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BT-202 · Mathematics II/Unsolved PYQ Paper

BT-202 Mathematics - II - Dec 2024 Question Paper

  1. 7 Marks
    aSolve $(1+y^2)dx = (\tan^{-1}y - x)dy$.
  2. 7 Marks
    bSolve $(D^2+3D+2)y = \sin 3x$.
  3. 7 Marks
    aSolve the simultaneous equations $\frac{dx}{dt}-7x+y=0$ and $\frac{dy}{dt}-2x-5y=0$.
  4. 7 Marks
    bSolve by the method of variation of parameter $(D^2+1)y = x$.
  5. 7 Marks
    aSolve $(1+x)^2\frac{d^2y}{dx^2}+(1+x)\frac{dy}{dx}+y = \cos\log(1+x)$.
  6. 7 Marks
    bShow that $J_n(-x) = (-1)^n J_n(x)$ when $n$ is positive or negative integer.
  7. 7 Marks
    aSolve by Charpit's method $px+qy = pq$.
  8. 7 Marks
    bSolve the Partial differential equation $\left(D^3-4D^2D'+4DD'^{2}\right)Z = \cos(2x+y)$.
  9. 7 Marks
    aConstruct a partial differential equation from the relation $f\left(x^2+y^2+z^2, z^2-2xy\right) = 0$
  10. 7 Marks
    bShow that $u = e^{-x}(x\sin y - y\cos y)$ is Harmonic.
  11. 7 Marks
    aDetermine P such that the function $f(z) = \frac{1}{2}\log(x^2+y^2) + i\tan^{-1}\left(\frac{px}{y}\right)$ be an analytic function.
  12. 7 Marks
    bEvaluate using Cauchy's theorem $\int_{c}\frac{z^3 e^{-z}}{(z-1)^3}dz$ where $c$ is $|z-1| = \frac{1}{2}$.
  13. 7 Marks
    aFind the poles and residues at each pole of $\frac{e^{iz}}{z^2+1}$.
  14. 7 Marks
    bFind the directional derivative of $\varnothing = x^2yz + 4xz^2$ at $(1,-2,-1)$ in the direction of $2\bar{i}-\bar{j}-2\bar{k}$.
  15. 14 Marks
    Verify Green's theorem for $\int_{C}\left[(xy+y^2)dx + x^2dy\right]$ where C is the boundary by $y = x$ and $y = x^2$.
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