1994

1) Examine the

(i) absolute convergence
(ii) uniform convergence

of the series \((1-x)+x(1-x)+x^{2}(1-x)+\ldots\) in \([- c , 1]\), where \(0< c <1\).

[10M]

2) Prove that \(S(x)=\sum \dfrac{1}{n^{p}+n^{q} x^{2}}, p>1\) is uniformly convergent for all values of \(x\) and can be differentiated term by term if \(q <3 p -2\)

[10M]

3) Let the function \(f\) be defined on \([0,1]\) by the condition \(f(x)=2 rx\) when \(\dfrac{1}{r+1}< x <\dfrac{1}{r}, r>0\). Show that \(f\) is Riemann integrable in \([0,1]\) and \(\int_{0}^{1} f(x) d x=\dfrac{\pi^{2}}{6}\).

[10M]

4) By means of the substitution \(x+y+z=u, y+z=u v, z=u v w\), evaluate \(\iiint(x+y+z)^{n} x y z d x d y d z\) taken over the volume bounded by \(x =0, y =0, z =0, x + y + z =1\).

[10M]

1993

1) Show that \(x =0\) is a point of non-uniform convergence of the series whose \(nth\) term is \(n ^{2} xe ^{n^{2}x^{2}}-( n -1)^{2} x e ^{-( n -1)} 2 x ^{2}\).

[10M]

2) Find all the maxima and minima of \(f(x, y)=x^{3}+y^{3}-63(x+y)+12 x y\)

[10M]

3) Examine for Riemann integrability over [0,2] of the function defined in [0,2] by \(f(x)=\left\{\begin{array}{r}x+x^{2}, \text { for rational values of } x \\ x^{2}+x^{3}, \text { for irrational values of } x\end{array}\right.\)

[10M]

4) Prove that \(\int_{0}^{\infty} \dfrac{\sin x}{x} d x\) converges and conditionally converges.

[10M]

5) Evaluate \(\iiint \dfrac{d x d y d z}{x+y+z+1}\) over the volume bounded by the coordinate planes and the plane \(x+y+z=1\).

[10M]

1992

1) Examine \(f(x, y, z)=2 x y z-4 z x-2 y z+x^{2}+y^{2}+z^{2}-2 x-4 y-4 z\) for extreme values.

[10M]

2) If \(U_{n}=\dfrac{1+n x}{n e^{n x}}-\dfrac{1+(n+1) x}{(n+1) e^{(n+1) x}}, \quad 0<x<1\) Prove that \(\dfrac{d}{d x} \Sigma U_{n}=\Sigma \dfrac{d}{d x} u_{n}\) Is the series uniformly convergent in \([ 0,1]\) ? Justify your claim.

[10M]

3) Find the upper and lower Riemann integral for the function defined in the interval [0,1] as follows: \(f(x)=\left\{\begin{array}{ll} \sqrt{1-x^{2}} & \text {when x rational } \\ 1- x & \text { when x is irrational } \end{array}\right.\) and show that \(f(x)\) is not Riemann integrable in [0,1]

[10M]

4) Discuss the convergence or divergence of \(\int_{0}^{\infty} \dfrac{x^{\beta} d x}{1+x \alpha \sin ^{2} x}, \quad \alpha>\beta>0\)

[10M]

5) Evaluate \(\iint \dfrac{\sqrt{\left(a^{2} b^{2}-b^{2} x^{2}-a^{2} y^{2}\right)}}{\sqrt{\left(a^{2} b^{2}+b^{2} x^{2}+a^{2} y^{2}\right)}} d x d y\) over the positive quadrant of the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\)

[10M]

1991

1) Examine whether the function \(y=\tan x\) is uniformly continuous in the open interval \(\left(0, \dfrac{\pi}{2}\right)\).

2) Evaluate \(\int_{0}^{\infty} \frac{\log \left(1+a^{2} x^{2}\right)}{1+b^{2} x^{2}} d x\).

3) If the rectangular axes \((x, y)\) are rotated through an angle \(\alpha\) about the origin and the new co-ordinates are \((\bar{x}, \bar{y})\) show that for any function \(u\), \(\dfrac{\partial^{2} u}{\partial x^{2}}+\dfrac{\partial^{2} u}{\partial y^{2}}=\dfrac{\partial^{2} u}{\partial \bar{x}^{2}}+\dfrac{\partial^{2} u}{\partial \bar{y}^{2}}\).

4) A rectangle is inscribed in the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\). What is the maximum possible area of the rectangle?

1990

1) Discuss the convergence of \(\int_{0}^{\pi/2} \log \sin x d x\) and evaluate it, if it is convergent.

2) Find the point on the parabola \(y^{2}=2 x\), \(z=0\), which is nearest the plane \(z=x+2 y+8\). Show that this minimum distance is \(\sqrt{6}\).

3) Show that \(\iiint\left(a^{2} b^{2} c^{2}-b^{2} c^{2} x^{2}-c^{2} a^{2} y^{2}-a^{2} b^{2} z^{2}\right)^{1 / 2} d x d y d z\) over the volume bounded by \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\) is equal to \(\dfrac{\neq^{2} a^{2} b^{2} c^{2}}{4}\).

TBC

1989

1) The function \(f:[a, \infty) \rightarrow \mathbb{R}\) is continuous and \(f(x) \rightarrow l\) as \(x \rightarrow \infty,\) prove that \(f\) is uniformly continuous on \(\left[a,\infty \right)\).

2) Find the values of \(x\) for which the series \(\sum_{n=1}^{\infty} \dfrac{x^{a}}{1+n^{2} x^{2}}, x \geq 0, \alpha \geq 0,\) converges uniformly on (i) \([0,1]\) and
(ii) \(\left[0,\infty \right)\)

3) Discuss the existence of the improper integral \(\int_{-\infty}^{\infty} e^{-x^{2} / 2} d x\).

4) Show that the value of \(\iint(1-x-y)^{2} x^{1 / 2} y^{1 / 2} d x d y\) taken over the interior of the triangle whose vertices are the origin and the points \((0,1)\) and \((1,0)\) is \(\dfrac{\neq}{410}\).

TBC