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 $f(x) = x + 2$ and $g(x) = x^3 - 3$ find $g(f(3))$.

2. Show that the area under the arch of the curve $y = \sin x$ is.

3. Test the convergence of the series $\lim_{n \to \infty} \frac{a-bn^3}{n^3-c}$.

4. Find the equation of the parabola with vertex at the origin and focus at $(0,2)$.

5. Find the angle between the planes $3x - 6y - 2z = 7$ and $2x + y - 2z = 5$

6. Evaluate $\int_1^2 \int_y^{y^2} dxdy$.

7. Find $\frac{df}{dx}$ and $\frac{df}{dy}$ if $f(x,y) = 10 - x^2 - y^2$.

8. Prove that $u_{xy} = u_{yx}$ is $u = 1 + n(2x + 3y)$

9. Show that $y = \frac{1}{2}e^x + be^{-x}$ of $\frac{dy}{dx} + y = e^x$

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

Group B (5×4=20)

11. Verify Rolle’s theorem for the function $f(x) = x^2 - 5x + 7$ in the interval $[2,3]$.

12. Find the Taylors series expression of $f(x) = \sin x$ at $x = 0$.

13. Obtain the polar equations for circles through the origin centered on $x$ and $y$ axis, with radius $a$.

14. Evaluate $\lim_{(x,y) \to (0,0)} \frac{2y^2}{x^2+xy}$.

15. Obtain the general solution of $(y - z) \frac{dz}{dx} + (x - y) \frac{dz}{dy} = z - x$

Group C (5×8=40)

16. State Lagrange’s mean value theorem and verify the theorem for

    $x = x^3 - x^2 - 5x + 3$ in $[0,4]$.

    Or

    Investigates the convergence of the integrals

    (a) $\int_0^{\infty} \frac{dx}{1+x^2}$

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

17. Define curvature of a curve. Show that the curvature of a (a) straight line on zero and (b) a circle of a radius $a$ is $l/a$

18. Find the volume enclosed between the surfaces $z = x^2 + 3y^2$ and $z = 8 - x^2 - y^2$

19. Find the maximum and minimum of the function $f(x,y) = x^3 + y^3 - 12x + 20$.

    OR

    Find the Point on the ellipse $x^2 + 2y^2 = 1$ where $f(x,y) = xy$ has its extreme values.

20. Define second order partial differential equation. What is initial boundary values problem? Solve $u_t = u_{xx} = u_{tt} = u_{xx}$