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## Engineering Mathematics 4 - Dec 2012

### Mechanical Engineering (Semester 4)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1 (a)** Using the Taylor's series method, solve the initial value problem $$ \dfrac {dy}{dx}=x^2 y-1, y(0)=1 $$ at the point x=0.1(6 marks)
**1 (b)** Employ the fourth order Runge-Kutta method to solve $$ \dfrac {dy}{dx}= \dfrac {y^2-x^2}{y^2+x^2} y(0)=1 $$ at the points x=0.2 and x=0.4. Take h=0.2.(7 marks)
**1 (c)** $$ Given \ \dfrac {dy}{dx}= xy+y^2, \ y(0)=1, \ y(0.1)=1.1169, \ y(0.2)=1.2773, \ y(0.3)=1.5049 $$ Find y(0.4) using the Milne's predictor-corrector method, Apply the corrector formula twice.(7 marks)
**2 (a)** Exmploying the Picard's method, obtain the second order approximate solution of the following problem at x=0.2. $$ \dfrac {dy}{dx}=x+yz, \ \dfrac {dz}{dx}= y+zx, \ y(0)=1, \ z(0)=-1 $$(6 marks)
**2 (b)** Using the Runge-kutta method, find the solution at x=0.1 of the differential equation $$ \dfrac {d^2y}{dx^3}- x^2 \dfrac {dy}{dx}- 2xy =1$$ under the conditions y(0)=1, y'(0)=0. Take step length h=0.1.(7 marks)
**2 (c)** Using the Milne's method, obtain an approximate solution at the point x=0.4 of the problem $$ \dfrac {d^2y}{dx^2}+3x \dfrac {dy}{dx}-6y=0, \ y(0)=1, \ y'(0)=0.1 $$ Given y(0.1)=1.03995, y'(0.1)=0.6955, y(0.2)=1.138036, y'(0.2)=1.258, y(0.3)=1.29865, y'(0.3)=1.873.(7 marks)
**3 (a)** If f(z)=u+iv is an analytic function then prove that $$ \left [ \dfrac {\partial }{\partial x} |f(z)| \right ]^2 + \left [ \dfrac {\partial }{\partial y} |f(z)| \right ]^2 =|f'(z)|^2 $$(6 marks)
**3 (b)** Find an analytic function whose imaginary part is v=e^{x}{(x^{2}-y^{2}) cos y-2xy sin y}(7 marks)
**3 (c)** If f(z)=u(r,?)+ iv(r,?) is an analytic function, show that u and v satisfy the equation $$ \dfrac {\partial^2 \varphi}{\partial r^2}+ \dfrac {1}{r} \dfrac {\partial \varphi}{ \partial r}+ \dfrac {1}{r^2} \dfrac {\partial^2 \varphi}{\partial \theta^2}=0 $$(7 marks)
**4 (a)** Find the bilinear transformation that maps the points 1, i, -1 onto the point i, 0, -i respectively.(6 marks)
**4 (b)** Discuss the transformation W=e^{x}.(7 marks)
**4 (c)** Evaluate $$ \int_C \dfrac {\sin \pi z^2 + \cos \pi z^3}{(z-1)^2 (z-2)}dz $$ where C is the circle |z|=3.(7 marks)
**5 (a)** Express the polynomial 2x^{3}-x^{2}-3x+2 in terms of Legendre polynomials.(6 marks)
**5 (b)** Obtain the series solution of Bessel's differential equation $$ x^2 \dfrac {d^2 y}{dx^2}+ x \dfrac {dy}{dx}+ (x^2-n^2)y=0 $$ in the form y=AJ_{n}(x) + BJ_{-n}(x).(7 marks)
**5 (c)** Derive Rodrique's formula $$ P_n(x)= \dfrac {1}{2^n n!} \dfrac {d^n}{dx^n}(x^2-1)^n. $$(7 marks)
**6 (a)** State the axioms of probability. For any two events A and B, Prove that P(A?B)=P(A)+P(B)-P(A?B).(6 marks)
**6 (b)** A bag contains 10 white balls and 3 red ball while another bag contains 3 white balls and 5 red balls. Two balls are drawn at random from the first bag and put in the second bag and then a ball is drawn at random from the second bag. What is the probability that it is a white ball?(7 marks)
**6 (c)** In a bolt factory there are four machines A, B, C, D manufacturing respectively 20%, 15%, 25%, 40% of the total production. Out of these 5%, 4%, 3% and 2% respectively are defective. A bolt is drawn at random from the production and is found to be defective. Find the probability that it was manufactured by A or D.(7 marks)
**7 (a)** The probability distribution of finite random variable x is given by the following table:

x_{i} | -2 | -1 | 0 | 1 | 2 | 3 |

p(x_{i}) | 0.1 | k | 0.2 | 2k | 0.3 | k |

Determine the value of k and find the mean, variance and standard deviation.(6 marks)

**7 (b)**The probability that a pen manufactured by a company will be defective is 0.1. If 12 such pens are selected, find the probability that (i) exactly 2 will be defective. (ii) at least 2 will be defective, (iii) none will be defective.(7 marks)

**7 (c)**In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and standard deviation, given that A(0.5)=0.19 and A(1.4)=0.42, where A(z) is the area under the standard normal curve from O to z>0.(7 marks)

**8 (a)**A biased coin is tossed 500 times and head turns up 120 times. Find the 95% confidence limits for the propertion of heads turning up in infinitely many tosses. (Given that z

_{c}=1.96)(6 marks)

**8 (b)**A certain stimulus administered to each of 12 patients resulted in the following changes in blood pressure;

5, 2, 8, -1, 3, 0, 6, -2, 1, 5, 0, 4 (in appropriate unit)

Can it be conclude that, on the whole, the stimulus will change the blood pressure. Use t

_{0.05}(11)=2.201.(7 marks)

**8 (c)**A die is thrown 60 times and the frequency distribution for the number appearing on the face x is given by the following table:

Test the hypothesis that the die is unbiased. (Given that x

^{2}

_{0.05}(5)=11.07 and x

^{2}

_{0.01}(5)=15.09)(7 marks)