Paper I PYQs-2016

1.(a)(i) Using elementary row operations, find the inverse of $A=\left[\begin{array}{ccc}1& 2& 1\\ 1& 3& 2\\ 1& 0& 1\end{array}\right]$.

[6M]

1.(a)(ii) If $A=\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right]$ then find $A^{14}+3A-2I$.

[4M]

1.(b)(i) Using elementary row operation find the condition that the linear equations have a solution: $x-2y+z=a$
$2x+7y-3z=b$
$3x+5y-2z=c$

[7M]

1.(b)(ii) If $w_1=\{(x,y,z)|x+y-z=0\}$, $w_2=\{(x,y,z)|3x+y-2z=0\}$, $w_3=\{(x,y,z)|x-7y+3z=0\}$, then find $\dim (w_1\cap w_2\cap w_3)$ and $\dim (w_1+w_2)$.

[3M]

1.(c) Evaluate $I={\int }_0^1\sqrt[3]{x\log \left({\displaystyle \frac{1}{x}}\right)}dx$

[10M]

1.(d) Find the equation of the sphere which passes though the circle $x^2+y^2=4$; $z=0$ and is cut by the plane $x+2y+2z=0$ in a circle of radius 3.

[10M]

1.(e) Find the shortest distance between the lines ${\displaystyle \frac{x-1}{2}}={\displaystyle \frac{y-2}{4}}=z-3$ and $y-mx=z=0$. For what value of $m$ will the two lines intersect?

[10M]

2.(a)(i) If $M_2(R)$ is space of real matrices of order $2\times 2$ and $P_2(x)$ is the space of real polynomials of degree at most 2, then find the matrix representation of $T:M_2(R)\to P_2(x)$ such that $T\left(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\right)=a+c+(a-d)x+(b+c)x^2$, with respect to the standard bases of $M_2(R)$ and $P_2(x)$, further find null space of $T$.

[10M]

2.(a)(ii) If $T:P_2(x)\to P_3(x)$ is such that $T(f(x))=f(x)+5{\int }_0^{x}f(t)dt$, then choosing $\{1,1+x,1-x^2\}$ and $\{1,x,x^2,x^3\}$ as bases of $P_2(x)\$and\$P_3(x)$ respectively, find the matrix of $T$.

[6M]

2.(b)(i) If $A=\left[\begin{array}{ccc}1& -1& 2\\ -2& 1& -1\\ 1& 2& 3\end{array}\right]$ is the matrix representation of a linear transformation $T:P_2(x)\to P_2(x)$ with respect to the bases $\{1-x,x(1-x),x(1+x)\}$ and $\{1,1+x,1+x^2\}$ then find $T$.

[6M]

2.(b)(ii) Prove that eigen values of a Hermitian matrix are all real.

[8M]

2.(c) If $A=\left[\begin{array}{ccc}1& 1& 0\\ 1& 1& 0\\ 0& 0& 1\end{array}\right]$, then find the eigen values and eigenvectors of $A$.

[6M]

3.(a) Find the maximum and minimum values of $x^2+y^2+z^2$ subject to the conditions ${\displaystyle \frac{x^2}{4}}+{\displaystyle \frac{y^2}{5}}+{\displaystyle \frac{z^2}{25}}=1$ and $x+y-z=0$.

[20M]

3.(b) Let $f(x,y)=\{\begin{array}{ll}{\displaystyle \frac{2x^{4}y-5x^{3s}{y}^2+y^5}{{(x^2+y^2)}^2}},& (x,y)\ne (0,0)\\ 0& ,(x,y)=(0,0)\end{array}$
Find a $\delta >0$ such that $|f(x,y)-f(0,0)|<0.01$ whenever $\sqrt{x^2+y^2}<\delta$.

[15M]

Remarks: Original question has printing mistake

3.(c) Find the surface area of the plane $x+2y+2z=12$ cut off by $x=0$, $y=0$ and $x^2+y^2=16$.

[15M]

4.(a) Find the surface generated by a line which intersects the line $y=a=z,x+3z=a=y+z$ and parallel to the plane $x+y=0$.

[10M]

4.(b) Show that the cone $3yz-2zx-2xy=0$ has an infinite set of three mutually perpendicular generators. If ${\displaystyle \frac{x}{1}}={\displaystyle \frac{y}{1}}={\displaystyle \frac{z}{z}}$ is a generator belonging to one such set, find the other two.

[10M]

4.(c) Evaluate ${\iint }_{R}f(x,y)dxdy$ over the rectangle $R=[0,1;0,1]$, where $f(x,y)=\{\begin{array}{c}x+y,\text{ if }{x}^2<y<2x^2\\ 0,\end{array}$

[15M]

4.(d) Find the locus of the point of intersection of three mutually perpendicular tangent planes to the conicoid $a{x}^2+b{y}^2+c{z}^2=1$.

[15M]

5.(a) Find a particular integral of ${\displaystyle \frac{d^{2}y}{d{x}^2}}+y=e^{x/2}\sin {\displaystyle \frac{x\sqrt{3}}{2}}$.

[10M]

5.(b) Prove that the vectors $\overrightarrow{a}=\hat{i}\hat{j}\hat{k}$, $\overrightarrow{b}=\hat{i}\hat{j}\hat{k}$, $\overrightarrow{c}=\hat{i}\hat{j}\hat{k}$ can form the sides of a triangle. Find the lengths of the medians of the triangle.

[10M]

5.(c) Solve:

[10M]

5.(d) Show that the family of parabolas $y^2=4cx+4c^2$ is self orthogonal.

[10M]

5.(e) A particle moves with a central acceleration which varies inversely as the cube of the distance. If it is projected from an apse at a distance $a$ from the origin with a velocity which is $\sqrt{2}$ times the velocity for a circle of radius $a$, then find the equation to the path.

[10M]

6.(a) Solve $\{y(1-x\tan x)+x^2\cos x\}dx-xdy=0$.

[10M]

6.(b) Using the method of variation of parameters, solve the differential equation $(D^2+2D+1)y=e^{-x}\log (x),[D\equiv {\displaystyle \frac{d}{dx}}]$.

[15M]

6.(c) Find the general solution of the equation $x^2{\displaystyle \frac{d^{3}y}{d{x}^3}}-4x{\displaystyle \frac{d^{2}y}{d{x}^2}}+6{\displaystyle \frac{dy}{dx}}=4$.

[15M]

6.(d) Using Laplace transformation, solve the following: $y^{\mathrm{\prime }\mathrm{\prime }}-2y^{\mathrm{\prime }}-8y=0$, $y(0)=3$, $y^{\mathrm{\prime }}(0)=6$.

[10M]

7.(a) A uniform rod $AB$ of length 2 a movable about a hinge at $A$ rests with other end against a smooth vertical wall. If $\alpha$ is inclination of the rod to the vertical, prove that the magnitude of reaction of the hinge is ${\displaystyle \frac{1}{2}}W\sqrt{4+{\tan }^2\alpha }$, where $W$ is the weight of the rod.

[15M]

7.(b) Two weights $P$ and $Q$ are suspended from a fixed point $O$ by strings $OA$, $OB$ and are kept apart by a light rod $AB$. If the strings $OA$ and $OB$ make angles $\alpha$ and $\beta$ with the rod $AB$, show that the angle $\theta$ which the rod makes with the vertical is given by $\tan \theta ={\displaystyle \frac{P+Q}{P\cot \alpha -Q\cot \beta }}$.

[15M]

7.(c) A square $ABCD$, the length of whose sides is $a$, is fixed in a vertical plane with two of its sides horizontal. An endless string of length $l(>4a)$ passes over four pegs at the angles of the board and through a ring of weight $W$ which is hanging vertically. Show that the tension of the string is ${\displaystyle \frac{W(l-3a)}{2\sqrt{l^2-6la+8a^2}}}$.

[20M]

8.(a) Find $f(r)$ such that $\mathrm{\nabla }f={\displaystyle \frac{\overrightarrow{r}}{r^5}}$ and $f(1)=0$.

[10M]

8.(b) Prove that the vector $\overrightarrow{a}=3\overrightarrow{i}+\hat{j}-2\hat{k}$, $\overrightarrow{b}=-\hat{i}+3\hat{j}+4\hat{k}$, $\overrightarrow{c}=4\hat{i}-2\hat{j}-6\hat{k}$ can from the sides of a triangle. Find the length of the medians of the triangle.

[10M]

8.(c) A particle moves in a straight line. Its acceleration is directed towards a fixed point $O$ in the line and is always equal to $\mu {\left({\displaystyle \frac{a^5}{x^2}}\right)}^{1/3}$ when it is at a distance $x$ from $O$. If it starts from rest at a distance $a$ from $O$, then find the time the particle will arrive at $O$.

[15M]

8.(d) For the cardioid $r=a(1+\cos \theta )$, show that the square of the radius of curvature at any point $(r,\theta )$ is proportion to $r$. Also find the radius of curvature if $\theta =0$, ${\displaystyle \frac{\pi }{4}}$, ${\displaystyle \frac{\pi }{2}}$.

[15M]