2000

1) Use the mean value theforem to prove that \(\dfrac{2}{7}<\log 1.4<\dfrac{2}{5}\).

[10M]

2) Let \(f(x)=\left\{\begin{array}{l}0, x \text { is irrational } \\ 1, x \text { is rational }\end{array}\right.\) show that \(f\) is not Riemann-integrable on \([a, b]\)

[10M]

3) Show that \(\dfrac{d^{n}}{d x^{n}}\left(\dfrac{\log x}{x}\right)=(-1)^{n} \dfrac{n !}{x^{n+1}}\left(\log x-1-\dfrac{1}{2}-\dfrac{1}{3}-\ldots-\dfrac{1}{n}\right)\)

[10M]

4) Find constants \(a\) and \(b\) for which \(F(a, b)=\int_{0}^{\pi}\left\{\sin x-\left(a x^{2}+b x\right)\right\}^{2} d x\) is a minimum.

[10M]

1999

1) Determine the set of all points where the function \(f(x)=\dfrac{x}{1+ \vert x \vert}\) is differentiable.

[10M]

2) Find three asymptotes of the curve \(x^3+2x^2y-4xy^2-8y^3-4x+8y-10=0\). Also find the intercept of one asymptote between the other two.

[10M]

3) Find the dimensions of a right circular cone of minimum volume which can be circumscribed about a sphere of radius \('a'\).

[10M]

4) If \(f\) is Riemann integrable over every interval of finite length and \(f(x+y)=f(x)+f(y)\) for every pair of real numbers \(x\) and \(y\), show that \(f(x)=cx\) where \(c=f(1)\).

[10M]

5) Show that the area bounded by cissoid \(x=a sin^2t,y=a\dfrac{sin^3t}{cos t}\) and its asymptote is \(\dfrac{3\pi a^2}{4}\).

[10M]

6) Show that \(\int\int x^{m-1}y^{n-1}\,dxdy\) over the positive quadrant of the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) is \(\dfrac{a^mb^n}{4}\dfrac{\Gamma(\dfrac{m}{2})\Gamma(\dfrac{n}{2})}{\Gamma(\dfrac{m}{2}+\dfrac{n}{2}+1)}\)

[10M]

1998

1) Find the asymptotes of the curve \((2x-3y+1)^2(x+y)-8x+2y-9=0\) and show that they intersect the curve again in three points which lie on a straight line.

[10M]

2) A thin closed rectangular box is to have one edge \(n\) times the length of another edge and the volume of the box is given to be \(v\). Prove that the least surface \(s\) is given by \(ns^3=54(n+1)^2v^2\).

[10M]

3) If \(x+y=1\), prove that \(\dfrac{d^n}{dx^n}(x^ny^n)=n![y^n-(\dfrac{n}{1})^2y^{n-1}x+(\dfrac{n}{2})^2y^{n-2}x^2+...+(-1)^nx^n]\)

[10M]

4) Show that \(\int^{\infty}_0 \dfrac{x^{p-1}}{(1+x)^{p+q}} dx=B(p,q)\).

[10M]

5) Show that: \(\iiint \dfrac{\,dx\,dy}{\sqrt{(1-x^2-y^2-z^2)}}=\dfrac{\pi^2}{8}\), integral being extended over all positive values of \(x,y,z\) for which the expression is real.

[10M]

6) The ellipse \(b^2x^2+a^2y^2=a^2b^2\) is divided into two parts by the line \(x=\dfrac{1}{2}\)a , the smaller part is rotated through four right angles about this line. Prove that the volume generated is

\(\pi a^2b\{\dfrac{3\sqrt{3}}{4}-\dfrac{\pi}{3}\}\).

[10M]

1997

1) Suppose

Determine \(\dfrac{d}{d x}\left(f^{-1}\right)\) at \(\mathrm{x}=-1\) if it exists.

[10M]

2) Prove that the volume of the greatest parallelopiped that can be inscribed in the ellipsoid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1 \text { is } \dfrac{8 a b c}{3 \sqrt{3}}\)

[10M]

3) Show that the asymptotes of the curve \(\left(x^{2}-y^{2}\right)\left(y^{2}-4 x^{2}\right)+6 x^{3}-5 x^{2} y-3 x y^{2}+z y^{3}-x^{2}+3 x y-1=0\) cut the curve again in eight points which lie on a circle of radius 1 .

[10M]

4) An area bounded by a quadrant of a circle of radius a and the tangents at its extremities revolves about one of the tangents. Find the volume so generated.

[10M]

5) Show how the change of order in the integral \(\int_{0}^{\infty} \int_{0}^{\infty} e^{-x y} \sin x d x d y\) leads to the evaluation of \(\int_{0}^{\infty} \dfrac{\sin x}{x} d x,\) Hence evaluate it.

[10M]

6) Show that in \(\sqrt{n} \sqrt{n+\dfrac{1}{2}}=\dfrac{\sqrt{\pi}}{2^{2 n-1}}\) where \(\mathrm{n}>0\) and \(\sqrt{n}\) denotes gamma function.

[10M]

1996

1) Find the asymptotes of the curve \(4\left(x^{4}+y^{4}\right)-17 x^{2} y^{2}-4 x\left(4 y^{2}-x^{2}\right)+2\left(x^{2}-2\right)=0\) and show that they pass through the points of intersection of the curve with the ellipse \(x^{2}+4 y^{2}=4\)

[15M]

2) Show that any continuous function defined for all real \(x\) and satisfying the equation \(f(x)=f\) \((2 x+1)\) for all \(x\) must be a constant function.

[15M]

3) Show that the maximum and minimum of the radii vectors of the sections of the surface \(\left(x^{2}+y^{2}+z^{2}\right)^{2}=\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}\) by the plane \(\lambda x+\mu y+v z=0\) are given by the equation \(\dfrac{a^{2} \lambda^{2}}{1-a^{2} r^{2}}+\dfrac{b^{2} \mu^{2}}{1-b^{2} r^{2}}+\dfrac{c^{2} v^{2}}{1-c^{2} r^{2}}=0\)

[15M]

4) If \(u=f\left(\dfrac{x}{y}, \dfrac{y}{z}, \dfrac{z}{x}\right),\) prove that \(x \dfrac{\partial u}{\partial x}+y \dfrac{\partial u}{\partial y}+z \dfrac{\partial u}{\partial z}=0\)

[12M]

5) Evaluate \(\int_{0}^{\infty} \int_{0}^{\infty} \dfrac{e^{-y}}{y} d x d y\)

[12M]

6) The area cut off from the parabola \(y^{2}=4 a x\) by the chord joining the vertex to an end of the latus rectum is rotated through four right angles about the chord. Find the volume of the solid so formed.

[12M]

1995

1) If \(\mathrm{g}\) is the inverse of \(\mathrm{f}\) and \(f^{\prime}(x)=\dfrac{1}{1+x^{3}}\), prove that \(g^{\prime}(x)=1+[g(x)]^{3}\).

[20M]

2) Taking the nth derivative of \(\left(\mathrm{x}^{n}\right)^{2}\) in two different ways, show that \(1+\dfrac{n^{2}}{1^{2}}+\dfrac{n^{2}(n-1)^{2}}{1^{2} \cdot 2^{2}}+\dfrac{n^{2}(n-1)^{2}(n-2)^{2}}{1^{2} \cdot 2^{2} \cdot 3^{2}}+\ldots \operatorname{to}(n+1)\) terms \(=\dfrac{(2 n) !}{(n !)^{2}}\)

[20M]

3) Let \(f(x, y),\) which possesses continuous partial derivatives of second order, be a homogeneous function of \(x\) and \(y\) of degree \(n\). Prove that \(\mathrm{x}^{2} \mathrm{f}_{\mathrm{xx}}+2 \mathrm{xy} \mathrm{f}_{\mathrm{xy}}+\mathrm{y}^{2} \mathrm{f}_{y y}=\mathrm{n}(\mathrm{n}-1) \mathrm{f}\)

[20M]

4) Find the area bounded by the curve \(\left(\dfrac{x^{2}}{4}+\dfrac{y^{2}}{9}\right)^{2}=\dfrac{x^{2}}{4}-\dfrac{y^{2}}{9}\)

[20M]

5) Let \(f(x), x \geq 1\) be such that the area bounded by the curve \(y=f(x)\) and the lines \(x=1, x=b\) is equal to \(\sqrt{1+b^{2}}-\sqrt{2}\) for all \(\mathrm{b} \geq 1\), Does \(f\) attain its minimum \(?\) If so what is its value?

[20M]

6) Show that \(\Gamma\left(\dfrac{1}{n}\right) \Gamma\left(\dfrac{2}{n}\right) \Gamma\left(\dfrac{3}{n}\right) \ldots . \Gamma\left(\dfrac{n-1}{n}\right)=\dfrac{(2 \pi)}{\sqrt{n}} \dfrac{n-1}{2}\)

[20M]