Paper I PYQs-2012

1.(a) Let $V={\mathbb{R}}^3$ and ${\alpha }_1=(1,1,2),{\alpha }_2=(0,1,3)$ ${\alpha }_3=(2,4,5)$ and ${\alpha }_4=(-1,0,-1)\cdot$ be the elements of $V$. Find a basis for the intersection of the subspace spanned by $\{{\alpha }_1,{\alpha }_2\}$ and $\{{\alpha }_3,{\alpha }_4\}$

[8M]

1.(b) Show that the set of all functions which satisfy the differential equation, ${\displaystyle \frac{d^{2}f}{d{x}^2}}+3{\displaystyle \frac{df}{dx}}=0$ is a vector space.

[8M]

1.(c) If the three thermodynamic variables $P,V,T$ are connected by a relation $f(P,V,T)=0$ show that, ${\left({\displaystyle \frac{\mathrm{\partial }P}{\mathrm{\partial }T}}\right)}_V\cdot {\left({\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }V}}\right)}_P{\left({\displaystyle \frac{\mathrm{\partial }\hat{V}}{\mathrm{\partial }P}}\right)}_T\cong -1$.

[8M]

1.(d) If $u=A{e}^{-gx}\sin (nt-gx),$ where $A,g,n$ are positive constants, satisfies the heat conduction equation, ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}=\mu {\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$ then show that $g=\sqrt{\left({\displaystyle \frac{n}{2\mu }}\right)}$.

[8M]

1.(e) Find the equations to the lines in which the plane $2x+y-z=0$ cuts the cone

[8M]

2.(a) Let $f:\mathbb{R}\to {\mathbb{R}}^3$ be a linear transformation defined by $f(a,b,c)=(a,a+b,0)$.

Find the matrices $A$ and $B$ respectively of the linear transformation $f$ with respect to the standard basis $(e_1,e_2,e_3)$ and the basis $(e_1^{\mathrm{\prime }},e_2^{\mathrm{\prime }},e_3^{\mathrm{\prime }})$ where $e_1^{\mathrm{\prime }}=(1,1,0),e_2^{\mathrm{\prime }}=(0,1,1)$ $e_3^{\mathrm{\prime }}=(1,1,1)$.

Also, show that there exists an invertible matrix $P$ such that

[10M]

2.(b) Verify Cayley-Hamilton theorem for the matrix $A=\left(\begin{array}{ll}1& 4\\ 2& 3\end{array}\right)$ and find its inverse. Also express $A^5-4A^4-7A^3+11A^2-A-10I$ as a linear polynomial in $A$.

[10M]

2.(c) Find the equations of the tangent plane to the ellipsoid $2x^2+6y_1^2+3z^2=27$ which passes through the line $x-y-z=0=x-y+2z-9$.

[10M]

2.(d) Show that there are three real values of $\lambda$ for which the equations: $(a-\lambda )x+by+cz=0$, $bx+(c-\lambda )y+az=0$, $cx+ay+(b-\lambda )z=0$ are simultaneously true and that the product of these values of $\lambda$ is $D=\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$.

[10M]

3.(a) Find the matrix representation of linear transförmation $T$ on $V_3(IR)$ defined as $T(a,b,c)=(2b+c,a-4b,3a)$ corresponding to the basis $B=\{(1,1,1),(1,1,0),(1,0,0)\}$.

[10M]

3.(b) Find the dimensions of the rectangular box, open at the top, of maximum capacity whose surface is 432 sq. $\mathrm{c}\mathrm{m}$.

[10M]

3.(c) If $2C$ is the shortest distance between the lines

and ${\displaystyle \frac{y}{m}}+{\displaystyle \frac{z}{n}}=1,x=0$

then show that

[10M]

3.(d) Show that the function defined as

has removable discontinuity at the origin.

[10M]

4.(a) Find by triple Integration the volume cut off from the cylinder $x^2+y^2=ax$ by the planes $z=mx$ and $z=nx$.

[10M]

4.(b) Show that afl the spheres, that can be drawn through the origin and each set of points where planes parallel to the plane ${\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}}+{\displaystyle \frac{z}{c}}=0$ cut the co-ordinate axes, form a system of spheres which are cut orthogonally by the sphere

if $af+bg+ch=0$

[10M]

4.(c) A plane makes equal intercepts on the positive parts of the axes and touches the ellipsoid $x^2+4y^2+9z^2=36$. Find its equation.

[10M]

4.(d) Evaluate the following in terms of Gamma function:

[10M]

5.(a) Solve ${\displaystyle \frac{dy}{dx}}-{\displaystyle \frac{\tan y}{1+x}}=(1+x)e^x\sec y$

[8M]

5.(b) Solve and find the singular solution of $x^3{p}^2+x^{2}py+a^3=0$.

[8M]

5.(c) A particle is projected vertically upwards from the earth’s surface with a velocity just sufficient to carry it to infinity. Prove that the time it takes to reach a height $h$ is ${\displaystyle \frac{1}{3}}\sqrt{\left({\displaystyle \frac{2a}{g}}\right)}[{(1+{\displaystyle \frac{h}{a}})}^{3/2}-1]$.

[8M]

5.(d) A triangle $ABC$ is immersed in a liquid with the vertex $C$ in the surface and the sides $AC$, $BC$ equally inclined to the surface. Show that the vertical $C$ divides the triangle into two others, the fluid pressures on which are as $b^3+3a{b}^2:a^3+3a^{2}b$ where $a$ and $b$ are the sides $BC$ & $AC$ respectively.

[8M]

5.(e) If $u=x+y+z$, $v=x^2+y^2+z^2$, $w=yz+zx+xy$, prove that grad $u$, grad $v$ and grad $w$ are coplanar.

[8M]

6.(a) Solve:

[10M]

6.(b) Find the value of ${\iint }_s(\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{F})\cdot \overrightarrow{ds}$ taken over the upper portion of the surface $x^2+y^2-2ax+az=0$ and the bounding curve lies in the plane $z=0$, when

[10M]

6.(c) A particle is projected with a velocity $u$ and strikes at right angle on a plane through the plane of projection inclined at an angle $\beta$ to the horizon. Show that the time of flight is $2u$ $\overline{g\sqrt{(1+3{\sin }^2\beta )}}$ range on the plane is ${\displaystyle \frac{2u^2}{g}}\cdot {\displaystyle \frac{\sin \beta }{1+3{\sin }^2\beta }}$ and the vertical height of the point struck is ${\displaystyle \frac{2u^2{\sin }^2\beta }{g(1+3{\sin }^2\beta )}}$ above the point of projection.

[10M]

6.(d) Solve ${\displaystyle \frac{d^{4}y}{d{x}^4}}+2{\displaystyle \frac{d^{2}y}{d{x}^2}}+y=x^2\mathrm{c}$.

[10M]

7.(a) A particle is moving with central acceleration $\mu [r^5-c^{4}r]$ being projected from an apse at. a distance $c$ with velocity $\sqrt{\left({\displaystyle \frac{2\mu }{3}}\right)c^3},$ show that its path is a curve, $x^4+y^4=c^4$.

[13M]

7.(b) A thin equilateral rectangular plate of uniform thickness and density rests with one end of its base on a rough horizontal plane and the other against a small vertical wall. Show that the least angle, its base can make with the horizontal plane is given by $\cot \theta =2\mu +{\displaystyle \frac{1}{\sqrt{3}}}$ $\mu$, being the coefficient of friction.

[14M]

7.(c) A semicircular area of radius $a$ is immersed vertically with its diameter horizontal at a depth $b.$ If the circumference be below the centre, prove that the depth of centre of pressure is ${\displaystyle \frac{1}{4}}{\displaystyle \frac{3\pi (a^2+4b^2)+32ab}{4a+3\pi b}}$

[13M]

8.(a) Solve $x=y{\displaystyle \frac{dy}{dx}}-{\left({\displaystyle \frac{dy}{dx}}\right)}^2$.

[10M]

8.(b) Find the value of the line integral aver a circular path given by $x^2+y^2=a^2,z=0$ where the vector field, $\overrightarrow{F}=(\sin y)\overrightarrow{i}+x(1+\cos y)\overrightarrow{j}$.

[10M]

8.(c) A heavy elastic string, whose natural length is $2\pi a,$ is placed round a smooth cone whose axis is vertical and whose semi vertical angle is $\alpha$. If $W$ be the weight and $\lambda$ the modulus of elasticity of the string, prove that it will be in equilibrium when in the form of a circle whose radius is

[10M]

8.(d) Solve $x^2{\displaystyle \frac{d^{2}y}{d{x}^2}}+3x{\displaystyle \frac{dy}{dx}}+y=(1-x{)}^{-2}$.

[10M]