2004

1) Using Lagrange’s mean value theorem, show that 1−x<e−x<1−x+x22,x>0.

[10M]

2) f(x,y)={xy(x2−y2)(x2+y2),(x,y)≠(0,0)0,(x,y)=(2,6)

show that fxy(0,0)≠fyx(0,0)

[10M]

3) Using Lagrange’s multipliers, find the volume of the greatest rectangular parallelopiped that can be inscribed in the sphere x2+y2+z2=1.

[10M]

4) Evaluate the integral ∬, where R is the triangular region in the first quadrant bounded by y=x and x=0 .

[10M]

5) Evaluate \int_{0}^{1} x^{m}\left(\ln \dfrac{1}{x}\right)^{n} d x, m, n>-1

[10M]

6) Find the volume cutt off the sphere x^{2}+y^{2}+z^{2}=a^{2} by the cone x^{2}+y^{2}=z^{2}.

[10M]

2003

1) Let \mathrm{f}, \mathrm{g}: \mathrm{[a}, \mathrm{b}] \rightarrow \mathrm{IR} be functions such that \mathrm{f}(\mathrm{x}) and \mathrm{g}^{\prime}(\mathrm{x}) exist for all \mathrm{x} \in[\mathrm{a}, \mathrm{b}] and \mathrm{g}^{\prime}(\mathrm{x}) \neq 0 for all x in (a, b). Prove that for some c \in(a, b)
\dfrac{f(c)-f(a)}{g(b)-g(c)}=\dfrac{f(c)}{g^{\prime}(c)}

[10M]

2) Let f:R^{2} \rightarrow R be defined by
f(x, y)=\dfrac{x y^{2}}{x^{2}+y^{2}} for (x, y) \neq(0,0) =0 for (x, y)=(0,0) Show that the partial derivatives D_{1} f(0,0) and D_{2} f(0,0) vanish but f is not differentiable at (0,0).

[10M]

3) Let f(x)=e^{-i x^{2}}(x \neq 0) =0 for x=0 Show that f(0)=0 and f^{\prime}(0)=0

Write \mathrm{f}^{(\mathrm{k})}(\mathrm{x}) as \mathrm{P}\left(\dfrac{1}{x}\right) f(x) for \mathrm{x} \neq 0, where \mathrm{P} is a polynomial and \mathrm{f}^{\mathrm{k} 0} denotes the \mathrm{k}^{\mathrm{th}} derivation of \mathrm{f}.

[10M]

[TBC]

4) Using Lagrange multipliers, show that a rectangular box with lid of volume 1000 cubic units and of least surface area is a cube of side 10 units.

[10M]

5) Show that the area of the surface of the solid obtained by revolving the arc of the curve y=c cosh \left(\dfrac{x}{c}\right) joining (0, c) and (x, y) about the x -axis is.

[10M]

[TBC]

6) Define \Gamma:(0, \infty) \rightarrow R by

\Gamma(x)=\int_{0}^{\infty} f^{x} e^{-t} d t. Show that this integral converges for all \mathrm{x}>0 and that \Gamma(\mathrm{x}+\mathrm{t})=\mathrm{x} \mathrm{\Gamma}(\mathrm{x}).

[10M]

2002

1) Find the extremum values of x^{2}+y^{2} subject to the condition 3 x^{2}+4 x y+6 y^{2}=140

[10M]

2) Prove that

[10M]

TBC

3) Find the volume and centroid of the region in the first octant bounded by 6 x+3 y+2 z=6.

[10M]

4) If f(x)=e^{-x \dfrac{1}{2}} and g(x)=xf(x) for all x, prove that
f(y)=\sqrt{\dfrac{2}{\pi}} \int_{0}^{\infty} f(x) \cos x y d x
g(y)=\sqrt{\dfrac{2}{\pi}} \int_{0}^{\infty} g(x) \sin x y d x

[10M]

TBC

5) If \sin ^{-1} x+\sin ^{-1} y and
v=x \sqrt{1-y^{2}}+\sqrt{1-x^{2}}
determine whether there is a functional relationship between u and v, and if so find it.

[10M]

6) If f(x) is monotonic in the interval 0<x<a, and the integral \int_{0}^{a} x^{p} f(x) d x exists, then show that \lim _{x \rightarrow 0} x^{p+1} f(x)=0.

[10M]

2001

1) Let f be a function defined on [0,1] by

and \mathrm{p}, \mathrm{q}, are relatively prime positive integers. Show that f is continuous at each irrational point and discontinuous at each rational point \dfrac{p}{q}.

[10M]

2) Show that the function [x], where [x] denotes the greatest integer not greater than x, is integrable in [0,3]. Also evaluate \int_{0}^{3}[x] d x.

[10M]

3) Examine the convergence of the integral \int_{0}^{1} x^{n-1} \log x d x.

[10M]

4) Examine the function

for continuity, partial derivability of the first order and differentiability at (0,0).

[10M]

5) Find the maximum and minimum values of the function f(x, y, z)=x y+2 x on the circle which is the intersection of the plane x+y+z=0 and the sphere x^{2}+y^{2}+z^{2}=24.

[10M]

6) Find the volume of the region \mathrm{R} lying below the plane z=3+2 y and above the paraboloid z=x^{2}+y^{2}.

[10M]

2000

1) Prove that the stationary values of u \equiv \dfrac{x^{2}}{a^{4}}+\dfrac{y^{2}}{b^{4}}+\dfrac{z^{2}}{c^{4}} subject to the conditions L x+m y+n z=0 and \dfrac{x^{2}}{2}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{a x^{2}}=1 are the roots of the equations \dfrac{l^{2} a^{4}}{1-a^{2} u}+\dfrac{m^{2} b^{4}}{1-b^{2} u}+\dfrac{n^{2} c^{4}}{1-c^{2} u}=0.

[10M]

2) Evaluate \int_{-1}^{1} x^{3} d x from first principles by using Riemann’s theory of integration.

[10M]

3) If f(x) is a continuous function of x satisfying f(x+y)=f(x)+f(y), for all real numbers x, y then prove that f(x)=A x, for all real numbers x, where A is a constant.

Express y=\left(x+\sqrt{1+x^{2}}\right) in ascending powers of x, by Taylor’s theorem.

[20M]

4) Using the transformations u=\dfrac{x^{2}+y^{2}}{x}, v=\dfrac{x^{2}+y^{2}}{y} evaluate \iint \dfrac{\left(x^{2}+y^{2}\right)^{2}}{x^{2} y^{2}} d x d y over the area common to the circles x^{2}+y^{2}-a x=0 and x^{2}+y^{2}-b y=0.

[10M]

5) Evaluate \iiint(1-x-y-z)^{t-1} x^{m-1} y^{n-1} z^{p-1} d x d y d z over the interior of the tetrahedron bounded by the planes x=0, y=0, z=0, x+y+z=1.

[10M]