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. If $f(x) = (x-1) + x$, then prove that $f(x).f(1-x) = 1$

2. Define critical point. Find the critical point of $f(x) = x^2$.

3. Evaluate $\lim_{n \to \infty} \frac{3-5n^6}{n^6-3}$.

4. Find the equation of the parabola with vertex at the origin and directrix at $y = 2$

5. Find the angle between the planes $x - 2y - 2z = 5$ and $5x - 2y - z = 0$

6. Evaluate $\int_0^3 \int_0^2 (4-y^2)dxdy$.

7. Find $\frac{dt}{dx}$ and $\frac{dt}{dy}$ if $f(x,y) = ye^2$.

8. Find the equation for the tangent plane to the surfaces $Z = f(x,y) = 9 - x^2 - y^2$ at the point $(1,2,3)$.

9. Show that $y = c_1xe^{-2x} + c_2e^{-2x}$ is the solution of $y” + y’ - 2y = 0$.

10. Solve $\frac{d^2y}{dx^2} + \frac{dy}{dzx} = 0$.

Group B (5×4=20)

11. Verify Rolle’s theorem for $f(x) = x^2$, $x \in [-1,1]$.

12. Find the Taylor series expansion of $f(x) = \cos \theta$ at $x = 1$.

13. Find the Cartesian equation of the polar equation $r \cos \left(\theta - \frac{\pi}{3}\right) = 3$

14. Show that the function $f(x,y) = \begin{cases} \frac{x^2}{x^2+y^2} & (x,y) \neq (0,0) , \ 0 & (x,y) = (0,0) \end{cases}$ is continuous at every point except the origin.

15. Solve $xz \frac{dz}{dx} + yz \frac{dz}{dy} = xy$

Group C (5×8=40)

16. Find the area bounded on right by the line $y = x - 2$ on the left by the parabola $x = y^2$ and below by the $x$-axis

    Or

    What is an improper integral? Evaluate

    (a) $\int_2^{\infty} \frac{dx}{\sqrt{x-1}}$ (b) $\int_2^{\infty} \frac{dx}{(x-1)^2}$

17. Define curvature of a curve. Find that the curvature of a helix

    $\vec{R}(t) = (a \cos wt)\vec{i} + (a \sin wt)\vec{j} + (bt)\vec{k}$

18. Find the area enclosed by $r^2 = 2a^2 \cos 2\theta$

19. Find the extreme values of $Z = x^3 - y^3 - 2xy + 6$.

    OR

    Find the extreme value of function $F(x,y) = xy$ takes on the ellipse $\frac{x^2}{8} + \frac{y^2}{2} = 1$

20. Define initial boundary values problems. Derive the heat equation or wave equation in one dimension