BT-401 Mathematics - III - Jun 2022 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aUse Newton's formula for interpolation to find the net premium at the age 25 from the table given below: Age 20 24 28 32 Net Premium 0.01427 0.01581 0.01772 0.01996
   How to answer

   • State Newton's forward/backward or divided-difference interpolation formula
   • Construct difference table from given Age vs Net Premium values
   • Substitute x=25 and compute interpolated value
   • Obtain net premium ~0.016 and check reasonableness
2. 7 Marks
   bUsing Newton-Raphson's method find the real root of $x^4 - x - 10 = 0$.
   How to answer

   • Write $f(x)=x^4-x-10$ and $f'(x)=4x^3-1$
   • Choose initial approximation by sign change
   • Apply iteration $x_{n+1}=x_n-f(x_n)/f'(x_n)$
   • Iterate to convergence and report real root
3. 7 Marks
   aSolve the simultaneous linear equations using Crout's method. $$x_1 + x_2 + x_3 = 1$$ $$3x_1 + x_2 - 3x_3 = 5$$ $$x_1 - 2x_2 - 5x_3 = 10$$
   How to answer

   • Write system $AX=B$ and set $A=LU$ with Crout's form
   • Compute L and U elements by Crout's formulae
   • Solve $LY=B$ for Y by forward substitution
   • Solve $UX=Y$ for $x_1,x_2,x_3$ by back substitution
4. 7 Marks
   bEvaluate $\int_0^1 \log x \cos x \, dx$ by (i) Trapezoidal rule (ii) Simpson 3/8 rule.
   How to answer

   • Define $f(x)=\log x \cos x$ on $[0,1]$ with suitable limit at 0
   • Apply Trapezoidal rule with chosen subintervals
   • Apply Simpson's 3/8 rule with $n$ multiple of 3
   • Compare/present both numerical values of integral
5. 7 Marks
   aFind $y(0.1)$ for differential equation $\frac{dy}{dx} = x^2 y - 1$, $y(0) = 1$ using Taylor's series method.
   How to answer

   • Write $y' = x^2y-1$, $y(0)=1$ and Taylor expansion at $x_0=0$
   • Compute $y', y'', y'''$ etc. at $x=0$
   • Substitute $h=0.1$ in Taylor series
   • Obtain $y(0.1)$ to required accuracy
6. 7 Marks
   bSolve $xy'' + y' + xy = 0$, where $y(0) = 1$, $y'(0) = 1$, for $x = 0$ to $x = 1.5$.
   How to answer

   • Identify second order ODE $xy''+y'+xy=0$ with $y(0)=1, y'(0)=1$
   • Assume power series / numerical stepping method
   • Compute coefficients / successive values
   • Extend solution from $x=0$ to $x=1.5$
7. 7 Marks
   aFind Laplace transform of $f(t) = \begin{cases} \sin t & 0 < t < 2\pi \\ 0 & 2\pi < t \end{cases}$
   How to answer

   • Write definition $L\{f(t)\}=\int_0^\infty e^{-st}f(t)dt$
   • Split integral $0$ to $2\pi$ and $2\pi$ to $\infty$ using given $f(t)$
   • Evaluate $\int_0^{2\pi} e^{-st}\sin t dt$
   • Simplify to $(1-e^{-2\pi s})/(s^2+1)$
8. 7 Marks
   bFind the Fourier series for periodic extension of $f(t) = \begin{cases} \sin t, & 0 \le t \le \pi \\ 0, & \pi \le t \le 2\pi \end{cases}$
   How to answer

   • Identify period $2\pi$ and piecewise definition on $[0,2\pi]$
   • Compute Fourier coefficients $a_0, a_n, b_n$ by integration
   • Use $\int_0^{\pi}\sin t \cos nt dt$ and $\int_0^{\pi}\sin t \sin nt dt$
   • Write full Fourier series and periodic extension
9. 7 Marks
   aIf '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 $$\frac{1}{2}\left[\frac{(b+a)^m-(b-a)^m}{(b+a)^m}\right]$$
   How to answer

   • Model each ball independently given to men with probability $a/(a+b)$
   • Probability men get $k$ balls is binomial $C(m,k)p^k q^{m-k}$
   • Sum over odd $k$ using $(p+q)^m$ and $(q-p)^m$
   • Simplify to $\frac12[1-(b-a)^m/(b+a)^m]$
10. 7 Marks
    bTwo 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 $P[Z + 1 \le 0]$.
    How to answer

    • Use linear combination of independent normals: mean $1-2=-1$, variance $9+16=25$
    • Write pdf of $N(-1,25)$
    • State median = mean = -1, s.d. = 5
    • Compute $P[Z+1\le 0]=P[Z\le -1]=0.5$
11. 7 Marks
    aProve that $$\Delta^n O^{n+1} = \frac{n(n+1)}{2} \Delta^n O^n$$
    How to answer

    • Recall $\Delta$ and $E$, $O$ operators and relations
    • Express $O^{n+1}$ in terms of $\Delta$
    • Apply $\Delta^n$ on both sides
    • Simplify combinatorial factor to $n(n+1)/2$
12. 7 Marks
    bGiven that $\log x$ for $x = 310, 320, 330, 340, 350$ and $360$ are according to following table. Find the value of $\log 3375$. $x$ 310 320 330 340 350 360 $\log x$ 2.4913617 2.5051500 2.5185139 2.5314789 3.5440680 2.5563025
    How to answer

    • Tabulate $x=310$ to $360$ and given $\log x$ values
    • Choose $x_0$ near 337.5 noting argument is 3375 = 10×337.5
    • Apply Newton's forward/backward interpolation
    • Evaluate and shift decimal to get $\log 3375$
13. 7 Marks
    aUse Runge-Kutta method to approximate $y$, when $x = 0.1$ and $x = 0.2$, given that $x = 0$ when $y = 1$ and $\frac{dy}{dx} = x + y$.
    How to answer

    • Write $f(x,y)=x+y$, $x_0=0, y_0=1$, $h=0.1$
    • Compute $k_1,k_2,k_3,k_4$ for each step
    • Apply $y_{n+1}=y_n+(k_1+2k_2+2k_3+k_4)/6$
    • Obtain $y(0.1)$ then $y(0.2)$
14. 7 Marks
    bUse Milne's method to solve $\frac{dy}{dx} = x + y$ with initial condition $y(0) = 1$, from $x = 0.20$ to $x = 0.30$.
    How to answer

    • Use RK values at $0,0.1,0.2$ as starting values
    • Write Milne's predictor $y_{n+1}^p$ formula
    • Apply corrector $y_{n+1}^c$ with $f=x+y$
    • Iterate corrector to get $y(0.3)$ from $x=0.20$
15. 7 Marks
    aProve that for normal distribution, the Quartile Deviation (QD), Mean Deviation (MD) and Standard Deviation (SD) follows QD : MD : SD :: 10 : 12 : 15.
    How to answer

    • Write normal pdf and definitions of QD, MD, SD
    • Evaluate $QD=0.6745\sigma$, $MD=\sigma\sqrt{2/\pi}$
    • Compute ratios $QD:MD:SD$
    • Show approximately $10:12:15$
16. 7 Marks
    bProve that $$\Delta^n \sin(ax+b) = \left(2\sin\frac{ah}{2}\right)^n \sin\left[ax+b+n\left(\frac{ah+\pi}{2}\right)\right]$$
    How to answer

    • Recall $\Delta f(x)=f(x+h)-f(x)$ and $E$ operator
    • Compute $\Delta \sin(ax+b)=2\sin(ah/2)\sin[ax+b+(ah+\pi)/2]$
    • Apply induction on $n$ for $\Delta^n$
    • Obtain factor $(2\sin ah/2)^n$ with phase shift