1988

1) If \(f(x)=\tan x\) prove that \(f^{\prime \prime}(0)={ }^{n} c_{2} f^{n-2}(0)+{ }^{n} c_{4} f^{n-4}(0) \ldots . .=\sin \dfrac{n \pi}{2}\).

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 \(x^{3}-x y^{2}-2 x y+2 x-y=0\) and show that they cut the curve again in points which lie on the line \(3 x-y=0\).

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

1987

1) If \(x_{1}=\dfrac{1}{2}\left(x+\dfrac{9}{x}\right)\) and for \(n>0, x_{n+1}=\dfrac{1}{2}\left(x_{n}+\dfrac{9}{x_{n}}\right)\), find the value of \(\lim _{n \rightarrow x} x_{n}(x>0\) is assumed).

2) If \(x=-a, h=2 a, f(x)=x^{\dfrac{1}{3}},\) 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 \(u = cosec^{-1} \left( \dfrac{x^{1/(n+1)} + y^{1/(n+1)} }{x+y} \right)^{1/2}\), show that \(x^{2} \dfrac{\partial^{2} u}{\partial x^{2}}+2 x y \dfrac{\partial^{2} u}{\partial x \partial y}+y^{2} \dfrac{\partial^{2} u}{\partial y^{2}}\) = \(\dfrac{1}{4 n^{2}} \tan u\left(2 n+\sec ^{2} u\right).\)

5) Show, by means of a suitable substitution that \(\int_{0}^{\pi / 2} \sin ^{2 x-1} \theta \cos ^{2 y-1} \theta d \theta\)=\(\dfrac{1}{2} \int_{0}^{\infty} \dfrac{t^{x-1}}{(1+t)^{x+y}} d t,\)x, y,>0\(. Establish the inequality\)\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 \(y^{2}=\dfrac{x^{3}}{2 a-x} ; a>0,\) about its asymptote \(x=2a\).

7) Evaluate \(\iint_{D} x^{1/2} y^{1/2}(1-x-y)^{2/3} d x d y,\) 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 \(f^{\prime}(x)\) exists and is continuous, find the value of \(L_{x \rightarrow 0} \dfrac{1}{x^{2}} \int_{0}^{x}(x-3 y) f(y) d y\)

3) Evaluate \(\int_{0}^{2} \int_{0}^{x}\left\{(x-y)^{2}+2(x+y)+1\right\}^{-\dfrac{1}{2}}\) 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) \(\begin{vmatrix}f(a) & f(b) \\ g(a) & g(b)\end{vmatrix}\)=\((b-a)\begin{vmatrix}f(a) & f^{\prime}(\xi) \\ g(a) & g^{\prime}(\xi)\end{vmatrix}\), \(a< \xi < b\). 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 \(u=\dfrac{\left(a x^{3}+b y^{3}\right)^{n}}{3 n(3 n-1)}+x f\left(\dfrac{y}{x}\right)\) find the value of \(x^{2} \dfrac{\partial^{2} u}{\partial x^{2}}+2 x y \dfrac{\partial^{2} u}{\partial x \partial y}+y^{2} \dfrac{\partial^{2} u}{\partial y^{2}}\).

6) Without evaluating the involved integrals, show that \(\int_{1}^{x} \dfrac{t d t}{1+t^{2}}+\int_{1}^{1 / 2} \dfrac{d t}{t\left(1+t^{2}\right)}=0\).

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

1985

1) If \(f(x, y)=\left\{\begin{array}{cl}\dfrac{x y}{x^{2}+y^{2}} ; & (x, y) \neq(0,0) \\ 0 & ; \quad(x, y)=(0,0)\end{array}\right.\) 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 \(^{\prime} \mathrm{n}^{\prime}, F(\lambda x, \lambda y)=\lambda^{n} F(x, y)\) identically, where \(\mathrm{F}\) is assumed differentiable, prove that \(x \dfrac{\partial F}{\partial x}+y \dfrac{\partial F}{\partial y}=n F\).

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

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 \(\exists\) atleast one real number “ \(\mathrm{c}^{\prime}, \mathrm{a}<\mathrm{c}\) \(<\mathrm{b}\) such that \(f^{\prime}(c)=0\).

5) Use Maclaurin’s expansion to show that \(\log (1+x)=x-\dfrac{x^{2}}{2}+\dfrac{x^{3}}{3}-\dfrac{x^{4}}{4}+\ldots \ldots . . .\). Hence find the value of \(\log \left(1+x+x^{2}+x^{3}+x^{4}\right)\).

1984

1) Show that \(\tan x\) is not continuous at \(x=\dfrac{\pi}{2}\).

2) Let \(f(x, y)=(x+y) \sin \left(\dfrac{1}{x}+\dfrac{1}{y}\right), x \uparrow 0, y \uparrow 0\) \(f(x, 0)=f(0, y)=0\) Examine whether (i) \(f(x, y)\) is continuous at (0,0) and (ii) \(L t f(x, y),\) for \((y \uparrow 0)\) and \(\operatorname{Lt}_{x \rightarrow 0} f(x, y)\) for \((x \uparrow 0),\) exist.

TBC

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

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