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. Define a relation and a function from a set into another set. Give suitable example.

2. Show that the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges by using integral test.

3. Investigate the convergence of the series $\sum_{n=0}^{\infty} \frac{2^n + 5}{3^x}$

4. Find the foci, vertices, center of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$.

5. Find the equation for the plane through $(-3,0,7)$ perpendicular to $\vec{n} = 5\vec{i} + 2\vec{j} - \vec{k}$

6. Define cylindrical coordinates $(r, v, z)$. Find an equation for the circular cylinder $4x^2 + 4y^2 = 9$ in cylindrical coordinates.

7. Calculate $\iint_R f(x, y)d4$ for $f(x,y) = 1 - 6x^2y$, $R: 0 \leq x \leq 2, -1 \leq y \leq 1$.

8. Define Jacobian determinant for $x = g(u, v, w)$, $y = h(u, v, w)$, $z = k(u, v, w)$.

9. What do you mean by local extreme points of $f(x,y)$? Illustrate the concept by graphs.

10. Define partial differential equations of the first index with suitable examples.

Group B (5×4=20)

11. State the mean value theorem for a differentiable function and verify it for the function $f(x) = \sqrt{1-x^2}$ on the interval $[-1,1]$.

12. Find the Taylor series and Taylor polynomials generated by the function $f(x) = \cos x$ at $x = 0$.

13. Find the length of the cardioid $r = 1 - \cos \theta$.

14. Define the partial derivative of $f(x,y)$ at a point $(x0, y0)$ with respect to all variables. Find the derivative of $f(x,y) = xe^y = \cos(x, y)$ at the point $(2, 0)$ in the direction of $A = 3i - 4j$.

15. Find a general solution of the differential equation $x^2 \frac{dz}{dx} + y^2 \frac{dz}{dy} = (x + y)z$.

Group C (5×8=40)

16. Find the area of the region in the first quadrant that is bounded above by $y = \sqrt{x}$ and below by the $x$-axis and the line $y = x - 2$.

    OR

    Investigate the convergence of the integrals

    (a) $\int_1^0 \frac{1}{1-x} dx$ (b) $\int_0^3 \frac{dx}{x-1^{2/3}}$

17. Calculate the curvature and torsion for the helix

    $r(t) = (a \cos t)i + (a \sin t)j + btk$, $a,b \geq 0$, $a^2 + b^2 \neq 0$.

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

19. Find the absolute maximum and minimum values of $f(x,y) = 2 + 2x + 2y - x^2 - y^2$ on the triangular plate in the first quadrant bounded by lines $x = 0$, $y = 0$ and $x + y = 9$.

    OR

    Find the points on the curve $xy^2 = 54$ nearest to the origin. How are the Lagrange multipliers defined?

20. Derive D’ Alembert’s solution satisfying the initials conditions of the one-dimensional wave equation.