1994

1) \(f(x)\) is defined as follows \(f(x)= \{\begin{array}{lll} \dfrac{1}{2}\left(b^{2}-a^{2}\right) & \text { for } & 0<x \leq a \\ \dfrac{1}{2} b^{2}-\dfrac{x^{2}}{6}-\dfrac{a^{3}}{3 x} & \text { for } & a<x \leq b \\ \dfrac{1}{3} \dfrac{b^{3}-a^{3}}{x} & \text { for } & x>b\end{array}\)
Prove that \(f(x)\) and \(f(x)\) are continuous but \(f ^{\prime}(x)\) is discontinuous.

[10M]

2) If \(\alpha\) and \(\beta\) lie between the least and greatest values of \(a , b , c\) prove that \(\begin{array}{|lll|} f(a) & f(b) & f(c) \\ \phi(a) & \phi(b) & \phi(c) \\ \psi(a) & \psi(b) & \psi(c) \end{array} =K \begin{array}{|lll|} f(a) & f^{\prime}(\alpha) & f^{\prime \prime}(\beta) \\ \phi(a) & \phi^{\prime}(\alpha) & \phi^{\prime \prime}(\beta) \\ \psi(a) & \psi^{\prime}(\alpha) & \psi^{\prime \prime}(\beta) \end{array}\) Where \(K =1 / 2( b - c )( c - a )( a - b )\)

[10M]

3) Prove that of all rectangular parallelopipeds of the same volume, the cube has the least surface.

[10M]

4) Show by means of beta function that \(\int_{t}^{z} \dfrac{d x}{(z-x)^{1-\alpha}(x-t)^{\alpha}}=\dfrac{\pi}{\sin \pi \alpha}(0<\alpha<1)\)

[10M]

5) Prove that the value of \(\iiint \dfrac{d x d y d z}{(x+y+z+1)^{3}}\) taken over the volume bounded by the co-ordinate planes and the plane \(x+y+z=1,\) is \(\dfrac{1}{2}\left(\log 2-\dfrac{5}{8}\right)\)

[10M]

6) The sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) is pierced by the cylinder \(\left(x^{2}+y^{2}\right)^{2}=a^{2}\left(x^{2}-y^{2}\right) .\) Prove by the cylinder \(\left(x^{2}+y^{2}\right)^{2}=a^{2}\left(x^{2}-y^{2}\right) .\) Prove that the volume of the sphere that lies inside the cylinder is \(\dfrac{8 a^{3}}{3}\left[\dfrac{\pi}{4}+\dfrac{5}{3}=\dfrac{4 \sqrt{2}}{3}\right]\)

[10M]

1993

1) Prove that \(f(x)=x^{2} \sin \dfrac{1}{x}, x \neq 0\) and \(f(x)=0\) for \(x=0\) is continuous and differentiable at \(x=0\) but its derivative is not continuous there.

[10M]

2) If \(f(x), \phi(x), \Psi(x)\) have derivatives when \(a \leq x \leq b,\) show that there is a value \(c\) of \(x\) lying between a and b such that

[10M]

3) Find the triangle of maximum area which can be inscribed in a circle.

[10M]

4) Prove that \(\int_{0}^{\infty} e^{-\omega x^{2}} d x=\dfrac{\sqrt{\pi}}{2 \sqrt{a}}(a>0)\) and deduce that \(\int_{0}^{\infty} x^{2 n} e^{-x^{2}} d x=\dfrac{\sqrt{\pi}}{2^{n+1}}[1,3.5 \quad(2 n-1)]\)]

[10M]

5) \(\Gamma{n\left(n+\dfrac{1}{2}\right)}=\dfrac{\sqrt{\pi}}{2^{2 n-1}} \Gamma{2 n}\)

[10M]

6) Show that the volume common to the sphere \(x ^{2}+ y ^{2}+z^{2}= a ^{2}\) and the cylinder \(x ^{2}+ y ^{2}= ax\) is \(\dfrac{2 a^{3}}{9}(3 \pi-4)\)

[10M]

1992

1) If \(y=e^{a x} \cos b x,\) prove that \(y_{2}-2 x y_{1}+\left(a^{2}+b^{2}\right) y=0\) and hence expand \(\mathrm{e}^{2 \mathrm{x}} \cos bx\) in powers of \(\mathrm{x}\). Deduce the expansion of \(\mathrm{e}^{\mathrm{ax}}\) and \(\cos bx\).

[10M]

2) If \(x=r \sin \theta \cos \phi\), \(y=r \sin \theta \sin \phi\), \(z=r \cos \theta\), then prove that
\(\mathrm{dx}^{2}+\mathrm{dy}^{2}+\mathrm{d} z^{2}=\mathrm{dr}^{2}+\mathrm{r}^{2} \mathrm{d} \theta^{2}+\mathrm{r}^{2} \sin ^{2} \theta \mathrm{d} \phi^{2}\)

[10M]

3) Find the dimensions of the rectangular parallelepiped inscribed in the ellipsoid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\) that has greatest volume.

[10M]

4) Prove that the volume enclosed by the cylinders \(x^{2}+y^{2}=2 a x, z^{2}=2 a x\)

is \(128 a^{3} / 15\).

[10M]

5) Find the centre of gravity of the volume formed by revolving the area bounded by the parabolas \(y^{2}=4 a x\) and \(x^{2}=4 b y\) about the \(x-axis\).

[10M]

6) Evaluate the following integral in terms of Gamma function: \(\int_{-1}^{+1}(1+x)^{P}(1-x)^{q} d x, \quad[p>-1, q>-1]\) and prove that \(\Gamma(1 / 3) \Gamma(2 / 3)=\dfrac{2}{\sqrt{3}} \pi\)

[10M]

1991

1) Sketch the curve \(\left(x^{2}-a^{2}\right)\left(y^{2}-b^{2}\right)=a^{2} b^{2}\)

2) Show that the function \(f(x, y)=y^{2}+x^{2} y+x^{4}\) has \((0,0)\) as the only critical point and that \(f(x, y)\) has a minimum at this point.

3) Evaluate \(\iint(1-x-y)^{t-1} x^{m-1} y^{n-1} d x d y,\) where \(D\) is the interior of the triangle formed by the lines \(x=0, y=0, x+y=1, l, m, n\) being all positive.

4) Find the equation of the cubic curve which has the same asymptotes as the curve \(x^{3}-6 x^{2} y+11 x y^{2}-6 y^{3}+x+y+1=0\) and which passes through the points \((0,0)\), \((1,0)\) and \((0,1)\).

5) Prove, by considering the integral \(\iint_{E} x^{2 m-1} y^{2 n-1} e^{-x^{2}-y^{2}} d x d y\) where \(E\) is the square [0\(\mathrm{R} ; 0, \mathrm{R}],\) or otherwise that \(B(m, n)=\dfrac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)}\).

1990

1.(a) If a function \(f(x)\) of the real variable \(x\) has the first 5 derivatives 0 at a given value \(x=a,\) show tht it has a maximum or a minimum at \(x=a\) according as the 6th derivative is negative or positive. What happens if only the first four derivatives are 0 but not the fifth?

1.(b) Show that \(f(x)=(x-2)^{2}\left(x^{2}+2 b x+c\right)(x+3)^{3}\) has a critical point at \(x=-1,\) if and only if, \(2 b+5 c=7\).

1.(c) Assuming that the condition in (2) holds, examine the nature of three critical points of \(f(x)\) detailing whether \(f(x)\) achieves maximum or minimum or else is stationary.

1.(d) Show that the function in (b) has atleast three (real) points of inflexion, irrespective of the condition in (2).

2) The functions \(f_{n}\) in on [0,1] are given by \(f_{n}(x)=\dfrac{n x}{1+n^{2} x^{p}},(p>0)\).

For what values of p does the sequence \(\{\mathrm{f}\}\) converge uniformly to its limit f? Examine whether \(\int_{0}^{1} f_{n} \rightarrow \int_{0}^{1} f\) for \(\mathrm{p}=2\) and \(\mathrm{p}=4\)

1989

1) If \(\mathrm{f}\) is at least thrice continuously differentiable then show that \(f(a+b)=f(a)+h f^{\prime}(a)+\dfrac{h^{2}}{2 !} f^{\prime \prime}(a+\theta h)\), where \(\theta\) lies between 0 and 1 and prove that \(\operatorname{Lt}_{h \rightarrow 0} \theta=\dfrac{1}{2}\)

2) Prove that the volume of a right circular cylinder of greatest volume which can be inscribed in a sphere, is \(\dfrac{\sqrt{3}}{3}\) times that of the sphere.

3) Find the surface of the solid generated by the rotation of the Astroid \(x=a \cos ^{3} t\); \(y=a \sin ^{3} t\) about the axis of \(x\).

4) Evaluate \(\iiint(1-z)^{1 / 2} d x d y d z\) over the interior of the tetrahedron with faces \(x=0, y=0, z=0, x+y+z=1\).