a) If the equation $f(x)$ is given as $x^3 - x^2 + 4x - 4 = 0$. Considering the initial approximation at $x = 2$ then find the value of next approximation correct upto 2 decimal places. (Unit 1)

b) What is the rate of convergence of Newton Raphson method? (Unit 1)

a) Find
i) $\Delta e^{ax}$
ii) $\Delta^2 e^x$
iii) $\Delta \log x$ (Unit 1)

b) Prove that $f(4) = f(3) + \Delta f(2) + \Delta^2 f(1) + \Delta^3 f(1)$ taking '1' as the interval of differencing. (Unit 1)

a) Using Simpson's 3/8 rule to solve the integral $\int_4^{5.2} \log_e x dx$ (Unit 2)

a) Apply Gauss Elimination method to solve the following equations:
$2x - y + 3z = 9$
$x + y + z = 6$
$x - y + z = 2$ (Unit 2)

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)

b) Find the real root of the equation $x \log_{10} x - 1.2 = 0$ correct to five places of decimal by Regula Falsi Method. (Unit 1)

a) Find the inverse Laplace transform of $\frac{2s+1}{s(s-1)}$. (Unit 4)

b) State convolution theorem and hence evaluate $L^{-1} \left[ \frac{s}{(s^2+1)(s^2+4)} \right]$. (Unit 4)

a) A binomial variable X satisfies the relation $9P(X = 4) = P(X = 2)$ when $n = 6$. Find the value of the parameter $p$ and $P(X = 1)$. (Unit 5)

b) A large number of measurement is normally distributed with a mean 65.5" and S.D. of 6.2". Find the percentage of measurements that fall between 54.8" and 68.8". (Unit 5)