Paper I PYQs-2009

1.(a) Let $V$ be the vector space of polynomials over $R$. Let $U$ and $W$ be the subspaces generated by $\{t^3+4t^2-t+3,t^3+5t^2+5,3t^3+10t^2-5t+5\}$ and $\{t^3+4t^2+6,t^3+2t^2-t+5,2t^3+2t^2-3t+9\}$ respectively. Find
(i) $\dim (U+W)$
(ii) $\dim (U\cap W)$

[10M]

1.(b) Find a linear map $T:R^3\to R^4$ whose image is generated by $(1,2,0,-4)$ and $(2,0,-1,-3)$.

[10M]

1.(c)(i) Find the difference between the maximum and the minimum of the function $(a-{\displaystyle \frac{1}{a}}-x)(4-3x^2)$ where $a$ is a constant and greater than zero.

[5M]

1.(c)(ii) If $f(h)=f(0)+h{f}^{\mathrm{\prime }}(0)+{\displaystyle \frac{h^2}{2!}}{f}^{\mathrm{\prime }\mathrm{\prime }}(\theta h)$, $0<\theta <1$

Find $\theta$, when $h=1$ and $f(x)=(1-x{)}^{5/2}$.

[5M]

1.(d) Evaluate:

(i) ${\int }_0^{\pi /2}{\displaystyle \frac{{\sin }^{2}xdx}{\sin x+\cos x}}$

[6M]

(ii) ${\int }_1^{\mathrm{\infty }}{\displaystyle \frac{x^{2}dx}{{(1+x^2)}^2}}$

[4M]

5.(d) A particle rests in equilibrium under the attraction of two centres of force which attract directly as the distance, their intensities being $\mu$ and ${\mu }^{\mathrm{\prime }}$. The particle is slightly displaced towards one of them, show that the time of small oscillation is ${\displaystyle \frac{2\pi }{\sqrt{(\mu +{\mu }^{\mathrm{\prime }})}}}$.

[10M]

5.(e) 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 $y=x^2$.

[10M]

6.(a) Solve:
$(y^3-2y{x}^2)dx+(2x{y}^2-x^3)dy=0$

[10M]

6.(b) Solve:
${\left({\displaystyle \frac{dy}{dx}}\right)}^2-2{\displaystyle \frac{dy}{dx}}\cos hx+1=0$

[8M]

6.(c) Solve:
${\displaystyle \frac{d^{3}y}{d{x}^3}}+3{\displaystyle \frac{d^{2}y}{d{x}^2}}+3{\displaystyle \frac{dy}{dx}}+y=x^2{e}^{-x}$

[10M]

6.(d) Show that $e^{x^2}$ is a solution of ${\displaystyle \frac{d^{2}y}{d{x}^2}}-4x{\displaystyle \frac{dy}{dx}}+(4x^2-2)y=0$. Find a second independent solution.

[12M]

7.(a) A solid hemisphere is supported by a string fixed to a point on its rim and to a point on a smooth vertical wall with which the curved surface of the sphere is in contact. If $\theta$ and $\varphi$ are the inclinations of the string and the plane base of the hemisphere to the vertical, prove that $\tan \varphi ={\displaystyle \frac{3}{8}}+\tan \theta$.

[10M]

7.(b) A particle moves with a central acceleration $\mu (\gamma +{\displaystyle \frac{a^4}{{\gamma }^3}})$ being projected from an apse at a distance $a$ with a velocity $2\sqrt{\mu }a$.

Prove that its path is ${\gamma }^2(2+\cos \sqrt{3}\theta )=3a^2$.

[10M]

7.(c) A shell, lying in a straight smooth horizontal tube, suddenly explodes and breaks into portions of masses $m$ and $m^{\mathrm{\prime }}$. If $d$ is the distance apart of the masses after a time $t$, show that the work done by the explosion is ${\displaystyle \frac{1}{2}}{\displaystyle \frac{m{m}^{\mathrm{\prime }}}{m+m^{\mathrm{\prime }}}}\cdot {\displaystyle \frac{d^2}{t^2}}$.

[10M]

7.(d) A hollow conical vessel floats in water with its vertex downwards and a certain depth of its axis immersed. When water is poured into it up to the level originally immersed, it sinks till its mouth is on a level with the surface of the water. What portion of axis was originally immersed?

[10M]

8.(a) Show that $\overline{A}=(6xy+z^3)\hat{i}+(3x^2-z)\hat{j}+(3x{z}^2-y)\hat{k}$ is irrotational. Find a scalar function $\varphi$ such that $\overline{A}=\mathrm{grad}\varphi$.

[10M]

8.(b) Let $\psi (x,y,z)$ be a scalar function. Find grad $\psi$ and ${\mathrm{\nabla }}^2\psi$ in spherical coordinates.

[8M]

8.(c) Verify Stokes’ theorem for $\overline{A}=(y-z+2)\hat{i}+(yz+4)\hat{j}-xz\hat{k}$ where $\dot{S}$ is the surface of the cube $x=0$, $y=0$, $z=0$, $x=2$, $y=2$, $z=2$ above the $xy$-plane.

[12M]

8.(d) Show that, if $\overline{r}=x(s)\hat{i}+y(s)\hat{j}+z(s)\hat{k}$ is a space curve, ${\displaystyle \frac{d\overline{r}}{ds}}\cdot {\displaystyle \frac{d^2\overline{r}}{d{s}^2}}\times {\displaystyle \frac{d^3\overline{r}}{d{s}^3}}={\displaystyle \frac{\tau }{{\rho }^2}},$ where $\tau$ is the torsion and $\rho$ the radius of curvature.

[10M]