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BT-301 · Mathematics-III/Unsolved PYQ Paper

BT-301 Mathematics - III - Jun 2025 Question Paper

  1. 7 Marks
    aIf the equation $f(x)$ is given as $x^3 - 2x - 5 = 0$. Considering the initial approximation at $x = 2$ then find the value of next approximation correct upto 2 decimal places.
  2. 7 Marks
    bWhat is the rate of convergence of Newton Raphson method?
  3. 7 Marks
    aFind i) $\Delta \tan x$ ii) $\Delta^2 e^{2x}$ iii) $\Delta \log x$
  4. 7 Marks
    bProve that $f(4) = f(3) + \Delta f(2) + \Delta^2 f(1) + \Delta^3 f(1)$ taking '1' as the interval of differencing.
  5. 7 Marks
    aUsing following table, by Lagrange's Method find $f(x)$ as a polynomial in $x$: $$ \begin{array}{|c|c|c|c|c|c|} \hline x & -1 & 0 & 3 & 6 & 7 \\ \hline f(x) & 3 & -6 & 39 & 822 & 1611 \\ \hline \end{array} $$
  6. 7 Marks
    bFind $f(3.5)$ from the following table using Newton's Forward interpolation formula. $$ \begin{array}{|c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 \\ \hline f(x) & 2.625 & 3.454 & 4.784 & 6.986 \\ \hline \end{array} $$
  7. 7 Marks
    aApply Gauss Elimination method to solve the following equations. $$\begin{aligned} 2x - y - 3z &= 9 \\ x - y - 2z &= 6 \\ x - y - z &= 2 \end{aligned}$$
  8. 7 Marks
    bFind missing values in the following table $$ \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 5 & 10 & 15 & 20 & 25 \\ \hline y & 6 & 10 & - & 17 & - & 31 \\ \hline \end{array} $$
  9. 7 Marks
    aFind $\int_0^6 \frac{e^x}{1 + x} \, dx$ approximately using Simpson's $3/8^{\text{th}}$ 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: $$ \begin{array}{|c|c|c|c|c|} \hline x & -1 & 0 & 2 & 3 \\ \hline f(x) & -8 & 3 & 1 & 12 \\ \hline \end{array} $$
  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
    aFind the inverse Laplace transform of $\frac{4s + 12}{s^2 + 8s + 16}$.
  14. 7 Marks
    bState convolution theorem and hence evaluate $$L^{-1}\left[\frac{1}{s(s^2 - a^2)}\right]$$
  15. 7 Marks
    aA binomial variable $X$ satisfies the relation $9P(X = 4) = P(X = 2)$ when $n = 6$. Find the value of the parameter $p$ and $P(X = 1)$.
  16. 7 Marks
    bA manufacturer knows from experience that the resistance of resistors he produces is with mean $\mu = 100\Omega$ and s.d. $\sigma = 2\,\Omega$. What percentage of resistors will have resistance between $98\Omega$ and $102\Omega$?
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