a) Find the real root of equation $f(x) = x^3 - x - 10$ by using Newton Raphson method correct up to 4 decimal places. (Unit 1)

b) Solve the given set of equations by using Gauss elimination method:
$x + y + z = 4$
$x + 4y + 3z = 8$
$x + 6y + 2z = 6$ (Unit 2)

a) Using modified Euler's method, find an approximate value of $Y$, when $x = 0.3$, Given that $\frac{dy}{dx} = x + y$ and $y = 1$, when $x = 0$. (Unit 3)

b) Consider an ordinary differential equation $\frac{dy}{dx} = x^2 + y^2, y(1) = 1.2$. Find $y(1.05)$ using the fourth order Runge-Kutta method. (Unit 3)

a) Find the inverse Laplace transform of the following functions.
$$F(s) = \frac{1}{(s+1)(s^2+1)}$$ (Unit 4)

b) Suppose that the function $y(t)$ satisfies the DE $y'' - 2y' - y = 1$, with initial values, $y(0) = -1, y'(0) = 1$. Find the laplace transform of $y(t)$. (Unit 4)

a) A device is used to measure the speed of cars on a highway. The speed is normally distributed with a 90 km/hr mean and a standard deviation of 10. What is the probability that the car picked was travelling at more than 100 km/hr? (Unit 5)

b) The mean and the standard deviation of a binomial distribution are 100 and 5. Find the binomial distribution. (Unit 5)

a) In a normal distribution, 10.03% of the items are under 25 kg. weight and 89.97% of the items are under 70 kg. weight. What are the mean and standard deviation of the distribution? (Unit 5)

a) Find the Laplace transform of the following function
i) $L(t^2 e^{-t} \cosh 2t)$
ii) $L(e^{3t} \sin^2 4t)$ (Unit 4)

b) Using Milne's method find $y(4.4.)$ given $5xy + y^2 - 2 = 0$
Given $y(4) = 1, y(4.1) = 1.0049, y(4.2) = 1.0097$ and $y(4.3) = 1.0143$. (Unit 3)

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

b) Write short note on Exponential distribution. (Unit 5)