Paper I PYQs-2014

1.(a) Show that ${\mathrm{u}}_1=(1,-1,0),{\mathrm{u}}_2=(1,1,0)$ and ${\mathrm{u}}_3=(0,1,1)$ form a basis for ${\mathbb{R}}^3.$ Express (5,3,4) in terms of ${\mathrm{u}}_1,{\mathrm{u}}_2$ and ${\mathrm{u}}_3$.

[8M]

1.(b) For the matrix $A=\left[\begin{array}{lll}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right].$ Prove that $A^n=A^{n-2}+A^2-I,n\ge 3$.

[8M]

1.(c) Show that the function given by

is continuous but not differentiable at $x=0$.

[8M]

1.(d) Evaluate ${\iint }_{\mathbf{R}}\mathrm{y}{\displaystyle \frac{\sin \mathrm{x}}{\mathrm{x}}}\mathrm{d}\mathrm{x}\mathrm{d}\mathrm{y}$ over $\mathrm{R}$ where $\mathrm{R}=\{(\mathrm{x},\mathrm{y}):\mathrm{y}\le \mathrm{x}\le \pi /2,0\le \mathrm{y}\le \pi /2\}$.

[8M]

1.(e) Prove that the locus of a variable line which intersects the three lines:

is the surface $y^2-m^2{x}^2=z^2-c^2$.

[8M]

2.(a) Let $B=\left[\begin{array}{rr}1& -1\\ 2& -1\end{array}\right]$. Find all eigen values and corresponding eigen vectors of $\mathrm{B}$ viewed as a matrix over: (i) the real field $R$ (ii) the complex field $C$.

[10M]

2.(b) If $\mathrm{x}\mathrm{y}\mathrm{z}={\mathrm{a}}^3$ then show that the minimum value of ${\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2$ is $3{\mathrm{a}}^2$.

[10M]

2.(c) Prove that every sphere passing through the circle $x^2+y^2+2ax+r^2=0$, $z=0$ cut orthogonally every sphere through the circle $x^2+z^2=r^2$, $y=0$.

[10M]

2.(d) Show that the mapping $\mathrm{T}:{\mathrm{V}}_2(\overline{\mathrm{R}})\to {\mathrm{V}}_3(\overline{\mathrm{R}})$ defined as $\mathrm{T}(\mathrm{a},\mathrm{b})=(\mathrm{a}+\mathrm{b},\mathrm{a}-\mathrm{b},\mathrm{b})$ is a linear transformation. Find the range, rank and nullity of $\overline{T}$.

[10M]

3.(a) Examine whether the matrix $A=\left[\begin{array}{rrr}-2& 2& -3\\ 2& 1& 6\\ -1& -2& 0\end{array}\right]$ is diagonalizable. Find all eigen values. Then obtain a matrix $P$ such that $P^{-1}AP$ is a digonal matrix.

[10M]

3.(b) A moving plane passes throughla fixed point $(2,2,2)$ and meets the coordinate axes at the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$, all away from the origin $\mathrm{O}$. Find the locus of the centre of the sphere passing through the points $\mathrm{O}$, $\mathrm{A},$\mathrm{B}, $\mathrm{C}$

[10M]

3.(c) Evaluate the integral $$ I=\int_{0}^{\infty} 2^{-a x^{2}} d x $$ using Gamma function.

[10M]

3.(d) Prove that the equation:

represents a cone with vertex at $(-1,-2,-3)$.

[10M]

4.(a) Let $f$ be a real valued function defined on [0,1] as follows:

where $a$ is an integer greater than $2$. Show that ${\int }_0^{1}f(x)dx$ exists and is equal to ${\displaystyle \frac{a}{a+1}}$.

[10M]

4.(b) Prove that the plane $ax+by+cz=0$ cuts the cone $yz+zx+xy=0$ in perpendicular lines if ${\displaystyle \frac{1}{a}}+{\displaystyle \frac{1}{b}}+{\displaystyle \frac{1}{c}}=0$.

[10M]

4.(c) Evaluate the integral ${\iint }_R{\displaystyle \frac{y}{\sqrt{x^2+y^2+1}}}dxdy$ over the region $R$ bounded between $0\le x\le {\displaystyle \frac{y^2}{2}}$ and $0\le y\le 2$.

[10M]

4.(d) Consider the linear mapping $\mathrm{F}:{\mathbb{R}}^2\to {\mathbb{R}}^2$ given as $\mathrm{F}(\mathrm{x},\mathrm{y})=(3\mathrm{x}+4\mathrm{y},2\mathrm{x}-5\mathrm{y})$ with usual basis.

Find the matrix associated with the linear transformation relative to the basis $\mathrm{S}=\{{\mathrm{u}}_1,{\mathrm{u}}_2\}$ where $u_1=(1,2),u_2=(2,3)$.

[10M]

5.(a) Solve the differential equation : $y=2px+p^{2}y,p={\displaystyle \frac{dy}{dx}}$

and obtain the non-singular solution.

[8M]

5.(b) Solve :

[8M]

5.(c) A particle whose mass is $m$, is acted upon by a force $m\mu (x+{\displaystyle \frac{a^4}{x^3}})$ towards the origin. If it starts from rest at a distance ‘a’ from the origin, prove that it will arrive at the origin in time ${\displaystyle \frac{\pi }{4\sqrt{\mu }}}$.

[8M]

5.(d) A hollow weightless hemisphere filled with liquid is suspended from a point on the rim of its base. Show that the ratio of the thrust on the plane base to the weight of the contained liquid is $12:\sqrt{73}$.

[8M]

5.(e) For three vectors show that:

[8M]

6.(a) Solve the following differential equation:

[10M]

6.(b) An engine, working at a constant rate $\mathrm{H},$ draws a load $\mathrm{M}$ against a resistance $\mathrm{R}.$ Show that the maximum speed is $\mathrm{H}/\mathrm{R}$ and the time taken to attain half of this speed is ${\displaystyle \frac{\mathrm{M}\mathrm{H}}{{\mathrm{R}}^2}}(\log 2-{\displaystyle \frac{1}{2}})$

[10M]

6.(c) Solve by the method of variation of parameters:

[10M]

6.(d) For the vector $\overline{A}={\displaystyle \frac{x\hat{i}+y\hat{j}+2\hat{k}}{x^2+y^2+z^2}}$ examine if $\overline{A}$ is an irrotational vector. Then determine $\varphi$ such that $\overline{\mathrm{A}}=\mathrm{\nabla }\varphi$.

[10M]

7.(a) A solid consisting of a cone and a hemisphere on the same base rests on a rough horizontal table with the hemisphere in contact with the table. Shoy that the largest height of the cone so that the equilibrium is stable is $\sqrt{3}\times$ radius of hemisphere.

[15M]

7.(b) Evaluate ${\iint }_{\mathrm{S}}\mathrm{\nabla }\times \overline{\mathrm{A}}\cdot \overline{\mathrm{n}}\mathrm{d}\mathrm{S}$ for $\overline{\mathrm{A}}=({\mathrm{x}}^2+\mathrm{y}-4)\hat{\mathrm{i}}+3\mathrm{x}\mathrm{y}\hat{\mathrm{j}}+(2\mathrm{x}\mathrm{z}+{\mathrm{z}}^2)\hat{\mathrm{k}}$ and $\mathrm{S}$ is the surface of hemisphere $x^2+y^2+z^2=16$ above $xy$ plane.

[15M]

7.(c) Solve the D.E.:

[10M]

8.(a) A semi circular disc rests in a vertical plane with its curved edge on a rough horizontal and equally rough yertical plane. If the coeff. of friction is $\mu ,$ prove that the greatest angle that the bounding diameter can make with the horizontal plane is:

[10M]

8.(b) A body floating in water has volumes $V_1,V_2$ and $V_3$ above the surface when the densities of the surrounding air are ${\rho }_1,{\rho }_2,{\rho }_3$ respectively. Prove that:

[15M]

8.(c) Verify the divergence theorem for $\overline{A}=4x\hat{i}-2y^2\hat{j}+z^2\hat{k}$ over the region $x^2+y^2=4$, $z=0$, $z=3$.

[10M]