b) Using Newton-Raphson's method find the real root of $x^4 - x - 10 = 0$. (Unit 1)

a) Solve the simultaneous linear equations using Crout's method.
$x_1 + x_2 + x_3 = 1$
$3x_1 + x_2 - 3x_3 = 5$
$x_1 - 2x_2 - 5x_3 = 10$ (Unit 2)

b) Evaluate $\int_0^1 \log x \cos x \, dx$ by (i) Trapezoidal rule (ii) Simpson 3/8 rule. (Unit 2)

a) Find $y(0.1)$ for differential equation $\frac{dy}{dx} = x^2 y - 1, y(0) = 1$ using Taylor's series method. (Unit 3)

b) Solve $xy'' + y' + xy = 0$, where $y(0) = 1, y'(0) = 0$, for $x = 0$ to $x = 1.5$. (Unit 3)

a) Find Laplace transform of $$f(t) = \begin{cases} \sin t & 0 < t < 2\pi \\ 0 & 2\pi < t \end{cases}$$ (Unit 4)

b) Find the Fourier series for periodic extension of $$f(t) = \begin{cases} \sin t, & 0 \le t \le \pi \\ 0, & \pi \le t \le 2\pi \end{cases}$$ (Unit 4)

a) If 'm' balls are distributed among 'a' men and 'b' women show that the probability that the number of balls received by men is odd, shall be
$$ \frac{1}{2} \left[ \frac{(b+a)^m - (b-a)^m}{(b+a)^m} \right] $$ (Unit 5)

b) Two independent random variable X and Y are both normally distributed with means 1 and 2 and standard deviations 3 and 4 respectively. If Z = X - Y, write the probability density function of Z. Also state the median, s.d. and mean of the distribution of Z. Find P[Z + 1 $\le$ 0]. (Unit 5)

a) Prove that
$$ \Delta^n 0^{n+1} = \frac{n(n+1)}{2} \Delta^n 0^n $$ (Unit 1)

a) Use Runge-Kutta method to approximate $y$, when $x = 0.1$ and $x = 0.2$, given that $x = 0$ when $y = 1$ and $\frac{dy}{dx} = x + y$. (Unit 3)

b) Use Milne's method to solve $\frac{dy}{dx} = x + y$ with initial condition $y(0) = 1$, from $x = 0.20$ to $x = 0.30$. (Unit 3)

a) Prove that for normal distribution, the Quartile Deviation (QD), Mean Deviation (MD) and Standard Deviation (SD) follows QD : MD : SD :: 10 : 12 : 15. (Unit 5)

b) Prove that
$$ \Delta^n \sin(ax + b) = \left( 2 \sin \frac{ah}{2} \right)^n \sin \left[ ax + b + n \left( \frac{ah + \pi}{2} \right) \right] $$ (Unit 1)