IFoS PYQs 4
2008
1) Obtain the values of the constants a, \(b\) and \(c\) for which the function defined by
\[f(x)=\left\{\begin{array}{cc} \dfrac{\sin (a+1) x+\sin x}{x} & , x<0 \\ c & , x=0 \\ \dfrac{\left(x+b x^{2}\right)^{1 / 2}-x^{1 / 2}}{b x^{3 / 2}}, & x>0 \end{array}\right.\]is continuous at \(x=0\).
[10M]
2) If \(u=\sin ^{-1}\left(\dfrac{x^{3}+y^{3}}{\sqrt{x}+\sqrt{y}}\right),\) prove that
\(x \dfrac{\partial u}{\partial x}+y \dfrac{\partial u}{\partial y}=\dfrac{5}{2} \tan u\).
[10M]
3) Determine the value of
\(\left(\int_{0}^{1} \dfrac{x^{2}}{\left(1-x^{4}\right)^{1 / 2}} d x\right)\left(\int_{0}^{1} \dfrac{d x}{\left(1+x^{4}\right)^{1 / 2}}\right)\)
[10M]
4) Obtain the value of the double integral \(\iint_{D}\left(x^{2}+y^{2}\right) d x d y\) where \(D\) represents the region bounded by the straight line \(y=x\) and the parabola \(y^{2}=4 x\).
[10M]
5) A wire of length \(b\) is cut into two parts which are bent in the form of a square and a circle respectively. Find the minimum value of the sum of the areas so formed.
[10M]
6) Calculate the volume cut off from the sphere \(x^{2}+y^{2}+z^{2}\) \(=a^{2}\) by the right circular cylinder given by \(x^{2}+y^{2}=b^{2}\).
[10M]
2007
1) If a function \(f\) is such that its derivative \(f\) is continuous on \([\mathrm{a}, \mathrm{b}]\) and derivable on \(]\mathrm{a}, \mathrm{b}[\), then show that there exists a number \(c\) between \(a\) and \(b\) such that \(\) f(b)=f(a)+(b-a)f’(a)+\dfrac{1}{2}(b-a)^{2} f^{\prime \prime}(c) \(\)
[10M]
2) If
\[f(x, y)=\left\{\begin{array}{cl} \dfrac{x^{2} y^{2}}{x^{4}+y^{4}}, & (x, y) \neq(0,0) \\ 0 & , \quad(x, y)=(0,0) \end{array}\right.\]Show that both the partial derivatives exist at (0,0) but the function is not continuous thereat.
[10M]
3) Find the values of \(a\) and \(b\), so that
\[\operatorname{lit}_{x \rightarrow 0} \dfrac{x(1+a \cos x)-b \sin x}{x^{3}}=1\]What are these conditions?
[10M]
4) Show that \(f(x y, z \quad 2 x)=0\) satisfies, under certain conditions, the equation
\[x \dfrac{\partial z}{\partial x}-y \dfrac{\partial z}{\partial y}=2 x\]What are these conditions?
[10M]
5) Find the surface area generated by the revolution of the cardioids \(r=a(\mathrm{I}+\cos \theta)\) about the initial line.
[10M]
6) The function \(f\) is defined on \(] 0,1[\) by
\[f(x)=(-1)^{n+1} n(n+1), \dfrac{1}{n+1} \leq x \leq \dfrac{1}{n}, n \in \mathrm{N}\]Show that
\[\int_{0}^{1} f(x) d x\]Does not converge.
[10M]
2006
1) Show that
\(\int_0^{\infty} \dfrac{\tan^{-1} \alpha x \tan^{-1} \beta x}{x^2} dx\) = \(\dfrac{\pi}{2} \dfrac{(\alpha+\beta)^{(\alpha+\beta)}}{\alpha^\alpha + \beta^\beta}\), \(\alpha\), \(\beta >0\)
[10M]
2) Show that \(f_{x}(0,0) \neq f_{y}(a, 0)\) where \(f(x, y)=0\), if \(x y=0\), \(f(x, y)=x^{2} \tan^{-1} \dfrac{y}{x}-y^{2} \tan^{-1} \dfrac{x}{y}\), if \(xy \neq 0\)
[10M]
3) Find the volume under the spherical surface \(x^{2}+y^{2}+z^{2}=a^{2}\) and over the lemniscate \(r^{2}=a^{2}\) \(\cos 2 \theta\).
[10M]
4) Find the centre of gravity of the volume common to a cone of vertical angle \(2 \alpha\) and a sphere of radius \(a\), the vertex of the cone being the centre of the sphere.
[10M]
5) Using Lagrange’s method of undetermined multipliers, find the stationary values of \(\mathrm{x}^{2}+\mathrm{y}^{2}+\) \(z^{2}\) subject to \(a x^{2}+b y^{2}+c z^{2}=1\) and \(lx+m y+n z=0\). Interpret geometrically.
[10M]
6) Find the extreme values of \(f(x, y)=2(x-y)^{2}-x^{4}-y^{4}\).
[10M]
2005
1) Let \(f(x)=\left\{\begin{array}{c}x^{2} \sin \dfrac{1}{x} \text { for } x \neq 0 \\ 0 . \text { for } x=0\end{array}\right.\)
Show that \(\mathrm{f}\) is differentiable at each point of reals but \(\mathrm{f}(\mathrm{x})\) is not continuous at \(\mathrm{x}=0\).
[10M]
2) Show that \(f: R^{2} \rightarrow R\) defined by \(f(x, y) 2 x^{2}-6 x y+3 y^{2}\) has a critical point at \((0,0)\) and that it is a saddle point.
[10M]
3.(i) Using Taylor’s theorem with remainder show that \(x-\dfrac{x^{3}}{6} \leq \sin x \leq x-\dfrac{x^{3}}{6}+\dfrac{x^{5}}{120} \text { for all } x \geq 0\)
[5M]
3.(ii) Let \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) be defined by \(\begin{aligned} f(x, y) &=\dfrac{x y}{x^{2}-y^{2}} \text { if } x \neq \pm y \\ &=0 \quad \text { if } x=\pm y \end{aligned}\)
[5M]
4) Show that the curve given by \(x^{3}-4 x^{2} y+5 x y^{2}-2 y^{3}+3 x^{2}-4 x y+2 y^{2}-3 x+2 y-1=0\) has only one asymptote given by \(y=\dfrac{1}{2} x+3\).
[10M]
5) Find the extremum values of \(f(x, y)=2 x^{2}-8 x y+9 y^{2} \text { on } x^{4}+y^{2}-1=0\) using Lagrange multiplier method.
[10M]
6) A solid cuboid in \(R^3\) given in spherical coordinates by \(R=[0,a]\), \(\theta=[0, \pi]\), \(\varphi=0, \pi/4\) has a density function \(\rho(\mathbf{R}, \theta, \varphi)=4 \mathbf{R} \sin \dfrac{\theta}{2} \cos \varphi .\) Find the total mass of \(\mathbf{C}\)
[10M]