• Attempt any five questions.
• All questions carry equal marks.
Use Newton's formula for interpolation to find the net premium at the age 25 from the given table.
Using Newton-Raphson's method find the real root of x⁴ - x - 10 = 0.
Solve the simultaneous linear equations using Crout's method: x₁+x₂+x₃=1, 3x₁+x₂-3x₃=5, x₁-2x₂-5x₃=10.
Evaluate ∫₀¹ log x cos x dx by (i) Trapezoidal rule (ii) Simpson 3/8 rule.
Find y(0.1) for differential equation dy/dx = x²y - 1, y(0) = 1 using Taylor's series method.
Solve xy'' + y' + xy = 0, where y(0) = 1, y'(0) = 0, for x = 0 to x = 1.5.
Find Laplace transform of piecewise function f(t) = sin t for 0 < t < 2π, and 0 for 2π < t.
Find the Fourier series for periodic extension of piecewise function f(t) = sin t for 0 ≤ t ≤ π, and 0 for π ≤ t ≤ 2π.
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 is (1/2)[((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 pdf of Z and find P[Z+1 ≤ 0].
Prove that Δⁿ 0^(n+1) = [n(n+1)/2] Δⁿ 0^n.
Given log x values for x = 310, 320, 330, 340, 350, 360, find the value of log 3375.
Use Runge-Kutta method to approximate y when x=0.1 and x=0.2, given x=0 when y=1 and dy/dx=x+y.
Use Milne's method to solve dy/dx = x + y with initial condition y(0)=1, from x=0.20 to x=0.30.
Prove that for normal distribution, QD:MD:SD :: 10:12:15.
Prove that Δⁿ sin(ax+b) = (2 sin(ah/2))^n sin[ax+b+n((ah+π)/2)].
