Paper I PYQs-2013

1.(a) Find the dimension and a basis of the solution space $\mathrm{W}$ of the system $x+2y+2z-s+3t=0$, $x+2y+3z+s+t=0$, $3x+6y+8z+s+5t=0$.

[8M]

1.(b) Find the characteristic equation of the matrix $A=\left[\begin{array}{lll}2& 1& 1\\ 0& 1& 0\\ 1& 1& 2\end{array}\right]$ and hence find the matrix represented by $A^8-5A^7+7A^6-3A^5+A^4-5A^3+8A^2-2A+I$.

[8M]

1.(c) Evaluate the integral ${\int }_0^{\mathrm{\infty }}{\int }_0^{x}x{\mathrm{e}}^{-{\mathrm{x}}^2/\mathrm{y}}\mathrm{d}\mathrm{y}\mathrm{d}\mathrm{x}$ by changing the order of integration.

[8M]

1.(d) Find the surface generated by the straight line which intersects the lines $y=z=a$ and $x+3z=a=y+z$ and is parallel to the plane $x+y=0$.

[8M]

1.(e) Find $C$ of the mean value theorem, if $f(x)=x(x-1)(x-2),a=0,b={\displaystyle \frac{1}{2}}$ and $C$ has usual meaning.

[8M]

2.(a) Let $\mathrm{V}$ be the vector space of $2\times 2$ matrices over $\mathbb{R}$ and let $\mathrm{M}=\left[\begin{array}{rr}1& -1\\ -2& 2\end{array}\right]$ Let $\mathrm{F}:\mathrm{V}\to \mathrm{V}$ be the linear map defined by $\mathrm{F}(\mathrm{A})=\mathrm{M}\mathrm{A}$. Find a basis and the dimension of
(i) the kernel of W of $\mathrm{F}$ (ii) the image U of F.

[10M]

2.(b) Locate the stationary points of the function $x^4+y^4-2x^2+4xy-2y^2$ and determine their nature.

[10M]

2.(c) Find an orthogonal transformation of co-ordinates which diagonalizes the quadratic form

[10M]

2.(d) Discuss the consistency and the solutions of the equations

for different values of $a$.

[10M]

3.(a) Prove that if $a_0,a_1,a_2,\dots \dots ,a_n$ are the real numbers such that

then there exists at least one real number $x$ between 0 and 1 such that

[10M]

3.(b) Reduce the following equation to its canonical form and determine the nature of the conic $4x^2+4xy+y^2-12x-6y+5=0$.

[10M]

3.(c) Let $\mathrm{F}$ be a subfield of complex numbers and $\mathrm{T}$ a function from ${\mathrm{F}}^3\to {\mathrm{F}}^3$ defined by $\mathrm{T}({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3)=({\mathrm{x}}_1+{\mathrm{x}}_2+3{\mathrm{x}}_3,2{\mathrm{x}}_1-{\mathrm{x}}_2,-3{\mathrm{x}}_1+{\mathrm{x}}_2-{\mathrm{x}}_3).$ What are the conditions on $(a,b,c)$ such that $(a,b,c)$ be in the null space of $T$? Find the nullity of $T$.

[10M]

3.(d) Find the equations to the tangent planes to the surface $7x^2-3y^2-z^2+21=0$, which pass through the line $7x-6y+9=0$, $z=3$.

[10M]

4.(a) Evaluate

[10M]

4.(b) Let $H=\left[\begin{array}{ccc}1& \text{ i }& 2+i\\ -i& 2& 1-i\\ 2-i& 1+i& 2\end{array}\right]$ be a Hermitian matrix. Find a non-singular matrix $P$ such that ${\mathrm{P}}^{\mathrm{t}}\mathrm{H}\overline{\mathrm{P}}$ is diagonal and also find its signature.

[10M]

4.(c) Find the magnitude and the equations of the line of shortest distance between the lines

[10M]

4.(d) Find all the asymptotes of the curve

[10M]

5.(a) Solve:

[8M]

5.(b) A particle is performing a simple harmopiç motion of period $\mathrm{T}$ about centre $\mathrm{O}$ and it passes through a point $P$, where $OP=(b$ with velocity $v$ in the direction of $OP$. Find the time which elapses before it returps to $\mathrm{P}$.

[8M]

5.(c) $\overrightarrow{\mathrm{F}}$ being a vector, prove that
curl curl $\overrightarrow{\mathrm{F}}=$ grad $\mathrm{div}\overrightarrow{\mathrm{F}}-{\mathrm{\nabla }}^2\overrightarrow{\mathrm{F}}$
where ${\mathrm{\nabla }}^2={\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}}$

[8M]

5.(d) A triangular lamina $ABC$ of density $\rho$ floats in a liquid of density $\sigma$ with its plane vertical, the angle $\mathrm{B}$ being in the surface of the liquid, and the angle $\mathrm{A}$ not immersed. Find $p/\sigma$ in terms of the lengths of the sides of the triangle.

[8M]

5.(e) A heavy uniform rod rests with one end against a smooth vertical wall and with a point in its length resting on a smooth peg. Find the position of equilibrium and discuss the nature of equilibrium.

[8M]

6.(a) Solve the differential equation

by changing the dependent variable.

[13M]

6.(b) Evaluate ${\int }_{\mathrm{S}}\overrightarrow{\mathrm{F}}\cdot \mathrm{d}\overrightarrow{\mathrm{s}},$ where $\overrightarrow{\mathrm{F}}=4\mathrm{x}\mathrm{i}-2{\mathrm{y}}^2\mathrm{j}+z^2\overrightarrow{\mathrm{k}}$ and $\mathrm{s}$ is the surface bounding the region $x^2+y^2=4,z=0$ and $z=3$.

[13M]

6.(c) Two bodies of weights $w_1$ and $w_2$ are placed on an inclined plane and are connected by a light string which coincides with a line of greatest slope of the plane; if the coefficient of friction between the bodies and the plane are respectively ${\mu }_1$ and ${\mu }_2$, find the inclination of the plane to the horizontal when both bodies are on the point of motion, it being assumed that smoother body is below the other.

[14M]

7.(a) Solve

[13M]

7.(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 respectively ${\rho }_1,{\rho }_2,{\rho }_3$. Find the value of:

[13M]

7.(c) A particle is projected vertically upwards with a velocity $\mathrm{u},$ in a resisting medium which produces a retardation $\mathrm{k}{v}^2$ when the velocity is $v$. Find the height when the particle comes to rest above the point of projection.

[14M]

8.(a) Apply the method of variation of parameters to solve

[13M]

8.(b) Verify the Divergence theorem for the vector function

taken over the rectangular parallelopiped

[14M]

8.(c) A particle is projected with a velocity $v$ along a smooth horizontal plane in a medium whose resistance per unit mass is double the cube of the velocity. Find the distance it will describe in time $t$.

[13M]