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. Verify Rolle’s theorem for the function $f(x) = \frac{x^3}{3} - 3x$ on the interval $[-3, 3]$.

2. Obtain the area between two curves $y = \sec^{2}x$ and $y = \sin x$ from $x = 0$ to $x = \pi/4$.

3. Test the convergence of $p$ - series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ for $p > 1$.

4. Find the eccentricity of the hyperbola $9x^2 - 16y^2 = 144$.

5. Find a vector perpendicular to the plane of $P(1, -1, 0)$, $Q(2, 1, -1)$ and $R(-1, 1, 2)$.

6. Find the area enclosed by the curve $r^2 = 4\cos 2\theta$.

7. Obtain the values of $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ at the point $(4, -5)$ if $f(x,y) = x^2 + 3xy + y - 1$.

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

9. Find the partial differential equation of the function $(x - a)^2 + (y - b)^2 + z^2 = c^2$.

10. Solve the partial differential equation $x^{2}p + q = z^2$.

Group B (5×4=20)

11. State and prove the mean value theorem for a differential function.

12. Find the length of the Asteroid $x = \cos^3t$, $y = \sin^3t$ for $0 \leq t \geq 2\pi$.

13. Define a curvature of a curve. Prove that the curvature of a circle of radius $a$ is $1/a$.

14. What is meant by direction derivative in the plain? Obtain the derivative of the function

    $f(x,y) = x^2 + xy$ at $P(1, 2)$ in the direction of the unit vector $v = \left(\frac{1}{\sqrt{2}}\right)i + \left(\frac{1}{\sqrt{2}}\right)j$

15. Find the center of mass of a solid of constant density $\delta$, bounded below by the disk: $x^2 + y^2 = 4$ in the plane $z = 0$ and above by the paraboloid $z = 4 - x^2 - y^2$.

Group C (5×8=40)

16. Graph the function $f(x) = -x^3 + 12x + 5$ for $-3 \leq x \leq 3$.

17. Define Taylor’s polynomial of order $n$. Obtain Taylor’s polynomial and Taylor’s series generated by the function $f(x) = e^x$ at $x = 0$.

18. Obtain the centroid and the region in the first quadrant that is bounded above by the line $y = x$ and below by the parabola $y = x^2$.

19. Find the maximum and the minimum values of $f(x, y) = 2xy - 2y^2 - 5x^2 + 4x - 4$. Also find the saddle point if it exists.

    OR

    Evaluate the integral $\int_0^{\sqrt{2}} \int_{0}^{3y} \int_{x^2 - 3y^2}^{6 - x^2 - y^2} dz dx dy$

20. What do you mean by d’ Alembert’s solution of the one-dimensional wave equation? Derive it.

    OR

    Find the particular integral of the equation $(D^2 - D^1)z = 2y - x^2$ where $D = \frac{\partial}{\partial x}$, $D’ = \frac{\partial}{\partial y}$.