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BT-202 · Mathematics II/Unsolved PYQ Paper

BT-202 Mathematics - II - Jun 2023 Question Paper

  1. 7 Marks
    aSolve: $\frac{dy}{dx} = \cos(x + y) + \sin(x + y)$.
  2. 7 Marks
    bSolve: $(1 + y^2)dx = (\tan^{-1} y - x)dy$.
  3. 7 Marks
    aSolve: $\frac{d^2y}{dx^2} + \frac{dy}{dx} = (1 + e^x)^{-1}$.
  4. 7 Marks
    bSolve: $\frac{dx}{dt} - y = e^t$, $\frac{dy}{dt} + x = \sin t$; $x(0) = 1$, $y(0) = 0$.
  5. 14 Marks
    Solve the differential equation $$x(1 - x)y'' + 2(1 - 2x)y' - 2y = 0$$ using Frobenius method.
  6. 7 Marks
    aProve that $J_{\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}} \sin x$
  7. 7 Marks
    bSolve by Charpit's method, the P.D.E $(p^2 + q^2)y = qz$.
  8. 7 Marks
    aSolve: $(D^2 - 6DD' + 9D'^2)z = 12x^2 + 36xy$.
  9. 7 Marks
    bProve that an analytic function with constant modulus is constant.
  10. 7 Marks
    aUse Cauchy Integral formula to solve $$\oint_C \frac{\sin \pi z^2 + \cos \pi z^2}{(z - 1)(z - 2)} dz$$ where $C$ is the circle $|z| = 3$.
  11. 7 Marks
    bUsing complex integration method, solve: $\int_0^{2\pi} \frac{\cos 4\theta}{5 + 4\cos\theta} d\theta$
  12. 7 Marks
    aSolve: $\int_0^{1+i} (x - y + ix^2) dz$ along the real axis from $z = 0$ to $z = 1$ and then along a line parallel to imaginary axis from $z = 1$ to $z = 1 + i$.
  13. 7 Marks
    bProve that: $\nabla^2 f(r) = f''(r) + \frac{2}{r} f'(r)$
  14. 7 Marks
    aFind the directional derivative of $f(x, y, z) = e^{2x}\cos yz$ at $(0, 0, 0)$ in the direction of the tangent to the curve $x = a\sin t$, $y = a\cos t$, $z = at$ at $t = \frac{\pi}{4}$.
  15. 7 Marks
    bUsing Green's theorem, find the area of the region in the first quadrant bounded by the curve $y = x$, $y = \frac{1}{x}$, $y = \frac{x}{4}$.
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