Paper I PYQs-2017

1.(a) Let $A=\left[\begin{array}{cc}2& 2\\ 1& 3\end{array}\right]$. Find a non-singular matrix $P$ such that $P^{-1}AP$ is diagonal matrix.

[10M]

1.(b) Show that similar matrices have the same characteristic polynomial.

[10M]

1.(c) Integrate the function $f(x,y)=xy(x^2+y^2)$ over the domain $R:\{-3\le x^2-y^2\le 3,1\le xy\le 4\}$

[10M]

1.(d) Find the equation of the tangent at the point (1,1,1) to the conicoid $3x^2-y^2=2z$.

[10M]

1.(e) Reduce the following equation to the standard form and hence determine the conicoid:

[15M]

2.(a) Find the volume of the solid above the $xy-plane$ and directly below the portion of the elliptic paraboloid $x^2+{\displaystyle \frac{y^2}{4}}=z$ which is cut off by the plane $z=9$.

[15M]

2.(b) A plane passes through a fixed point $(a,b,c)$ and cuts the axes at the points $A$, $B$, $C$ respectively. Find the locus of the center of the sphere which passes through the origin $O$ and $A$, $B$, $C$.

[15M]

2.(c) Show that the plane $2x-2y+z+12=0$ touches the sphere $x^2+y^2+z^2-2x-4y+2z-3=0$. Find the point of contact.

[10M]

2.(d) Suppose $U$ and $W$ are distinct four dimensional subspaces of a vector space $V$, where $dimV=6$. Find the possible dimensions of subspace $U\cap W$.

[10M]

3.(a) Consider the matrix mapping $A:R^4\to R^3$, where $A=\left[\begin{array}{cccc}1& 2& 3& 1\\ 1& 3& 5& -2\\ 3& 8& 13& -3\end{array}\right]$. Find a basis and dimension of the image of $A$ and those of the kernel $A$.

[15M]

3.(b) Prove that distance non-zero eigenvectors of a matrix are linearly independent.

[10M]

3.(c) If $f(x,y)=\{\begin{array}{l}{\displaystyle \frac{xy(x^2-y^2)}{x^2+y^2}},(x,y)\ne (0,0)\\ 0,(x,y)=(0,0)\end{array}$
Calculate ${\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }x\mathrm{\partial }y}}$ and ${\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }y\mathrm{\partial }x}}$ at (0,0).

[15M]

3.(d) Find the locus of the points of intersection of three mutually perpendicular tangent planes to $a{x}^2+b{y}^2+c{z}^2=1$.

[10M]

4.(a) Reduce the following equation to the standard form and hence determine the nature of the conicoid: $x^2+y^2+z^2-yz-zx-xy-3x-6y-9z+21=0$.

[15M]

4.(b) Consider the following system of equation in $x$, $y$, $z$:
$x+2y+2z=1$
$x+ay+3z=3$
$x+11y+az=b$

i) For which values of $a$ does the system have a unique solution?
ii) For which of values $(a,b)$ does the system have more than one solution?

[15M]

4.(c) Examine if the improper integral ${\int }_0^3{\displaystyle \frac{2xdx}{{(1-x^2)}^{2/3}}}$ exists.

[10M]

4.(d) Prove that ${\displaystyle \frac{\pi }{3}}\le \iint {\displaystyle \frac{dxdy}{\sqrt{x^2+(y-2{)}^2}}}\le \pi$, where $D$ is the unit disc.

[10M]

5.(a) Find the differential equation representing the entire circle in the $xy-plane$.

[10M]

5.(b) Suppose that the streamlines of the fluid are given by a family of curves $xy=c$. Find the equipotential lines, that is, the orthogonal trajectories of the family of curves representing the streamlines.

[10M]

5.(c) A fixed wire is in the shape of the cardiod $r=a(1+\cos \theta )$, the initial line being the downward vertical. A small ring of mass $m$ can slide on the wire and is attached to point $r=0$ of the cardiod by an elastic string of natural length $a$ and modulus of elasticity 4 $\mathrm{m}\mathrm{g}$. The string is released from rest when the string is horizontal. Show by laws of conservation of energy that $a{\theta }^2(1+\cos \theta )-g\cos \theta (1-\cos \theta )=0$, $g$ being the acceleration due to gravity.

[10M]

5.(d) For what values of the constant $a$, $b$ and $c$ the vector $\overline{V}=(x+y+az)\hat{i}+(bx+2y-z)j+(-x+cy+2z)k$ is irrotational. Find the divergence in cylindrical coordinates of the vector with these values.

[10M]

5.(e) The position vector of a moving point at time $t$ is $\overline{r}=\sin t\hat{i}+\cos 2tj+(t^2+2t)k$. Find the components of acceleration $\overline{a}$ in the direction parallel to the velocity vector $\overline{v}$ and perpendicular to the plane of $\overline{r}$ and $\overline{v}$ at time $t=0$.

[10M]

6.(a)(i) Solve the following simultaneous liner differential equations:
$(D+1)y=z+e^x$ and $(D+1)z=y+e^x$
where $y$ and $z$ are functions of independent variable $x$ and $D\equiv {\displaystyle \frac{d}{dx}}$.

[8M]

6.(a)(ii) If the growth rate of the population of bacteria at time $t$ is proportional to the amount present at the time $t$ and population doubles in one week, then how much bacteria can be expected after 4 weeks?

[8M]

6.(b)(i) Consider the differential equation $xy{p}^2-(x^2+y^2-1)p+xy=0$ where $p={\displaystyle \frac{dy}{dx}}$ substituting $u=x^2$ and $v=y^2$ reduce the equation to Clairaut’s form in terms of $u$, $v$ and $p^{\mathrm{\prime }}={\displaystyle \frac{dv}{du}}$ hence or otherwise solve the equation.

[10M]

6.(b)(ii) Solve the following initial value problem using Laplace transform: ${\displaystyle \frac{d^{2}y}{d{x}^2}}+9y=r(x)$, $y(0)=0$, $y^{\mathrm{\prime }}(0)=4$ where $r(x)=\{\begin{array}{ll}8\sin x& \text{ if }0<x<\pi \\ 0& \text{ if }x\ge \pi \end{array}$.

[17M]

6.(c) A uniform solid hemisphere rests on a rough plane inclined to the horizon at an angle $\varphi$ with its curved surface touching the plane. Find the greatest admissible value of the inclination $\varphi$ for equilibrium. If $\varphi$ be less than this value, is the equilibrium stable?

[17M]

7.(a) Find the curvature vector and its magnitude at any point $\overline{r}=(\theta )$ of the curve $\overline{r}=(a\cos \theta ,a\sin \theta ,a\theta )$. Show that the locus of the feet of the perpendicular from the origin to the tangent is a curve that completely lies on the hyperboloid $x^2+y^2-z^2=a^2$.

[16M]

7.(b)(i) Solve the differential equation: $x{\displaystyle \frac{d^{2}y}{d{x}^2}}-{\displaystyle \frac{dy}{dx}}-4x^{3}y=8x^3\sin \left(x^2\right)$

[8M]

7.(b)(ii) Solve the following initial value differential equations: $20y^{\mathrm{\prime }\mathrm{\prime }}+4y^{\mathrm{\prime }}+y=0$, $y(0)=3.2$, $y^{\mathrm{\prime }}(0)=0$.

[7M]

7.(c) A particle is free to move on a smooth vertical circular wire of radius $a$. At time $t=0$ it is projected along the circle from its lowest point $A$ with velocity just sufficient to carry it to the highest point $B$. Find the time $T$ at which the reaction between the particle and the wire is zero.

[17M]

8.(a) A spherical shot of $W$ gm weight and radius $r$ cm, lies at the bottom of cylindrical bucket of radius $R\mathrm{c}\mathrm{m}$. The bucket is filled with water up to a depth of $\mathrm{h}\mathrm{c}\mathrm{m}(\mathrm{h}>2\mathrm{r})$. Show that the minimum amount of work done in lifting the shot just clear of the water must be $\left[W(h-{\displaystyle \frac{4r^3}{3R^2}})\right]+W^{\mathrm{\prime }}(r-h+{\displaystyle \frac{2r^3}{3R^2}})\mathrm{c}\mathrm{m}\mathrm{g}\mathrm{m}$. ${\mathrm{W}}^{\mathrm{\prime }}\mathrm{g}\mathrm{m}$ is the weight of water displaced by the shot.

[17M]

8.(b) Solve the following initial value problem using Laplace transform:

where

[17M]

8.(c)(i) Evaluate the integral ${\iint }_{S}Fnds$, where $\overline{F}=3x{y}^2\hat{i}+(y{x}^2-y^3)j+3z{x}^{2}K$ and $S$ is a surface of the cylinder $y^2+z^2\le 4,-3\le x\le 3$ using divergence theorem.

[9M]

8.(c)(ii) Using Green theorem, evaluate the ${\int }_{C}F(\overrightarrow{r}).d\overrightarrow{r}$ counterclockwise where $F(r)=(x^2+y^2)\hat{i}+(x^2-y^2)j$ and $d\overrightarrow{r}=x\hat{i}+dyj$ and the curve $C$ is the boundary off the region $R=\{(x,y)|1\le y\le 2-x^2\}$.

[8M]