Paper I PYQs-2016

1.(a) Let $\mathrm{T}:\mathrm{I}{\mathrm{B}}^3\to {\mathrm{B}}^4$ be given by
$T(x,y,z)=(2x-y,2x+z,(z)+2z,x+y+z)$.

Find the matrix of $T$ with respect to standard basis of ${\mathrm{E}\mathrm{B}}^3$ and ${\mathrm{R}}^4$ (i.e., $(1,0,0)$, $(0,1,0)$, etc. Examine if $\mathrm{T}$ is a linear map.

[8M]

1.(b) Show that $\frac{x}{(1)+x)}<\log (1+x)<x\$for\$x>0$.

[8M]

1.(c) Examine if the function $f(x,y)={\displaystyle \frac{xy}{x^2+y^2}},(x,y)\ne (0,0)$ and $f(0,0)=0$ is continuous at $(0,0)$. Find ${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}}$ and ${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }y}}$ at points other than origin.

[8M]

1.(d) If the point (2,3) is the mid-point of a chord of the parabola $y^2=4x$, then obtain the equation of the chord.

[8M]

1.(e) For the matrix $A=\left[\begin{array}{rrr}-1& 2& 2\\ 2& -1& 2\\ 2& 2& -1\end{array}\right],$ obtain the eigen value and get the value of $A^4+3A^3-9A^2$.

[8M]

2.(a) After changing the order of integration of ${\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{e}^{-xy}\sin nxdxdy$, show that ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{\sin nx}{x}}dx={\displaystyle \frac{\pi }{2}}$.

[10M]

2.(b) A perpendicular is drawn from the centre of ellipse $Q{\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$ to any tangent. Prove that the locus of the foot of the perpendicular is given-by ${(x^2+y^2)}^2=a^2{x}^2+b^2{y}^2$.

[10M]

2.(c) Using mean value theorem, find a point on the curve $y=\sqrt{x-2},$ defined on $[2,3],$ where the tangent $(\mathrm{i}\mathrm{s},$ parallel to the chord joining the end points of the curve.

[10M]

2.(d) Let $\mathrm{T}$ be a linear map such that $\mathrm{T}:{\mathrm{v}}_3\to {\mathrm{v}}_2$ defined by $\mathrm{T}\left({\mathrm{e}}_1\right)=2{\mathrm{f}}_1-{\mathrm{f}}_2$, $\mathrm{T}\left({\mathrm{e}}_2\right)={\mathrm{f}}_1+2{\mathrm{f}}_2\mathrm{T}\mathrm{T}\left({\mathrm{e}}_3\right)=0{\mathrm{f}}_1+0{\mathrm{f}}_2,$ where ${\mathrm{e}}_1,{\mathrm{e}}_2,{\mathrm{e}}_3$ and ${\mathrm{f}}_1,{\mathrm{f}}_2$ are standard basis in ${\mathrm{v}}_3$ and ${\mathrm{V}}_2$. Find the matrix of $\mathrm{T}$ relative to these basis. Further take two other basis ${\mathrm{B}}_1[(1,1,0)(1,0,1)(0,1,1)]$ and ${\mathrm{B}}_2[(1,1)(1,-1)]$. Obtain the matrix ${\mathrm{T}}_1$ relative to ${\mathrm{B}}_1$ and ${\mathrm{B}}_2$.

[10M]

3.(a) For the matrix $A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$, find two non-singular matrices $P$ and $Q$ such that $\mathrm{P}\mathrm{A}\mathrm{Q}=\mathrm{I}$. Hence find $A^{-1}$.

[10M]

3.(b) Using Lagrange’s method of multipliers, find the point on the plane $2x+3y+4z=5$ which is closest to the point $(1,0,0)$.

[10M]

3.(c) Obtain the area between the curve $x=3(\sec \theta +\cos \theta )$ and its asymptote $x=3$.

[10M]

3.(d) Obtain the equation of the sphere on which the intersection of the plane $5x-2y+4z+7=0$ with the sphere which has $(0,1,0)$ and $(3,-5,2)$ as the end points of its diameter is a great circle.

[10M]

4.(a) Examine whether the real quadratic form $4x^2-y^2+2z^2>2xy-2yz-4xz$ is a positive definite or not. Reduce it to its Aiagonal form and determine its signature.

[10M]

4.(b) Show that the integral ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x}{\mathrm{x}}^{\alpha -1}\mathrm{d}\mathrm{x},\alpha >0$ exists, by separately taking the cases for $\alpha \ge 1$ and $0<\alpha <\mathbf{1}$.

[10M]

4.(c) Prove that $\mathrm{\Gamma }(2z)$ = ${\displaystyle \frac{2^{2z-1}}{\sqrt{\pi }}}\mathrm{\Gamma }(z)\mathrm{\Gamma }(z+{\displaystyle \frac{1}{2}})$

[10M]

4.(d) A plane ${\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}}+{\displaystyle \frac{z}{c}}=a_2$ cuts the coordinate plane at $A$, $B$, $C$. Find the equation of the cone with vertex at origin and guiding curve as the circle passing through $\mathrm{A}$, $\mathrm{B},$\mathrm{C}$$.

[10M]

5.(a) Obtain the curve which passes through (1,2) and has a slope $={\displaystyle \frac{-2xy}{x^2+1}}$ Dbtain one asymptote to the curve.

[8M]

5.(b) Solve the de to get the particular integral of ${\displaystyle \frac{{\mathrm{d}}^4\mathrm{y}}{{\mathrm{d}\mathrm{x}}^4}}+2{\displaystyle \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{d}\mathrm{x}}^2}}+\mathrm{y}={\mathrm{x}}^2\cos \mathrm{x}$.

[8M]

5.(c) A weight $W$ is hanging with the help of two strings of length $l$ and $2l$ in such a way that the other ends A and $B$ of those strings lie on a horizontal line at a distance $2l$. Obtain the tension in the two strings.

[8M]

5.(d) From a point in a smooth horizontal plane, a particle is projected with velocity u at angle $\alpha$ to the horizontal from the foot of a plane, inclined at an angle $\beta$ with respect to the horizon. Show that if will strike the plane at right angles, if cot $\beta =2\tan (\alpha -\beta )$.

[8M]

5.(e) If $\mathrm{E}$ be the solid bounded by the xy plane and the paraboloid the volume $E$ and $\overline{F}$=$(zx\sin yz+x^3)\hat{i}$+ $\cos yz\hat{j}$ + $(3z{y}^2-e^{{\lambda }^2+y^2})\hat{k}$.

[8M]

6.(a) A stone is thrown vertically with the velocity which would just carry it to a height of $40\mathrm{m}$. Two seconds later another stone is projected vertically from the same place with the same velocity. When and where will they meet?

[10M]

6.(b) Using the method of variation of parameters, solve

[10M]

6.(c) Water is flowing through a pipe of $80\mathrm{m}\mathrm{m}$ diameter under a gauge pressure of $60\mathrm{k}\mathrm{P}\mathrm{a}$, with a mean velocity of 2 $m/s$. Find the total head, if the pipe is 7 $m$ above the datum line.

[10M]

6.(d) Evaluate ${\iint }_{\mathrm{S}}(\mathrm{\nabla }\times \overline{f}),\hat{\mathrm{n}}\mathrm{d}\mathrm{S}$ for $\overline{\mathrm{f}}=(2\mathrm{x}-\mathrm{y})\hat{\mathrm{i}}-{\mathrm{y}\mathrm{z}}^2\hat{\mathrm{j}}-{\mathrm{y}}^2\mathrm{z}\hat{\mathrm{k}}$ where $\mathrm{S}$ is the upper half surface of the sphere $x^2+y^2+z^2=1$ bounded by its projection on the $xy$ plane.

[10M]

7.(a) State Stokes’ theorem. Verify the Stokes’ theorem for the function $\overline{\mathrm{f}}=x\hat{i}+z\hat{j}+2y\hat{k}$, where $\mathrm{c}$ is the curve obtained by the intersection of the plane $z=x$ and the cylinder $x^2+y^2=1$ and $S$ is the surface inside the interesected cone.

[15M]

7.(b) A uniform rod of weight $\mathrm{W}$ is resting against an equally rough horizon and a wall, at an angle $\alpha$ with the wall. At this condition, a horizontal force $P$ is stopping them from sliding, implemented at the mid-point of the rod. Prove that $P=W\tan (\alpha -2\lambda ),$ where $\lambda$ is the angle of friction. Is there any condition on $\lambda$ and $\alpha$?

[15M]

7.(c) Obtain the singular solution of the differential equation

[10M]

8.(a) A body immersed in a liquid is balanced by a weight $\mathrm{P}$ to which it is attached by a thread passing over a fixed pulley and when half immersed, is balanced in the same manner by weight $2\mathrm{P}$. Prove that the density of the body and the liquid are in the ratio 3:2?

[10M]

8.(b) Solve the differential equation

[10M]

8.(c) Prove that $\overline{a}\times (\overline{b}\times \overline{c})=(\overline{a}\times \overrightarrow{b})\times \overrightarrow{c}$, if and only if either $\overline{b}=\overline{0}$ or $\overline{c}$ is collinear with $\overline{a}$ or $\overline{b}$ is perpendicular to both $\overline{a}$ and $\overline{c}$.

[10M]

8.(d) A particle is acted on a force parallel to the axis of $y$ whose acceleration is $\lambda y$, initially projected with a velocity $a\sqrt{\lambda }$ parallel to $x$ -axis at the point where $y=a$. Prove that it will describe a catenary.

[10M]