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BT-202 · Mathematics II/Unsolved PYQ Paper

BT-202 Mathematics - II - Nov 2022 Question Paper

  1. 7 Marks
    aSolve $\cos x\,dy = y(\sin x - y)\,dx$ using Bernoulli's.
  2. 7 Marks
    bSolve the Linear differential equation $$\sin 2x\frac{dy}{dx} - y = \tan x$$
  3. 7 Marks
    aSolve $(r + \sin\theta - \cos\theta)dr + r(\sin\theta + \cos\theta)d\theta = 0$.
  4. 7 Marks
    bSolve the differential equation. $$(D^3 - 7D^2 + 14D - 8)y = e^x \cos 2x$$
  5. 14 Marks
    Solve $(D^2 + 4)y = \tan 2x$ by using method of variation of parameters.
  6. 7 Marks
    aShow that $\frac{\vec{r}}{r^3}$ is solenoidal.
  7. 7 Marks
    bShow that the vector $$(x^2 - yz)\hat{i} + (y^2 - zx)\hat{j} + (z^2 - xy)\hat{k}$$ is Irrotational. Find it's scalar potential.
  8. 14 Marks
    Verify Green's theorem for $\oint_C [(3x^2 - 8y^2)\,dx + (4y - 6xy)\,dy]$, Where $C$ is the region bounded by $x = 0, y = 0$ and $x + y = 1$.
  9. 7 Marks
    aShow that $f(Z) = z\bar{z}$ is differentiable but not analytic at origin.
  10. 7 Marks
    bShow that $u(x,y) = e^{-2x}\sin 2y$ is harmonic and determine it's Harmonic conjugate.
  11. 7 Marks
    aBy Residue theorem, Evaluate $\oint_C \frac{\tan z}{z^2 - 1}\,dz$, where $C:|Z|=2$.
  12. 7 Marks
    bUsing Cauchy integral theorem, to evaluate the integral $$\int_C \frac{e^{2z}}{(z-1)^2(z-3)}\,dz$$, where $C$ is the circle $|Z|=2$.
  13. 7 Marks
    aSolve $x^2 p^2 + y^2 q^2 = z^2$.
  14. 7 Marks
    bSolve $(D^2 - 4DD' + 4{D'}^2)Z = \cos(x - 2y)$
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