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BT-202 · Mathematics II/Unsolved PYQ Paper

BT-202 Mathematics - II - Jun 2025 Question Paper

  1. 7 Marks
    aSolve $(e^y + 1)\cos x\,dx + e^y\sin x\,dy = 0$.
  2. 7 Marks
    bSolve $(D^2 - 5D + 6)y = 4e^x + 5$.
  3. 7 Marks
    aSolve $x^2 \frac{d^2 y}{dx^2} + 5x \frac{dy}{dx} + 4y = x \log x$.
  4. 7 Marks
    bSolve in series Legendre's differential equation $$(1 - x^2)\frac{d^2 y}{dx^2} - 2x\frac{dy}{dx} + 2y = 0.$$
  5. 14 Marks
    Solve $(D^2 + a^2)y = \tan ax$ by using method of variation of parameters.
  6. 7 Marks
    aEliminate the arbitrary function $f$ from the relation $$z = y^2 + 2f\left(\frac{1}{x} + \log y\right).$$
  7. 7 Marks
    bSolve the partial differential equation $$\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial y^2} = x^2 y.$$
  8. 7 Marks
    aSolve $(y+z)p + (x+z)q = x+y$.
  9. 7 Marks
    bFind all values of $K$ such that $f(z) = e^x(\cos ky + i\sin ky)$ is analytic.
  10. 7 Marks
    aEvaluate $\int_{(0,0)}^{(1,1)} (3x^2 + 4xy + ix^2)\,dz$ along $y = x^2$.
  11. 7 Marks
    bIf $f(z) = \frac{1}{(z-1)(z-2)^4}$, find residue of all poles.
  12. 7 Marks
    aIf $\bar{r}$ is the position vector of any point in space, then prove that $r^n \bar{r}$ is irrotational.
  13. 7 Marks
    bFind the workdone by the force $\bar{F} = z\bar{i} + x\bar{j} + y\bar{k}$, when it moves a particle along the arc of the curve $\bar{r} = \cos t\,\bar{i} + \sin t\,\bar{j} - t\bar{k}$ from $t = 0$ to $t = 2\pi$.
  14. 14 Marks
    Verify stokes theorem for $\bar{F} = (x^2 - y^2)\bar{i} + 2xy\bar{j}$ over the box bounded by the planes $x = 0$, $x = a$, $y = 0$, $y = b$.
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