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BT-401 · Mathematics-III/Unsolved PYQ Paper

BT-401 Mathematics - III - Jun 2022 Question Paper

  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
  2. 7 Marks
    bUsing Newton-Raphson's method find the real root of $x^4 - x - 10 = 0$.
  3. 7 Marks
    aSolve the simultaneous linear equations using Crout's method. $$x_1 + x_2 + x_3 = 1$$ $$3x_1 + x_2 - 3x_3 = 5$$ $$x_1 - 2x_2 - 5x_3 = 10$$
  4. 7 Marks
    bEvaluate $\int_0^1 \log x \cos x \, dx$ by (i) Trapezoidal rule (ii) Simpson 3/8 rule.
  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.
  6. 7 Marks
    bSolve $xy'' + y' + xy = 0$, where $y(0) = 1$, $y'(0) = 1$, for $x = 0$ to $x = 1.5$.
  7. 7 Marks
    aFind Laplace transform of $f(t) = \begin{cases} \sin t & 0 < t < 2\pi \\ 0 & 2\pi < t \end{cases}$
  8. 7 Marks
    bFind the Fourier series for periodic extension of $f(t) = \begin{cases} \sin t, & 0 \le t \le \pi \\ 0, & \pi \le t \le 2\pi \end{cases}$
  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 $$\frac{1}{2}\left[\frac{(b+a)^m-(b-a)^m}{(b+a)^m}\right]$$
  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 $P[Z + 1 \le 0]$.
  11. 7 Marks
    aProve that $$\Delta^n O^{n+1} = \frac{n(n+1)}{2} \Delta^n O^n$$
  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
  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 $\frac{dy}{dx} = x + y$.
  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 $x = 0.30$.
  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.
  16. 7 Marks
    bProve that $$\Delta^n \sin(ax+b) = \left(2\sin\frac{ah}{2}\right)^n \sin\left[ax+b+n\left(\frac{ah+\pi}{2}\right)\right]$$
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