2000

1) Given that the terms of a sequence \(\left\{a_{n}\right\}\) are such that each \(a_{k}\),\(k \leq 3\), is the arithmetic mean of its two immediately preceding terms. Show that the sequence converges. Also find the limit of the sequence.

[12M]

2) Determine the values of \(x\) for which the infinite product \(\prod_{n=0}^{\infty}\left(1+\dfrac{1}{x^{2^{n}}}\right)\) converges absolutely. Find its value whenever it converges.

[12M]

3) Suppose \(f\) is twice differentiable real valued function in \((0, \infty)\) and \(M_{0}, M_{1}\) and \(M_{2}\) the least upper bounds of \(\vert f(x)\vert ,\vert f^{\prime}(x)\vert\) and \(\vert f^{\prime \prime}(x)\vert\) respectively in \((0, \infty)\). Prove for each \(x>0, h>0\) that \(f^{\prime}(x)=\dfrac{1}{2 h}[f(x+2 h)-f(x)]-h f^{\prime}(u)\) for some \(\mathbf{u} \in(\mathbf{x}, \mathbf{x}+2 \mathbf{h}) .\) Hence show that \(\mathrm{M}_{1}^{2} \leq 4 \mathrm{M}_{0} \mathrm{M}_{2}\)

[15M]

4) Evaluate \(\iint_{S}\left(x^{3} d y d z+x^{2} y d z d x+x^{2} z d x d y\right)\) by transforming into triple integral where \(S\) is the closed surface formed by the cylinder \(x^{2}+y^{2}=a^{2}, 0 \leq z \leq b\) and the circular discs \(x^{2}+y^{2} \leq a^{2}\), \(z=0\) and \(x^{2}+y^{2} \leq a^{2}, z=b\)

[15M]

5) Suppose \(\mathrm{f}(\zeta)\) is continuous on a circle \(\mathrm{C}\). Show that \(\int_{C} \dfrac{f(\zeta) d \zeta}{(\zeta-z)}\) as \(z\) varies inside of \(\mathrm{C},\) is differentiable under the integral sign. Find the derivative. Hence or otherwise, derive an integral representation for \(f^{\prime}(z)\) if \(f(z)\) is analytic on and inside of \(C\).

[15M]

1999

1) A sequence \(\left\{\mathrm{S}_{n}\right\}\) is defined by the recursion formula \(S_{n+1}=\sqrt{3 S_{n}} ; S_{1}=1 .\) Does this sequence converge? If so, find limit \(\mathrm{S}_{\mathrm{n}}\).

2) Find the shortest distance from the origin to the hyperbola \(x^2+8xy+7y^2=225\), \(z=0\).

[20M]

3) Show that the double \(\int\int_R \dfrac{x-y}{(x+y)^3} dxdy\) does not exist over \(R=[0,1;0,1]\).

[20M]

4) Show that the double integral \(\iint_{R} \frac{x-y}{(x+y)^{3}} d x d y\) does not exist over \(R=[0,1 ; 0,1]\).

1998

1) Show that the function \(f(x,y)=2x^4-3x^2y+y^2\) has (0,0) as the only critical point but the function neither has a minima nor a maxima at \((0,0)\).

2) Test the convergence of the integral \(\int^{\infty}_0 e^{-ax}\dfrac{sin x}{x} dx\), \(a\geq 0\).

3) Test the series \(\sum^{\infty}_{n=1} \dfrac{x}{(n+x^2)^2}\) for uniform convergence.

4) Let \(f(x)=x\) and \(g(x)=x^2\). Does \(\int^1_0 f dg\) exist? If it exist, then find its value.s

1997

1) Show that a non-empty set P in \(\mathrm{R}^{n}\) each of whose points is a limit-point is uncountable.

[10M]

2) Show that \(\iiint_{D} x yd x d y d z=\dfrac{a^{2} b^{2} c^{2}}{6}\) where domain \(D\) is given by \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}} \leq 1\).

[10M]

3) If \(u=\sin ^{-1}\left[\left(x^{2}+y^{2}\right)^{1 / 5}\right],\) prove that \(x^{2} \dfrac{\partial^{2} u}{\partial x^{2}}+2 x y \dfrac{\partial^{2} u}{\partial y \cdot \partial x}+y^{2} \dfrac{\partial^{2} x}{\partial y^{2}}=\dfrac{2}{25} \tan u\left(2 \tan ^{2} u-3\right)\)

[10M]

1996

1) A function \(\mathrm{f}\) is defined in the interval \((\mathrm{a}, \mathrm{b})\) as follows: \(\begin{array}{l} f(x)=q^{-2}, \quad \text { when } \quad x=p q^{-1} \\ f(x)=q^{-3}, \quad \text { when } \quad x=\left(p q^{-1}\right)^{1 / 2} \end{array}\) where \(\mathrm{p}, \mathrm{q}\) are relatively prime integers; \(f(\mathrm{x}) =0,\) for all other values of \(\mathrm{x}\). Is \(\mathrm{f}\) Riemann integrable ? Justify your answer.

[15M]

2) Test for uniform convergence, the series \(\sum_{n=1}^{\infty} \dfrac{2^{n} x^{\left(2^{n}-1\right)}}{1+x^{2 n}}\)

[15M]

3) Evaluate \(\int_{0}^{\pi/2} \int_{0}^{\pi/2} \sin x \sin ^{-1}(\sin x \sin y) d x d y\)

[15M]

1995

1) Let \(f\) be a continuous real function on \(R\) such that \(f\) maps open interval onto open intervals. Prove that \(f\) is monotonic.

[15M]

2) Suppose \(f\) maps an open ball \(U \subset R^{ n }\) into \(R^{m}\) and \(f\) is differentiable on \(U\). Suppose there exists a real number \(M >0\) such that \(\vert f'(x) \vert \leq M\) for all \(x \in U\). Prove that \(\vert f(b)- f(a) \vert \leq M \vert b - a \vert\) for all \(a, b \in U\)

[15M]

3) Find and classify the extreme values of the function \(f(x, y)=x^{2}+y^{2}+x+y+x y\)

[15M]

4) Suppose \(\alpha\) is a real number not equal to \(n \pi\) for any integer \(n\). Prove that \(\int_{0}^{\infty} \int_{0}^{\infty} e^{-\left(x^{2}+2 x y \cos \alpha+y^{2}\right)} d x d y=\dfrac{\alpha}{2 \sin \alpha}\)

[15M]