2008

1) Obtain the values of the constants a, b and c for which the function defined by

is continuous at x=0.

[10M]

2) If u=sin−1(x3+y3√x+√y), prove that
x∂u∂x+y∂u∂y=52tanu.

[10M]

3) Determine the value of
(∫10x2(1−x4)1/2dx)(∫10dx(1+x4)1/2)

[10M]

4) Obtain the value of the double integral ∬ where D represents the region bounded by the straight line y=x and the parabola y^{2}=4 x.

[10M]

5) A wire of length b is cut into two parts which are bent in the form of a square and a circle respectively. Find the minimum value of the sum of the areas so formed.

[10M]

6) Calculate the volume cut off from the sphere x^{2}+y^{2}+z^{2} =a^{2} by the right circular cylinder given by x^{2}+y^{2}=b^{2}.

[10M]

2007

1) If a function f is such that its derivative f is continuous on [\mathrm{a}, \mathrm{b}] and derivable on ]\mathrm{a}, \mathrm{b}[, then show that there exists a number c between a and b such that f(b)=f(a)+(b-a)f’(a)+\dfrac{1}{2}(b-a)^{2} f^{\prime \prime}(c)

[10M]

2) If

Show that both the partial derivatives exist at (0,0) but the function is not continuous thereat.

[10M]

3) Find the values of a and b, so that

What are these conditions?

[10M]

4) Show that f(x y, z \quad 2 x)=0 satisfies, under certain conditions, the equation

What are these conditions?

[10M]

5) Find the surface area generated by the revolution of the cardioids r=a(\mathrm{I}+\cos \theta) about the initial line.

[10M]

6) The function f is defined on ] 0,1[ by

Show that

Does not converge.

[10M]

2006

1) Show that

\int_0^{\infty} \dfrac{\tan^{-1} \alpha x \tan^{-1} \beta x}{x^2} dx = \dfrac{\pi}{2} \dfrac{(\alpha+\beta)^{(\alpha+\beta)}}{\alpha^\alpha + \beta^\beta}, \alpha, \beta >0

[10M]

2) Show that f_{x}(0,0) \neq f_{y}(a, 0) where f(x, y)=0, if x y=0, f(x, y)=x^{2} \tan^{-1} \dfrac{y}{x}-y^{2} \tan^{-1} \dfrac{x}{y}, if xy \neq 0

[10M]

3) Find the volume under the spherical surface x^{2}+y^{2}+z^{2}=a^{2} and over the lemniscate r^{2}=a^{2} \cos 2 \theta.

[10M]

4) Find the centre of gravity of the volume common to a cone of vertical angle 2 \alpha and a sphere of radius a, the vertex of the cone being the centre of the sphere.

[10M]

5) Using Lagrange’s method of undetermined multipliers, find the stationary values of \mathrm{x}^{2}+\mathrm{y}^{2}+ z^{2} subject to a x^{2}+b y^{2}+c z^{2}=1 and lx+m y+n z=0. Interpret geometrically.

[10M]

6) Find the extreme values of f(x, y)=2(x-y)^{2}-x^{4}-y^{4}.

[10M]

2005

1) Let f(x)=\left\{\begin{array}{c}x^{2} \sin \dfrac{1}{x} \text { for } x \neq 0 \\ 0 . \text { for } x=0\end{array}\right.
Show that \mathrm{f} is differentiable at each point of reals but \mathrm{f}(\mathrm{x}) is not continuous at \mathrm{x}=0.

[10M]

2) Show that f: R^{2} \rightarrow R defined by f(x, y) 2 x^{2}-6 x y+3 y^{2} has a critical point at (0,0) and that it is a saddle point.

[10M]

3.(i) Using Taylor’s theorem with remainder show that x-\dfrac{x^{3}}{6} \leq \sin x \leq x-\dfrac{x^{3}}{6}+\dfrac{x^{5}}{120} \text { for all } x \geq 0

[5M]

3.(ii) Let f: \mathbf{R}^{2} \rightarrow \mathbf{R} be defined by \begin{aligned} f(x, y) &=\dfrac{x y}{x^{2}-y^{2}} \text { if } x \neq \pm y \\ &=0 \quad \text { if } x=\pm y \end{aligned}

[5M]

4) Show that the curve given by x^{3}-4 x^{2} y+5 x y^{2}-2 y^{3}+3 x^{2}-4 x y+2 y^{2}-3 x+2 y-1=0 has only one asymptote given by y=\dfrac{1}{2} x+3.

[10M]

5) Find the extremum values of f(x, y)=2 x^{2}-8 x y+9 y^{2} \text { on } x^{4}+y^{2}-1=0 using Lagrange multiplier method.

[10M]

6) A solid cuboid in R^3 given in spherical coordinates by R=[0,a], \theta=[0, \pi], \varphi=0, \pi/4 has a density function \rho(\mathbf{R}, \theta, \varphi)=4 \mathbf{R} \sin \dfrac{\theta}{2} \cos \varphi . Find the total mass of \mathbf{C}

[10M]