Paper I PYQs-2014

1.(a) Find one vector in $R^3$ which generates the intersection of $V$ and $W$, where $V$ is the $xy-plane$ and $W$ is the space generated by the vectors $(1,2,3)$ and $(1,-1,1)$.

[10M]

1.(b) Using elementary row or column operations, find the rank of the matrix $\left[\begin{array}{cccc}0& 1& -3& -1\\ 0& 0& 1& 1\\ 3& 1& 0& 2\\ 1& 1& -2& 0\end{array}\right]$

[10M]

1.(c) Prove that between two real roots of $e^x\cos x+1=0$, a real root of $e^x\sin x+1=0$ lies.

[10M]

1.(d) Evaluate ${\int }_0^1{\displaystyle \frac{{\log }_e(1+x)}{1+x^2}}dx$

[10M]

1.(e) Examine whether the plane $x+y+z=0$ cuts the cone $yz+zx+xy=0$ in perpendicular lines.

[10M]

2.(a) Let $V$ and $W$ be the following subspaces of $R^4:V=\{(a,b,c,d):b-2c+d=0\}$ and $W=\{(a,b,c,d):a=d,b=2c\}$. Find a basis and the dimension of:
(i) $V$
(ii) $V\cap W$.

[15M]

2.(b)(i) Investigate the values of $\lambda$ and $\mu$ so that the equations $x+y+z=6$, $x+2y+3z=10$, $x+2y+\lambda z=\mu$ have:
(i) no solution
(ii) a unique solution,
(iii) an infinite number of solutions.

[10M]

2.(b)(ii) Verify Cayley-Hamilton theorem for the matrix $A=\left[\begin{array}{cc}1& 4\\ 2& 3\end{array}\right]$ and hence find its inverse. Also, find the matrix represented by $A^5-4A^4-7A^3+11A^2-A-10I$.

[10M]

2.(c) By using the transformation $x+y=u$, $y=uv$, evaluate the integral $\iint \{xy(1-x-y){\}}^{1/2}dxdy$ taken over the area enclosed by the straight lines $x=0$, $y=0$ and $x+y=1$.

[15M]

3.(a) Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $a$.

[15M]

3.(b) Find the maximum or minimum values of $x^2+y^2+z^2$ subject to the condition $a{x}^2+b{y}^2+c{z}^2=1$ and $lx+my+nz=0$ interpret result geometrically.

[20M]

3.(c)(i) Let $A=\left[\begin{array}{ccc}-2& 2& -3\\ 2& 1& -6\\ -1& -2& 0\end{array}\right]$. Find the Eigen values of $A$ and the corresponding Eigen vectors.

[8M]

3.(c)(ii) Prove that eigen values of a unitary matrix have absolute value 1.

[7M]

4.(a)(i) Find the co-ordinates of the points on the sphere $x^2+y^2+z^2-4x+2y=4$ the tangent planes at which are parallel to the plane $2x-y+2z=1$.

[10M]

4.(a)(ii) Prove that equation $a{x}^2+b{y}^2+c{z}^2+2ux+2vy+d=0$ represents a cone if ${\displaystyle \frac{u^2}{a}}+{\displaystyle \frac{v^2}{b}}+{\displaystyle \frac{w^2}{c}}=d$.

[10M]

4.(b) Show that the lines drawn from the origin parallel to the normals to the central conicoid $a{x}^2+b{y}^2+c{z}^2=1$, at its points of intersection with the plane $lx+my+nz=p$ generate the cone $p^2({\displaystyle \frac{x^2}{a}}+{\displaystyle \frac{y^2}{b}}+{\displaystyle \frac{z^2}{c}})={({\displaystyle \frac{lx}{a}}+{\displaystyle \frac{my}{b}}+{\displaystyle \frac{nz}{c}})}^2$.

[15M]

4.(c) Find the equations of the two generating lines through any point $(a\cos \theta ,b\sin \theta ,0)$ of the principal elliptic section ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$, $z=0$ of the hyperboloid by the plane $z=0$.

[15M]

5.(a) Justify that a differential equation of the form: $[y+xf(x^2+y^2)]dx+[yf(x^2+y^2)-x]dy=0$ where $f(x^2+y^2)$ is an arbitrary function of $(x^2+y^2)$, is not an exact differential equation and ${\displaystyle \frac{1}{x^2+y^2}}$ is an integrating factor for it. Hence solve this differential equation for $f(x^2+y^2)={(x^2+y^2)}^2$.

[10M]

5.(b) Find the curve for which the part of the tangent cut-off by the axes is bisected at the point of tangency.

[10M]

5.(c) A particle is performing a simple harmonic motion (SHM) of period $T$ about a centre $O$ with amplitude $a$ and it passes through a point $P$, where $OP=b$ in the direction $OP$. Prove that the time which elapses before it returns to $P$ is ${\displaystyle \frac{T}{\pi }}{\cos }^{\mathrm{\prime }}\left({\displaystyle \frac{b}{a}}\right)$.

[10M]

5.(d) Two equal uniform rods $AB$ and $AC$, each of length $l$, are freely jointed at $A$ and rest on a smooth fixed vertical circle of radius $r$. If 2$\theta$ is the angle between the rods, then find the relation between $l$, $r$ and $\theta$, by using the principle of virtual work.

[10M]

5.(e) Find the curvature vector at any point of the curve $\overline{r}(t)=t\cos t\hat{i}+t\sin t\hat{j}$, $0\le t\le 2\pi$. Give its magnitude also.

[10M]

6.(a) Solve by the method of variation of parameters: ${\displaystyle \frac{dy}{dx}}-5y=\sin x$.

[10M]

6.(b) Solve the differential equation: $x^3{\displaystyle \frac{d^{3}y}{d{x}^3}}+3x^2{\displaystyle \frac{d^{2}y}{d{x}^2}}+x{\displaystyle \frac{dy}{dx}}+8y=65\cos ({\log }_{e}x)$.

[20M]

6.(c) Evaluate by Stokes’ theorem:

where $$ is the curve given by $x^2+y^2+z^2-2ax-2ay=0$, $x+2y=2a$, starting from $(2a,0,0)$ and then going below the z-plane.

[20M]

7.(a) Solve the following differential equation: $x{\displaystyle \frac{d^{2}y}{d{x}^2}}-2(x+1){\displaystyle \frac{dy}{dx}}+(x+2)y=(x-2)e^{2x}$, when $e^x$ is a solution to its corresponding homogeneous differential equation.

[15M]

7.(b) A particle of mass $\mathrm{m}$, hanging vertically from a fixed point by a light inextensible cord of length $l$, is struck by a horizontal blow which imparts to it a velocity 2$\sqrt{gl}$. Find the velocity and height of the particle from the level of its inition when the cord becomes slack.

[15M]

7.(c) A regular pentagon $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}\mathrm{E}$, formed of equal heavy uniform bars jointed together, is suspended from the joint $\mathrm{A}$, and is maintained in form by a light rod joining the middle points of $\mathrm{B}\mathrm{C}$ and $DE$. Find the stress in this rod.

[20M]

8.(a) Find the sufficient condition for the differential equation $M(x,y)dx+N(x,y)dy=0$ to have an integrating factor as a function of $(x+y)$. What will be the integrating factor in that case? Hence find the integrating factor for the differential equation of $(x^2+xy)dx+(y^2+xy)dy=0$ and solve it.

[15M]

8.(b) A particle is acted on by a force parallel to the axis of $y$ whose acceleration (always towards the axis of $x$) is $\mu y-2$ and when $y=a$, it is projected parallel to the axis of $X$ with velocity $\sqrt{{\displaystyle \frac{2\mu }{a}}}$. Find the parametric equation of the path of the particle. Here $\mu$ is a constant.

[15M]