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BT-401 · Mathematics-III/Unsolved PYQ Paper

BT-401 Mathematics III - Nov 2023 Question Paper

  1. 7 Marks
    aEvaluate $\sqrt{12}$ to four decimal places by Newton Raphson Method.
  2. 7 Marks
    bWhat is the rate of convergence of bisection method?
  3. 7 Marks
    aProve that i) $\Delta = \frac{1}{2}\delta^{2} + \delta\sqrt{1+\left(\frac{\delta^{2}}{4}\right)}$ ii) $\Delta + \nabla = \frac{\Delta}{\nabla} - \frac{\nabla}{\Delta}$
  4. 7 Marks
    bConstruct a backward difference table for $y = \log x$ given that | $x$ | 10 | 20 | 30 | 40 | 50 | | $y$ | 1 | 1.3010 | 1.4771 | 1.6021 | 1.6990 | And find values of $\nabla^{3} \log 40$ and $\nabla^{4} \log 50$
  5. 7 Marks
    aUsing following table, by Lagrange's Method find $f(x)$ as a polynomial in $x$:
  6. 7 Marks
    bUsing Newton's divided difference formula, calculate the value of $f(6)$ from the following data:
  7. 7 Marks
    aFind $f'(1.1)$ and $f''(1.1)$ from the following table:
  8. 7 Marks
    bFind missing values in the following table:
  9. 7 Marks
    aFind $\int_{0}^{6} \frac{e^{x}}{1+x} dx$ approximately using simpson's $\frac{3}{8}$th rule on integration.
  10. 7 Marks
    bSolve the equation $\frac{dy}{dx} = 1 - y$ with initial condition $y(0) = 0$ using Euler's modified method and tabulate the solution at $x = 0.1, 0.2, 0.3$.
  11. 7 Marks
    aUsing Lagrange's interpolation formula to fit a polynomial to the data:
  12. 7 Marks
    bUsing Regula-Falsi method, find the real root of $x \log_{10} x = 1.2$, correct to four decimal places.
  13. 7 Marks
    aSolve $\frac{dy}{dx} = x + y^{2}$ by using Runge-Kutta method of fourth order to find an approximate value of $y$ for $x = 0.2$, given that $y = 1$ when $x = 0$. (Take $h = 0.1$)
  14. 7 Marks
    bEvaluate $$L^{-1}\left[\frac{s+7}{s^{2}+4s+8}\right]$$
  15. 7 Marks
    aFind the inverse Laplace transform of $\frac{s+4}{s(s-1)(s^{2}+4)}$.
  16. 7 Marks
    bProve that if $L\{f(t)\} = F(s)$ then $L\left\{\frac{1}{t}f(t)\right\} = \int_{s}^{\infty} F(s) ds$ hence, evaluate $\frac{e^{at} - \cos bt}{t}$.
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