a) Find the real root of the equation $x^3 - 3x - 5 = 0$ correct to four places of decimals by Newton Raphson method. (Unit 1)

b) Evaluate
$\Delta^6 (ax - 1)(bx^2 - 1)(cx^3 - 1); h=1$ (Unit 1)

b) Find the real root of the equation $xe^x - 3 = 0$ correct to three decimal places by Regula Falsi method. (Unit 1)

a) Use Simpson's 1/3 and 3/8 rule to evaluate the following.
$\int_0^1 \frac{dx}{1+x^2}$
Hence obtain the approximate value of $\pi$ in each case. (Unit 2)

b) Compute the value of the following integral by trapezoidal Rule.
$\int_{0.2}^{1.4} (\sin x - \log_e x + e^x) dx$ (Unit 2)

a) Solve the following system by Gauss Siedel iteration method.
$27x + 6y - z = 85$
$6x + 15y + 2z = 72$
$x + y + 54z = 110$ (Unit 2)

b) Use Simpson's rule to find approximate value of the following integral.
$\int_1^2 \frac{dx}{x}$ (Unit 2)

a) Use Euler's method compute the solution of
$\frac{dy}{dx} = y^2 - x^2$ with $y(0) = 1$ at $x = 0.1$. (Unit 3)

b) Use Euler's Modified method to solve the equation
$\frac{dy}{dx} = x + y$, $y(0) = 1$, $h = 0.2$ compute $y(1)$ (Unit 3)

a) Use Laplace transform to solve the Initial Value Problem.
$y' + 2y = 26 \sin 3t, y(0) = 3$ (Unit 4)

b) Evaluate
$L(t \cos at)$ (Unit 4)

a) Evaluate
$L^{-1} \left[ \frac{1}{(1+s)^3} \right]$ (Unit 4)

b) Evaluate
$L \{ t^2 e^{-3t} \}$ (Unit 4)

a) Find mean of Poisson distribution. (Unit 5)

b) Write Short note on Exponential Distribution. (Unit 5)