1988

1) If $f(x)=\tan x$ prove that $f^{\mathrm{\prime }\mathrm{\prime }}(0)={}^n{c}_2{f}^{n-2}(0)+{}^n{c}_4{f}^{n-4}(0)\dots ..=\sin {\displaystyle \frac{n\pi }{2}}$.

2) Find the minimum value of $x^2+y^2+z^2$ when $x+y+z=k$.

3) Find the a symptotes of the cubic $x^3-x{y}^2-2xy+2x-y=0$ and show that they cut the curve again in points which lie on the line $3x-y=0$.

4) Evaluate $\iint x^{3/}2y^2(1-x^2-y^2)dxdy$ over the positive quadrant of the circle $x^2+y^2=1$.

1987

1) If $x_1={\displaystyle \frac{1}{2}}(x+{\displaystyle \frac{9}{x}})$ and for $n>0,x_{n+1}={\displaystyle \frac{1}{2}}(x_n+{\displaystyle \frac{9}{x_n}})$, find the value of $\underset{n\to x}{lim}{x}_n(x>0$ is assumed).

2) If $x=-a,h=2a,f(x)=x^{{\displaystyle \frac{1}{3}}},$ find $\theta$ from the mean value theorem $$ f(x+h)=f(x)+h f^{\prime}(x+\theta h) $$

3) If $u=x+y-z,v=x-y+z$ and $w=x^2+(y-z{)}^2,$ examine whether or not there exists any functional relationship between $\mathrm{u},\mathrm{v}$ and $\mathrm{w},$ and find the relation, if any.

4) If $u=cose{c}^{-1}{\left({\displaystyle \frac{x^{1/(n+1)}+y^{1/(n+1)}}{x+y}}\right)}^{1/2}$, show that $x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}+2xy{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }y}}+y^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}}$ = ${\displaystyle \frac{1}{4n^2}}\tan u(2n+{\sec }^{2}u).$

5) Show, by means of a suitable substitution that ${\int }_0^{\pi /2}{\sin }^{2x-1}\theta {\cos }^{2y-1}\theta d\theta$=${\displaystyle \frac{1}{2}}{\int }_0^{\mathrm{\infty }}{\displaystyle \frac{t^{x-1}}{(1+t{)}^{x+y}}}dt,$x, y,>0$.Establishtheinequality$\dfrac{1}{2}<\int_{0}^{1} \dfrac{d x}{\left(4-x^{2}+x^{3}\right)^{\dfrac{1}{2}}}<\dfrac{\pi}{6}$$.

6) Find the volume of the solid generated by revolving the curve $y^2={\displaystyle \frac{x^3}{2a-x}};a>0,$ about its asymptote $x=2a$.

7) Evaluate ${\iint }_D{x}^{1/2}{y}^{1/2}(1-x-y{)}^{2/3}dxdy,$ where $D$ is the domain bounded by the lines $x=0$, $y=0$, $x+y=1$.

1986

1) A function $f(x)$ is defined as follows:

Examine whether or not $f(x)$ is differentiable at $x=0$.

2) If $f^{\mathrm{\prime }}(x)$ exists and is continuous, find the value of $L_{x\to 0}{\displaystyle \frac{1}{x^2}}{\int }_0^x(x-3y)f(y)dy$

3) Evaluate ${\int }_0^2{\int }_0^x{\{(x-y{)}^2+2(x+y)+1\}}^{-{\displaystyle \frac{1}{2}}}$ dxdy by using the transformation $x=u(1+v),y=v(1+u)$. Assume $u$, $v$ are positive in the region concerned.

4) Use Rolle’s theorem to establish that under suitable conditions (to be stated) $\left|\begin{array}{cc}f(a)& f(b)\\ g(a)& g(b)\end{array}\right|$=$(b-a)\left|\begin{array}{cc}f(a)& f^{\mathrm{\prime }}(\xi )\\ g(a)& g^{\mathrm{\prime }}(\xi )\end{array}\right|$, $a<\xi <b$. Hence or otherwise deduce the inequality $n{b}^{n-1}(a-b)<a^n-b^n<n{a}^{n-1}(a-b)$ where $a>b$ and $n>1$.

5) If $u={\displaystyle \frac{{(a{x}^3+b{y}^3)}^n}{3n(3n-1)}}+xf\left({\displaystyle \frac{y}{x}}\right)$ find the value of $x^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}+2xy{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }y}}+y^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}}$.

6) Without evaluating the involved integrals, show that ${\int }_1^x{\displaystyle \frac{tdt}{1+t^2}}+{\int }_1^{1/2}{\displaystyle \frac{dt}{t(1+t^2)}}=0$.

7) If $f(x)$ is periodic of period $\mathrm{T},$ show that ${\int }_a^{a+T}f(t)dt$ is independent of $a$.

1985

1) If $f(x,y)=\{\begin{array}{cl}{\displaystyle \frac{xy}{x^2+y^2}};& (x,y)\ne (0,0)\\ 0& ;(x,y)=(0,0)\end{array}$ Show that both the partial derivatives $f_{\mathrm{x}}$ and $f_{\mathrm{y}}$ exist at (0,0) but the function is not continuous there.

2) If for all values of the parameter $\lambda$ and for some constant ${}^{\mathrm{\prime }}{\mathrm{n}}^{\mathrm{\prime }},F(\lambda x,\lambda y)={\lambda }^{n}F(x,y)$ identically, where $\mathrm{F}$ is assumed differentiable, prove that $x{\displaystyle \frac{\mathrm{\partial }F}{\mathrm{\partial }x}}+y{\displaystyle \frac{\mathrm{\partial }F}{\mathrm{\partial }y}}=nF$.

3) Prove the relation between beta and gamma functions $\beta (m,n)={\displaystyle \frac{\mathrm{\Gamma }(m)\mathrm{\Gamma }(n)}{\mathrm{\Gamma }(m+n)}}$.

4) If a function $\mathrm{f}$ defined on $[\mathrm{a},\mathrm{b}]$ is continuous on $[\mathrm{a},$ b] and differentiable on $(a,b)$ and $f(a)=f(b)$, then prove that $\mathrm{\exists }$ atleast one real number “ ${\mathrm{c}}^{\mathrm{\prime }},\mathrm{a}<\mathrm{c}$ $<\mathrm{b}$ such that $f^{\mathrm{\prime }}(c)=0$.

5) Use Maclaurin’s expansion to show that $\log (1+x)=x-{\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{x^3}{3}}-{\displaystyle \frac{x^4}{4}}+\dots \dots ...$. Hence find the value of $\log (1+x+x^2+x^3+x^4)$.

1984

1) Show that $\tan x$ is not continuous at $x={\displaystyle \frac{\pi }{2}}$.

2) Let $f(x,y)=(x+y)\sin ({\displaystyle \frac{1}{x}}+{\displaystyle \frac{1}{y}}),x\uparrow 0,y\uparrow 0$ $f(x,0)=f(0,y)=0$ Examine whether (i) $f(x,y)$ is continuous at (0,0) and (ii) $Ltf(x,y),$ for $(y\uparrow 0)$ and ${\mathrm{Lt}}_{x\to 0}f(x,y)$ for $(x\uparrow 0),$ exist.

TBC

3) If $B(p,q)$ be the Beta function, show that $pB(n,q)=(q-1)B(p+1,q-1),$ where $\mathrm{p},\mathrm{q}$ are real $\mathrm{p}>0,\mathrm{q}>1$. Hence or otherwise find $\mathrm{B}(\mathrm{p},\mathrm{n})$ when ${}^{\mathrm{\prime }}{\mathrm{n}}^{\mathrm{\prime }}$ is an integer $(>0)$.

4) If $u={\displaystyle \frac{x+y}{1-xy}}$ and $v={\tan }^{-1}x+{\tan }^{-1}y,$ find ${\displaystyle \frac{\mathrm{\partial }(u,v)}{\mathrm{\partial }(x,y)}}$. Are $\mathrm{u}$ and $\mathrm{v}$ functionally related? If so, find the relationship.