a) Using Newton-Raphson method find the real root of the equation $3x = \cos x + 1$ correct to five decimal places. (Unit 1)

a) Apply the Simpson's $\frac{3}{8}$ rule to evaluate the following integral for six digit $\int_{1.0}^{1.30} \sqrt{x} \cdot dx$. (Unit 2)

b) Solve the following system of equation using Gauss-Seidel method.
$27x + 6y - z = 85$
$6x + 15y + 2z = 72$
$x + y + 54z = 110$ (Unit 2)

a) Using Runge-Kutta method of fourth order solve
$\frac{dy}{dx} = \frac{y^2 - x^2}{y^2 + x^2}, y(0) = 1$ at $x = 0.4$ in step of $0.2$ (Unit 3)

b) Find $y(2.2)$ using Euler's method for $\frac{dy}{dx} = -xy^2$ where $y(2) = 1$. (Unit 3)

a) Find the Inverse Laplace transform of $\frac{1}{p^3 (p^2 + a^2)}$. (Unit 4)

b) Using Laplace transform solve the differential equation
$\frac{d^3y}{dt^3} + 2 \frac{d^2y}{dt^2} - \frac{dy}{dt} - 2y = 0$
Where $y = 1, \frac{dy}{dt} = 2, \frac{d^2y}{dt^2} = 2$ at $t = 0$. (Unit 4)

b) Calculate the mean and standard deviation of Binomial Distribution. (Unit 5)

b) Evaluate $\int_0^\infty \frac{e^{-t} \cdot \sin t}{t} dt$ (Unit 4)

a) Show that Newton Raphson method is quadratic convergent. (Unit 1)

a) Solve the system of equations
$3x + y - z = 3$
$2x - 8y + z = -5$
$x - 2y + 9z = 8$
Using Gauss elimination method. (Unit 2)

b) Find Fourier sine transform of $\frac{1}{x}$. (Unit 4)