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

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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.

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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.

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2 (a)
Exmploying the Picard's method, obtain the second order approximate solution of the following problem at x=0.2. \[ \dfrac {dy}{dx}=x+yz, \ \dfrac {dz}{dx}= y+zx, \ y(0)=1, \ z(0)=-1 \]

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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.

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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.

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3 (a)
If f(z)=u+iv is an analytic function then prove that \[ \left [ \dfrac {\partial }{\partial x} |f(z)| \right ]^2 + \left [ \dfrac {\partial }{\partial y} |f(z)| \right ]^2 =|f'(z)|^2 \]

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3 (b)
Find an analytic function whose imaginary part is v=e

^{x}{(x^{2}-y^{2}) cos y-2xy sin y}
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3 (c)
If f(z)=u(r,?)+ iv(r,?) is an analytic function, show that u and v satisfy the equation \[ \dfrac {\partial^2 \varphi}{\partial r^2}+ \dfrac {1}{r} \dfrac {\partial \varphi}{ \partial r}+ \dfrac {1}{r^2} \dfrac {\partial^2 \varphi}{\partial \theta^2}=0 \]

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4 (a)
Find the bilinear transformation that maps the points 1, i, -1 onto the point i, 0, -i respectively.

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4 (b)
Discuss the transformation W=e

^{x}.
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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.

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5 (a)
Express the polynomial 2x

^{3}-x^{2}-3x+2 in terms of Legendre polynomials.
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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).
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5 (c)
Derive Rodrique's formula \[ P_n(x)= \dfrac {1}{2^n n!} \dfrac {d^n}{dx^n}(x^2-1)^n. \]

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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).

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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?

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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.

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7 (a)
The probability distribution of finite random variable x is given by the following table:

Determine the value of k and find the mean, variance and standard deviation.

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.

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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.

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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.

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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)
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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

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.
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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

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)
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