1988

1) Show that the Cauchy product of \(\sum_{n=0}^{\infty} \dfrac{(-1)^{n}}{\sqrt{n+1}}\) with itself diverges.

2) Prove that the sequence \(\left\{S_{n}(x)\right\}\) where \(S_{n}(x)=n x e^{-n x^{2}}\) is not uniformly convergent in any interval \((0, \mathrm{k}), \mathrm{k}>0\).

3) Evaluate
\(\iint\left(1-\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}\right)^{1 / 2} d x d y\) over the area of the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\).

4) Discuss the convergence of the improper integral \(\int_{0}^{x} t^{x-1} e^{-t} d t\).

5) Show that a local extreme value of f given by \(f(\bar{x})=x_{1}^{k}+\ldots .+x_{n}^{k}\), \(\bar{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right)\)
subject to the condition \(x_{1}+x_{2}+\ldots \ldots+x_{n}=a,\) is \(a^{k} n^{1-k}\).

6) The function \(f: R^{2} \rightarrow R^{1}\) is given by
\(f(x, y)=\left\{\begin{array}{cc}\dfrac{x^{2} y}{x^{2}+y^{2}} & \text { when }(x, y) \neq(0,0) \\ 0 & \text { when }(x, y)=(0,0)\end{array}\right.\) Prove that at (0,0) \(\mathrm{f}\) is continuous and possesses all directional derivatives but is not differentiable.

1987

1) Let

2) If \(f\) is continuous on a closed and finite interval \([a,b]\) then show that \(f\) is uniformly continuous on \([\mathrm{a}, \mathrm{b}]\).

3) Test for convergence \(\int_{0}^{-x^{2}} d x\).

4) If \(\mathrm{f}\) is monotonic and \(\mathrm{g}\) is bounded and real valued function Riemann integrable over \([\mathrm{a}, \mathrm{b}]\), then Prove that there exists a \(c \in[a, b]\) such that

\(\int_{0}^{b} f(x) g(x) d x\)=\(f(a) \int_{0}^{c} g(x) d x+f(b) \int_{c}^{b} g(x) d x\)

5) Test for uniform convergence of the sequence \(\left\{S_{n}(x)\right\}\) where \(S_{n}(x)=n x(1-x)^{n}\) when \(0^{\prime \prime} x^{\prime \prime \prime} 1\).

6) Find the maximum and minimum value of \(f(x)=x y\) subject to the condition that \(x^{2}+y^{2}+x y=a^{2}\).

1986

1) If \(f(x)=x^{2}\) for all \(x \in R,\) then show that \(f\) is uniformly continuous on every closed and finite interval, but is not uniformly continuous on \(\mathrm{R}\).

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

3) If fand \(\mathrm{g}\) are differentiable on \([\mathrm{a}, \mathrm{b}]\) and \(f^{\prime}, \mathrm{g}^{\prime}\) are Riemann integrable over \([\mathrm{a}, \mathrm{b}]\), then show that \(\int_{0}^{b} f(x) d g(x)+\int_{a}^{b} g(x) d f(x)=f(b) g(b)-f(a) g(a)\).

4) If \(\mathrm{f}\) is monotonic and \(\mathrm{g}\) is Riemann integrable over \([\mathrm{a}, \mathrm{b}],\) then show that there exists a \(C \in[a, b]\) such that

\(\int_{0}^{b} f(x) g(x) d x\)=\(f(a) \int g(x) d x+f(b) \int_{c}^{b} g(x) d x\)

5) Find the maximum and minimum values of \(f(x, y)=7 x^{2}+8 x y+y^{2}\) where \(x\), \(y\) are connected by the relation \(x^{2}+y^{2}=1\).

1985

1) Examine the convergence of the integral \(\int_{0}^{\infty} \dfrac{\sin x^{\infty}}{x^{n}} d x\).

2) State and prove the second mean value theorem for Riemann integrals.

3) Show that for the function \(f(x, y)=\left\{\begin{array}{cc}\dfrac{x^{2} y^{2}}{x^{2}+y^{2}} & ;(x, y) \neq(0,0) \\ 0 & ; x=0, y=0\end{array}\right.\)
(i) \(f_{x}\) is not differentiable at \((0,0)\)
(ii) \(f_{y x}\) is not continuous at \((0,0)\)
(iii) \(f_{x y}(0,0)=f_{y x}(0,0)\)

1984

1) If \(f\) is monotonic on \([a, b]\) and if \(\alpha\) is continuous on \([\mathrm{a}, \mathrm{b}],\) then prove that \(\int_a^b f d x\) exists.

2) If \(f\) and \(\alpha^{\prime}\) are integrable in the sense of Riemann on \([\mathrm{a}, \mathrm{b}],\) then prove that \(\int f d \alpha=\int_{0}^{b} f(x) \alpha^{\prime}(x) d x\).

3) Show that the maximum and minimum values of the function \(u=x^{2}+y^{2}+x y\), where \(a x^{2}+b y^{2}=a b(a>b>0)\) are given by \(4(u-a)(u-b)=a b\).

4) Discuss the continuity and differentiability at (0,0) of the function \(f(x, y)=x^{2} \tan ^{-1}(y / x)-y^{2} \tan ^{-1}(x / y)\); \(f(0,0)=0\).

Also examine if \(f x y\) and \(f y x\) are equal at \((0,0)\).

1983

1) If \(f(x) \geq 0\) and \(f\) decreases monotonically for \(x \geq 1,\) then prove that \(\int_{1}^{\infty} f(x) d x\) converges iff \(\sum_{n=1}^{\infty} f(n)\) converges.

2) Examine the convergence of the integral \(\int_{0}^{1}\left(x^{p}+x^{-p}\right) \log (1+x) \dfrac{d x}{x}\).

3) Obtain a set of sufficient conditions such that for a function \(f(x, y)\), \(\dfrac{\partial^{2} f}{\partial x \partial y}\)=\(\dfrac{\partial^{2} f}{\partial y \partial x}\).

4) Find the maximum and minimum values of \(x^{2}+y^{2}+z^{2}\) subject to the conditions \(x+y+z=1\); \(xyz+1=0\).