Using the relation between the operators, prove that, $(1 + ∆) (1 - \nabla) = 1$ .

Q2 (c ( ii )) State Simpson’s $\frac{3}{8} th$ rule and hence evaluate $\int_{0}^{3} \frac{1}{1 + x} d x$ with $n = 6 .$

Q3 (a) Use Trapezoidal rule to estimate $\int_{0.5}^{1.3} e^{x^{2}} d x$ using a strip of width $0.2$ .

Q3 (c ( i )) Using Euler’s method find $y (0.2)$ , given that $\frac{dy}{dx} = y - \frac{2 x}{y}; y (0) = 1$ taking $h = 0.1$ .

Q3 (c ( ii )) State the formula for Runge-Kutta method of fourth order and use it to calculate $y (0.2)$ , given that $y^{'} = x + y, y (0) = 1$ taking $h = 0.1$ .

OR

Q3 (a)
Define the following.

Favorable event

Random variable

Probability density function

Q3 (c ( ii ))
The joint probability density function of two random variables $x$ and $y$ is given by $f (x, y) = \{\begin{cases} k (x + 2 y); & 0 < x < 1, 0 < y < 2 \\ 0; & elsewhere \end{cases}$ . Find the marginal density function of $x$ and $y$ .

Q4 (a)
Define the following.

Mutually exclusive events

Probability

Compound events

OR

Q4 (a)
Evaluate $\int_{0}^{1} \exp (- x^{2}) d x$ by Gauss integration formula with $n = 3$ .

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

Q4 (c ( i ))
Attempt the following.

Solve the Ricatti’s equation $y^{'} = x^{2} + y^{2}$ using the Taylor’s series method for the initial condition $y (0) = 0$ . Where, $0 \leq x \leq 0.2$ .

Q4 (c ( ii ))
Find a positive root of the equation $x - \cos x = 0$ using bisection method correct to two places of decimals.

Q5 (b) Find the first four moments about mean $x = 5, 10, 8, 13, 4 .$

OR

Q5 (a)
The pH solution is measured eight times using the same instrument and 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.

Q5 (c ( i ))
Attempt the following.

Find the coefficient of quartile deviation for the following data: $6, 8, 10, 4, 20, 18, 16, 14, 12, 10 .$ .

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