2000

1) Given that the terms of a sequence {an} are such that each ak,k≤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 ∏∞n=0(1+1x2n) converges absolutely. Find its value whenever it converges.

[12M]

3) Suppose f is twice differentiable real valued function in (0,∞) and M0,M1 and M2 the least upper bounds of |f(x)|,|f′(x)| and |f′′(x)| respectively in (0,∞). Prove for each x>0,h>0 that f′(x)=12h[f(x+2h)−f(x)]−hf′(u) for some u∈(x,x+2h). Hence show that M21≤4M0M2

[15M]

4) Evaluate ∬ 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]