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BT-202 · Mathematics II/Unsolved PYQ Paper

BT-202 Mathematics - II - Jun 2022 Question Paper

  1. 7 Marks
    aSolve $\left(1+e^{\frac{x}{y}}\right)dx + e^{\frac{x}{y}}\left(1-\frac{x}{y}\right)dy = 0$.
  2. 7 Marks
    bSolve the Linear differential equation $$\left(1+y^{2}\right) + \left(x - e^{-\tan^{-1}y}\right)\frac{dy}{dx} = 0$$
  3. 7 Marks
    aSolve $\frac{dy}{dx} + y \tan x = y^{2}\sec x$ using Bernoulli's.
  4. 7 Marks
    bSolve the differential equation $$(D^{2} - 2D - 3)y = x^{3}e^{-3x}.$$
  5. 14 Marks
    Solve $(D^{2} + a^{2})y = \tan ax$ by using method of variation of parameters.
  6. 7 Marks
    aFind the angle between the surfaces $x^{2} + y^{2} + z^{2} = 9$ and $Z = x^{2} + y^{2} - 3$ at the point $(2, -1, 2)$.
  7. 7 Marks
    bProve that $r^{n}\overline{r}$ is Solenoidal if $n = -3$.
  8. 14 Marks
    Verify Gauss divergence theorem for $\overline{F} = x^{2}\overline{i} + y^{2}\overline{j} + z^{2}\overline{k}$ over the cube formed by the planes $x = 0, x = a, y = 0, y = b, z = 0, z = c$.
  9. 7 Marks
    aProve that $\left[\frac{d^{2}}{dx^{2}} + \frac{d^{2}}{dy^{2}}\right] |f(z)|^{2} = 4|f'(z)|^{2}$.
  10. 7 Marks
    bConstruct the analytic function $f(z)$, whose real part is $e^{x}\cos y$.
  11. 7 Marks
    aUsing Cauchy's integral formula, find $\int_{C} \frac{e^{2z}}{(z+1)^{3}} dz$ where $C$ is the curve $|z| = 2$.
  12. 7 Marks
    bEvaluate $\int_{C} \frac{1}{(z+4)z^{8}} dz$ where $C$ is the circle $|z| = 2$.
  13. 7 Marks
    aForm the partial differential equation by eliminating arbitrary function from $Z = f(y/x)$.
  14. 7 Marks
    bSolve the $(D^{3} - 3D^{2}D' + 4{D'}^{3})z = e^{x+2y}$.
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