Full Marks: 80
Pass Marks: 32
Time: 3 hours

Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks.

Attempt all questions.

Group A (10×2=20)

1. Find the length of the curve $y = x^{3/2}$ from $x = 0$ to $x = 4$.

2. Find the critical points of the function $f(x) = x^{3/2}(x-4)$.

3. Does the following series converge?

   $\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots$

4. Find the polar equation of the circle $(x+2)^2 + y^2 = 4$

5. Find the area of the parallelogram where vertices are $A(0,0)$, $B(7,3)$, $C(9,8)$ and $D(2,5)$.

6. Evaluate the integral $\int_t^{2t} \int_0^1 (\sin x + \cos y)dxdy$

7. Evaluate the limit $\lim_{(x,y) \to (0,0)} \frac{x^2-xy}{\sqrt{x} + \sqrt{y}}$

8. Find $\left(\frac{\partial w}{\partial x}\right)_{y,z}$ if $w = x^2 + y - z + \sin t$ and $x + y = t$.

9. Solve the partial differential equation $p + q = x$.

10. Find the general integral of the linear partial differential equation $z(xp - yq) = z^2 - x^2$.

Group B (5×4=20)

11. State and prove Rolle’s Theorem.

12. Find the length of the cardioid $r = 1 + \cos \theta$.

13. Define unit tangent vector of a differentiable curve. Find the unit tangent vector of the curve $r(t) = (\cos t + t \sin t)i + (\sin t - t \cos t)j$, $t > 0$.

14. What do you mean by critical point of a function $f(x,y)$ in a region? Find local extreme values of the function $f(x,y) = xy - x^2 - y^2 - 2x - 2y + 4$.

15. Find a particular integral of the equation

    $\frac{\partial^2 z}{\partial x^2} - \frac{\partial z}{\partial y} = 2y - x^2$

Group C (5×8=40)

16. Graph the function $y = x^{4/3} - 4x^{1/3}$

17. What do you mean by Taylor’s polynomial of order $n$? Obtain Taylor’s polynomial and Taylor’s series generated by the function $f(x) = \cos x$ at $x = 0$.

18. Find the volume of the region enclosed by the surface $z = x^2 + 3y^2$ and $z = 8 - x^2 - y^2$.

19. Obtain the absolute maximum and minimum values of the function.

    $f(x,y) = 2 + 2x + 2y - x^2 - y^2$ on the triangular plate in the first quadrant bounded by lines $x = 0$, $y = 0$, $y = 9 - x$.

    OR

    Evaluate the integral $\int_0^1 \int_0^{3-3x} \int_0^{3-3x-y} dz dydx$

20. Show that the solution of the wave equation $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$, $c^2 = \frac{T}{\rho}$ is $u(x,t) = \frac{1}{2}[f(x + ct) + f(x - ct)] + \frac{1}{2c} \int_{x-ct}^{x+ct} g(s)ds$ and deduce the result if the velocity is zero.

    OR

    Find a particular integral of the equation $(D^2 - D^1) = A \cos(lx + my)$ where $A$, $l$, $m$ are constants.