IFoS PYQs 5
2004
1) Using Lagrange’s mean value theorem, show that \(1-x<e^{-x}<1-x+\dfrac{x^{2}}{2}, x>0\).
[10M]
2) \(f(x, y)=\left\{\begin{array}{cc}\dfrac{x y\left(x^{2}-y^{2}\right)}{\left(x^{2}+y^{2}\right)}, & (x, y) \neq(0,0) \\ 0, & (x, y)=(2,6)\end{array}\right.\)
show that \(f_{xy}(0,0) \neq f_{yx}(0,0)\)
[10M]
3) Using Lagrange’s multipliers, find the volume of the greatest rectangular parallelopiped that can be inscribed in the sphere \(x^{2}+y^{2}+z^{2}=1\).
[10M]
4) Evaluate the integral \(\iint_{R} \dfrac{x e^{-y^{4}}}{y} d x d y\), where \(R\) is the triangular region in the first quadrant bounded by \(y=x\) and \(x=0\) .
[10M]
5) Evaluate \(\int_{0}^{1} x^{m}\left(\ln \dfrac{1}{x}\right)^{n} d x, m, n>-1\)
[10M]
6) Find the volume cutt off the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) by the cone \(x^{2}+y^{2}=z^{2}\).
[10M]
2003
1) Let \(\mathrm{f}, \mathrm{g}: \mathrm{[a}, \mathrm{b}] \rightarrow \mathrm{IR}\) be functions such that \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}^{\prime}(\mathrm{x})\) exist for all \(\mathrm{x} \in[\mathrm{a}, \mathrm{b}]\) and \(\mathrm{g}^{\prime}(\mathrm{x}) \neq 0\) for all \(x\) in \((a, b)\). Prove that for some \(c \in(a, b)\)
\(\dfrac{f(c)-f(a)}{g(b)-g(c)}=\dfrac{f(c)}{g^{\prime}(c)}\)
[10M]
2) Let \(f:R^{2} \rightarrow R\) be defined by
\(f(x, y)=\dfrac{x y^{2}}{x^{2}+y^{2}}\) for \((x, y) \neq(0,0)\)
\(=0\) for \((x, y)=(0,0)\)
Show that the partial derivatives \(D_{1} f(0,0)\) and \(D_{2} f(0,0)\) vanish but \(f\) is not differentiable at \((0,0)\).
[10M]
3) Let \(f(x)=e^{-i x^{2}}(x \neq 0)\) \(=0\) for \(x=0\) Show that \(f(0)=0\) and \(f^{\prime}(0)=0\)
Write \(\mathrm{f}^{(\mathrm{k})}(\mathrm{x})\) as \(\mathrm{P}\left(\dfrac{1}{x}\right) f(x)\) for \(\mathrm{x} \neq 0,\) where \(\mathrm{P}\) is a polynomial and \(\mathrm{f}^{\mathrm{k} 0}\) denotes the \(\mathrm{k}^{\mathrm{th}}\) derivation of \(\mathrm{f}\).
[10M]
[TBC]
4) Using Lagrange multipliers, show that a rectangular box with lid of volume 1000 cubic units and of least surface area is a cube of side 10 units.
[10M]
5) Show that the area of the surface of the solid obtained by revolving the arc of the curve \(y=c\) cosh \(\left(\dfrac{x}{c}\right)\) joining \((0, c)\) and \((x, y)\) about the \(x\) -axis is.
[10M]
[TBC]
6) Define \(\Gamma:(0, \infty) \rightarrow R\) by
\(\Gamma(x)=\int_{0}^{\infty} f^{x} e^{-t} d t\). Show that this integral converges for all \(\mathrm{x}>0\) and that \(\Gamma(\mathrm{x}+\mathrm{t})=\mathrm{x} \mathrm{\Gamma}(\mathrm{x})\).
[10M]
2002
1) Find the extremum values of \(x^{2}+y^{2}\) subject to the condition \(3 x^{2}+4 x y+6 y^{2}=140\)
[10M]
2) Prove that
\[2^{2 x-1} \Gamma(x) \Gamma\left(x+\dfrac{1}{2}\right) \sqrt{\pi} \Gamma(2 x)\][10M]
TBC
3) Find the volume and centroid of the region in the first octant bounded by \(6 x+3 y+2 z=6\).
[10M]
4) If \(f(x)=e^{-x \dfrac{1}{2}}\) and \(g(x)=xf(x)\) for all \(x,\) prove that
\(f(y)=\sqrt{\dfrac{2}{\pi}} \int_{0}^{\infty} f(x) \cos x y d x\)
\(g(y)=\sqrt{\dfrac{2}{\pi}} \int_{0}^{\infty} g(x) \sin x y d x\)
[10M]
TBC
5) If \(\sin ^{-1} x+\sin ^{-1} y\) and
\(v=x \sqrt{1-y^{2}}+\sqrt{1-x^{2}}\)
determine whether there is a functional relationship between \(u\) and \(v\), and if so find it.
[10M]
6) If \(f(x)\) is monotonic in the interval \(0<x<a,\) and the integral \(\int_{0}^{a} x^{p} f(x) d x\) exists, then show that \(\lim _{x \rightarrow 0} x^{p+1} f(x)=0\).
[10M]
2001
1) Let \(f\) be a function defined on [0,1] by
\[f(x)=\left\{\begin{array}{l} 0, \text{ if } x \text { is irrational } \\ \dfrac{1}{q}, \text{ if } x=\dfrac{p}{q}, q \neq 0 \end{array}\right.\]and \(\mathrm{p}, \mathrm{q},\) are relatively prime positive integers. Show that \(f\) is continuous at each irrational point and discontinuous at each rational point \(\dfrac{p}{q}\).
[10M]
2) Show that the function \([x],\) where \([x]\) denotes the greatest integer not greater than \(x,\) is integrable in \([0,3]\). Also evaluate \(\int_{0}^{3}[x] d x\).
[10M]
3) Examine the convergence of the integral \(\int_{0}^{1} x^{n-1} \log x d x\).
[10M]
4) Examine the function
\[f(x, y)=\left\{\begin{array}{ll}\dfrac{x y}{\sqrt{x^{2}+y^{2}}},(x, y) \neq(0,0) \\ 0, (x, y)=(0,0)\end{array}\right.\]for continuity, partial derivability of the first order and differentiability at \((0,0)\).
[10M]
5) Find the maximum and minimum values of the function \(f(x, y, z)=x y+2 x\) on the circle which is the intersection of the plane \(x+y+z=0\) and the sphere \(x^{2}+y^{2}+z^{2}=24\).
[10M]
6) Find the volume of the region \(\mathrm{R}\) lying below the plane \(z=3+2 y\) and above the paraboloid \(z=x^{2}+y^{2}\).
[10M]
2000
1) Prove that the stationary values of \(u \equiv \dfrac{x^{2}}{a^{4}}+\dfrac{y^{2}}{b^{4}}+\dfrac{z^{2}}{c^{4}}\) subject to the conditions \(L x+m y+n z=0\) and \(\dfrac{x^{2}}{2}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{a x^{2}}=1\) are the roots of the equations \(\dfrac{l^{2} a^{4}}{1-a^{2} u}+\dfrac{m^{2} b^{4}}{1-b^{2} u}+\dfrac{n^{2} c^{4}}{1-c^{2} u}=0\).
[10M]
2) Evaluate \(\int_{-1}^{1} x^{3} d x\) from first principles by using Riemann’s theory of integration.
[10M]
3) If \(f(x)\) is a continuous function of \(x\) satisfying \(f(x+y)=f(x)+f(y),\) for all real numbers \(x\), \(y\) then prove that \(f(x)=A x,\) for all real numbers \(x\), where \(A\) is a constant.
Express \(y=\left(x+\sqrt{1+x^{2}}\right)\) in ascending powers of \(x\), by Taylor’s theorem.
[20M]
4) Using the transformations \(u=\dfrac{x^{2}+y^{2}}{x}, v=\dfrac{x^{2}+y^{2}}{y}\) evaluate \(\iint \dfrac{\left(x^{2}+y^{2}\right)^{2}}{x^{2} y^{2}} d x d y\) over the area common to the circles \(x^{2}+y^{2}-a x=0\) and \(x^{2}+y^{2}-b y=0\).
[10M]
5) Evaluate \(\iiint(1-x-y-z)^{t-1} x^{m-1} y^{n-1} z^{p-1} d x d y d z\) over the interior of the tetrahedron bounded by the planes \(x=0\), \(y=0\), \(z=0\), \(x+y+z=1\).
[10M]