BT-401 (GS) – Mathematics-III

Use Newton's formula for interpolation to find the net premium at the age 25 from the table given below:

Using Newton-Raphson's method find th rea root of $x^3 -x=10=0$.

Solve the simultaneous linear equations using Crouts method. $x_1 +x_2 +x_3 =1$ $3x_1 +x_2 -3x_3 =5$ $x_1 -2x_2 -x_3 =10$

Evaluate $\int_1^2 \log x \cos x \, dx$ by (i) Trapezoidal rule (ii) Simpson 3/8 rule.

Find $y(0.1)$ for differential equation $y' = x^2y-1$, $y(0)=1$ using Taylor’s series method.

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

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

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

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 $\frac{1}{2} \frac{(b+a)^m -(b-a)^m}{(b+a)^m}$

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 < 0].

Prove that $\Delta \log f(x) = \log \left[ 1 + \frac{\Delta f(x)}{f(x)} \right]$

Given that log for x = 310, 320, 330, 340, 350 and 360 are according to following table. Find the value of log 337.5.

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

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.

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

Prove that $\Delta^n f(x) = \sum_{k=0}^n (-1)^{n-k} \binom{n}{k} f(x+kh)$