1988

1) If f(x)=tanx prove that f′′(0)=nc2fn−2(0)+nc4fn−4(0)…..=sinnπ2.

2) Find the minimum value of x2+y2+z2 when x+y+z=k.

3) Find the a symptotes of the cubic x3−xy2−2xy+2x−y=0 and show that they cut the curve again in points which lie on the line 3x−y=0.

4) Evaluate ∬x3/2y2(1−x2−y2)dxdy over the positive quadrant of the circle x2+y2=1.

1987

1) If x1=12(x+9x) and for n>0,xn+1=12(xn+9xn), find the value of lim 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.