Paper I PYQs-2012

1.(a) Define a function $f$ of two real variables in the plane by: $f(x,y)=\{\begin{array}{c}{\displaystyle \frac{x^3\cos {\displaystyle \frac{1}{y}}+y^3\cos {\displaystyle \frac{1}{x}}}{x^2+y^2}},\text{ for }x,y\ne 0\\ 0,\text{ otherwise }\end{array}$

Check the continuity and differentiability of $f$ at (0,0).

[12M]

1.(b) Let $p$ and $q$ be positive real numbers such that ${\displaystyle \frac{1}{p}}$ + ${\displaystyle \frac{1}{q}}$ =1. Show that for real numbers $a$, $b\ge 0$,

[12M]

1.(c) Prove or disprove the following statement: If $B=\{b_1,b_2,b_3,b_4,b_5\}$ is a basis for ${\mathbb{R}}^5$ and $V$ is a two-dimensional subspace of ${\mathbb{R}}^5$, then $V$ has a basis made of two members of $B$.

[12M]

1.(d) Let $T:{\mathbb{R}}^3\to {\mathbb{R}}^3$ be the linear transformation defined by $T(\alpha ,\beta ,\gamma )=(\alpha +2\beta -3\gamma ,2\alpha +5\beta -4\gamma ,\alpha +4\beta +\gamma$. Find a basis and the dimension of the image of $T$ and the kernel of $T$.

[12M]

1.(e) Prove that two of the straight lines represented by the equation $x^3+b{x}^{2}y+cx{y}^2+y^3=0$ will be at right angles, if $b+c=-2$.

[12M]

2.(a)(i) Let $V$ be the vector space of all $2\times 2$ matrices over the field of real numbers. Let $W$ be the set consisting of all matrices with zero determinant. Is $W$ a subspace of $V$? Justify your answer?

[8M]

2.(a)(ii) Find the dimension and a basis for the space $W$ of all solutions of the following homogeneous system using matrix notation:

[12M]

2.(b)(i) Consider the linear mapping $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ by $f(x,y)=(3x+4y,2x-5y)$. Find the matrix $A$ relative to the basis $\{(1,0),(0,1)\}$ and the matrix $B$ relative to the basis $\{(1,2),(2,3)\}$.

[12M]

2.(b)(ii) If $\lambda$ is a characteristic root of a non-singular matrix $A$, then prove that ${\displaystyle \frac{|A|}{\lambda }}$ is a characteristic root of $\mathrm{Adj}A$.

[8M]

2.(c) Let $H=\left[\begin{array}{ccc}1& i& 2+i\\ -i& 2& 1-i\\ 2-i& 1+i& 2\end{array}\right]$ be a Hermitian matrix. Find a non-singular matrix $P$ such that $D=P^{T}H\overline{P}$ is diagonal.

[20M]

3.(a) Find the points of local extrema and saddle points of the function $f$ for two variable defined by $f(x,y)=x^3+y^3-63(x+y)+12xy$.

[20M]

3.(b) Define a sequence $s_n$ of real numbers by $s_n=\sum _{i=1}^n{\displaystyle \frac{(\log (n+i)-\log n{)}^2}{n+i}}$. Does $\underset{n\to \mathrm{\infty }}{lim}{s}_n$ exist? If so, compute the value of this limit and justify your answer.

[20M]

3.(c) Find all the real values of $p$ and $q$ so that the integral ${\int }_0^1{x}^p{(\log {\displaystyle \frac{1}{x}})}^{q}dx$ converges.

[20M]

4.(a) Compute the volume of the solid enclosed between the surfaces $x^2+y^2=9$ and $x^2+z^2=9$.

[20M]

4.(b) A variable plane is parallel to the plane ${\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}}+{\displaystyle \frac{z}{c}}=0$ and meets the axes in $A$, $B$, $C$ respectively. Prove that circle $ABC$ lies on the cone $yz({\displaystyle \frac{b}{c}}+{\displaystyle \frac{c}{b}})+zx({\displaystyle \frac{c}{a}}+{\displaystyle \frac{a}{c}})+xy({\displaystyle \frac{a}{b}}+{\displaystyle \frac{b}{a}})=0$.

[20M]

4.(c) Show that locus of a point from which three mutually perpendicular tangent lines can be drawn to the paraboloid $x^2+y^2+2z^2=0$ is $x^2+y^2+4z=1$.

[20M]

5.(a) Solve ${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{2xy{e}^{(x/y{)}^2}}{y^2(1+e^{(x/y{)}^2})+2x^2{e}^{(x/y{)}^2}}}$.

[12M]

5.(b) Find the orthogonal trajectories of the family of curves $x^2+y^2=ax$.

[12M]

5.(c) Using Laplace transforms, solve the initial value problem $y^{\mathrm{\prime }\mathrm{\prime }}+2y^{\mathrm{\prime }}+y=e^{-t}$, $y(0)=-1$, $y^{\mathrm{\prime }}(0)=1$.

[12M]

5.(d) A particle moves with an acceleration

towards the origin. It it starts from rest at a distance $a$ from the origin, find its velocity when its distance from the origin is ${\displaystyle \frac{a}{2}}$.

[12M]

5.(e) If $\overrightarrow{A}=x^{2}yz\overrightarrow{i}-2x{z}^3\overrightarrow{j}+x{z}^2\overrightarrow{k}$, $\overrightarrow{B}=2z\overrightarrow{i}+y\overrightarrow{j}-x^2\overrightarrow{k}$, find the value of ${\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }x\mathrm{\partial }y}}(\overrightarrow{A}+\overrightarrow{B})$ at $(1,0,-2)$.

[12M]

6.(a) Show that the differential equation $(2xy\log y)dx+(x^2+y^2\sqrt{y^2+1})dy=0$ is not exact. Find an integrating factor and hence, the solution of the equation.

[20M]

6.(b) Find the general solution of the equation $y^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}-y^{\mathrm{\prime }\mathrm{\prime }}=12x^2+6x$.

[20M]

6.(c) Solve the ordinary differential equation $x(x-1)y^{\mathrm{\prime }\mathrm{\prime }}-(2x-1)y^{\mathrm{\prime }}+2y=x^2(2x-3)$.

[20M]

7.(a) A heavy ring of mass $m$, slides on a smooth vertical rod and is attached to a light string which passes over a small pulley distant $a$ from the rod and has a mass $M(>m)$ fastened to its other end. Show that if the ring be dropped from a point in the rod in the same horizontal plane as the pulley, it will descend a distance ${\displaystyle \frac{2Mma}{M^2-m^2}}$ before coming to rest.

[20M]

7.(b) A heavy hemispherical shell of radius $a$ has a particle attached to a point on the rim, and rests with the curved surface in contact with a rough sphere of radius $b$ at the highest point. Prove that if ${\displaystyle \frac{b}{a}}>\sqrt{5}-1$, the equilibrium is stable, whatever be the weight of the particle.

[20M]

7.(c) The end links of a uniform chain slide along a fixed rough horizontal rod. Prove that the ratio of the maximum span to the length of the chain is $\mu \log \left[{\displaystyle \frac{1+\sqrt{1+{\mu }^2}}{\mu }}\right]$, where $\mu$ is the coefficient of friction.

[20M]

8.(a) Derive the Frenet-Serret formulae. Define the curvature and torsion for a space curve. Compute them for the space curve $x=t$, $y=t^2$, $z={\displaystyle \frac{2}{3}}{t}^3$. Show that the curvature and torsion are equal for this curve.

[20M]

8.(b) Verify Green’s theorem in the plane for ${\oint }_C[xy+y^{2}dx+x^{2}dy]$ where $C$ is the closed curve of the region bounded by $y=x$ and $y=x^2$.

[20M]

8.(c) If $\overrightarrow{F}=y\overrightarrow{i}+(x-2xz)\overrightarrow{j}-xy\overrightarrow{k}$, evaluate ${\iint }_S(\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{F}).\overrightarrow{n}d\overrightarrow{s}$, where $S$ is the surface of the sphere $x^2+y^2+z^2=a^2$ above the $xy-plane$.

[20M]