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

Attempt all questions.

Group A (10×2=20)

1. Verify the men value theorem for the function $f(x) = \sqrt{x(x-1)}$ in the interval $[0, 1]$.

2. Find the length of the curve $y = \frac{4\sqrt{2}}{3}x^{3/2} - 1$ for $0 \leq x \leq 1$.

3. Test the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n!}$ By comparison test.

4. Obtain the semi-major axis, semi-minor axis, foci, vertices $\frac{x^2}{25} + \frac{y^2}{16} = 1$.

5. Find the angle between the vectors $2\vec{i} + \vec{j} + \vec{k}$ and $-4\vec{i} + 3\vec{j} + \vec{k}$.

6. Obtain the area of the region R bounded by $y = x$ and $y = x^2$ in the first quadratic

7. Show that the function

   $f(x,y) = \begin{cases} \frac{2xy}{x^2 + y^2} & (x,y) \neq (0,0), \ 0 & (x,y) = (0,0) \end{cases}$

   Is continuous at every point in the plane except the origin.

8. Using partial derivatives, find $\frac{dy}{dx}$ if $2xy + \tan y - 4y^2 = 0$.

9. Verify that the partial differential equation $\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial y^2} = \frac{2z}{x}$ is satisfied by

   $z = \frac{1}{x}\phi(y - x) + \phi’(y - x)$

10. Find the general solution of the equation

    $x^2 \frac{dz}{dx} + y^2 \frac{dz}{dy} = (x,y)z$.

Group B (5×4=20)

11. State and prove mean value theorem for definite integral.

12. Find the area of the region that lies in the plane enclosed by the cardioid $r = 2(1 + \cos \theta)$.

13. What do you mean by principle unit normal vector? Find unit tangent vector and principle unit vector for the circular motion $\vec{r}(t) = (\cos 2t)\vec{i} + (\sin 2t)\vec{j}$.

14. Define partial derivative of a function $f(x,y)$ with respect to $x$ at the point $(x_0,y_0)$. State Euler’s theorem, verify if it for the function $f(x, y) = x^2 + 5xy + \sin x + 7e^x$, $x = \left(\frac{y}{2}\right){+1}$

15. Find the particular integral of the equation $\frac{d^2z}{dz^2} - \frac{dz}{dy} = 2y - x^2$

Group C (5×8=40)

16. Graph the function $y = x^{1/2} - 5x^{2/3}$

17. What is mean by maclaurin series? Obtain the maclaurin series for the function $f(x) = e^{-x}$.

18. Evaluate the double integral $\int_0^4 \int_{x=\frac{y}{2}}^{x=\frac{y}{2}+1} \frac{2x-y}{2} dxdy$ by applying the transformation $u = \frac{2x-y}{2}$, $v = \frac{y}{2}$ and integrating over an appropriate region in the uv-plane.

19. Define maximum and minimum of a function at a point. Final the local maximum and local minimum of the function $f(x,y) = 2xy - 5x^2 - 2y^2 + 4x + 4y - 4$.

20. Find the solution of the equation $\frac{d^2z}{dx^2} - \frac{d^2z}{dy^2} = x - y$

    Or

    Find the particular integral of the equation

    $(D^2 - D)z = 2y - x^2$ Where $D = \frac{d}{dx}$, $D’ = \frac{d}{dy}$