Paper I PYQs-2019

1.(a) Let $T:R^3\to R^3$ be a linear operator on $R^3$ defined by $T(x,y,z)=$ $(2y+z,x-4y,3x)$. Find the matrix of $T$ in the basis $\{(1,1,1)$, $(1,1,0)$, $(1,0,0)\}$.

[8M]

1.(b) The eigenvalues of a real symmetric matrix $A$ are -1, 1 and -2. The corresponding eigenvectors are ${\displaystyle \frac{1}{\sqrt{2}}}(-1,1,0{)}^T$, $(0,0,1{)}^T$ and ${\displaystyle \frac{1}{\sqrt{2}}}(-1,-1,0{)}^T$ respectively. Find the matrix $A^4$.

[8M]

1.(c) Find the volume lying inside the cylinder $x^2+y^2-2x=0$ and outside the paraboloid $x^2+y^2=2z$, while bounded by $xy$-plane.

[8M]

1.(d) Justify by using Rolle’s theorem or mean value theorem that there is no number $k$ for which the equation $x^3-3x+k=0$ has two distinct solutions in the interval $[-1,1]$.

[8M]

1.(e) If the coordinates of the points $A$ and $B$ are respectively $(b\cos \alpha ,b\sin \alpha )$ and $(a\cos \beta ,a\sin \beta )$ and if the line joining $A$ and $B$ is produced to the point $M(x,y)$ so that $AM:MB=b:a$, then show that $x\cos {\displaystyle \frac{\alpha +\beta }{2}}+y\sin {\displaystyle \frac{\alpha +\beta }{2}}=0$

[8M]

2.(a) Determine the extreme values of the function $f(x,y)=3x^2-6x+2y^2-4y$ in the region $\{(x,y)\in {\mathbb{R}}^2:3x^2+2y^2\le 20\}$

[10M]

2.(b) Consider the singular matrix

Given that one eigenvalue of $A$ is 4 and one eigenvector that does not correspond to this eigenvalue 4 is $(1100{)}^T.$ Find all the eigenvalues of $A$ other than 4 and hence also find the real numbers $p$, $q$, $r$ that satisfy the matrix equation $A^4+p{A}^3+q{A}^2+rA=0$

[15M]

2.(c) A line makes angles $\alpha ,\beta ,\gamma ,\delta$ with the four diagonals of a cube. Show that

[15M]

3.(a) Consider the vectors $x_1=(1,2,1,-1)$, $x_2=(2,4,1,1)$, $x_3=(-1,-2,0,-2)$ and $x_4=(3,6,2,0)$ in ${\mathbb{R}}^4$. Justify that the linear span of the set $\{x_1,x_2,x_3,x_4\}$ is a subspace of ${\mathbb{R}}^4$, defined as

Can this subspace be written as $\{(\alpha ,2\alpha ,\beta ,2\alpha -3\beta ):\alpha ,\beta \in \mathrm{R}\}?$ What is the dimension of this subspace?

[15M]

3.(b) The dimensions of a rectangular box are linear functions of time- $l(t)$, $w(t)$ and $h(t)$. If the length and width are incréasing at the rate $2\mathrm{c}\mathrm{m}/\mathrm{s}\mathrm{e}\mathrm{c}$ and the height is decreasing at the rate $3\mathrm{c}\mathrm{m}/\mathrm{s}\mathrm{e}\mathrm{c}$ find the rates at which the volume $V$ and with respect to time. If $l(0)=10$, $w(0)=8$ and the surface area $S$ are changing $h(0)=20,$ is $V$ increasing or decreasing, when $t=5$ sec? What about $S$, when $t=5$ sec?

[10M]

3.(c) Show that the shortest distance between the straight lines

and

is $3\sqrt{30}$. Find also the equation of the line of shortest distance.

[15M]

4.(a) Using elementary row operations, reduce the matrix $A=\left[\begin{array}{llll}2& 1& 3& 0\\ 3& 0& 2& 5\\ 1& 1& 1& 1\\ 2& 1& 1& 3\end{array}\right]$ to reduced echelon form and find the inverse of $A$ and hence solve the system of linear equations $AX=b,$ where $X=(x,y,z,u{)}^T$ and $b=(2,1,0,4{)}^T$

[15M]

4.(b) Find the centroid of the solid generated by revolving the upper half of the cardioid $r=a(1+\cos \theta )$ bounded by the line $\theta =0$ about the initial line. Take the density of the solid as uniform.

[10M]

4.(c) A variable plane is parallel to the plane ${\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}}+{\displaystyle \frac{z}{c}}=0$ and meets the axes at the points $A,B$ and $C$. Prove that the circle $ABC$ lies on the cone

[15M]

5.(a) Solve the differential equation $(D^2+1)y=x^2\sin 2x;D\equiv {\displaystyle \frac{d}{dx}}$.

[8M]

5.(b) Solve the differential equation $(px-y)(py+x)=h^{2}p$, where $p=y^{\mathrm{\prime }}$.

[8M]

5.(c) A 2 metres rod has a weight of $2\mathrm{N}$ and has its centre of gravity at $120\mathrm{c}\mathrm{m}$ from one end. At $20\mathrm{c}\mathrm{m},100\mathrm{c}\mathrm{m}$ and $160\mathrm{c}\mathrm{m}$ from the same end are hing loads of $3\mathrm{N}$, $7\mathrm{N}$ and $10\mathrm{N}$ respectively. Find the point at which the rod must be supported if it is to remain horizontal.

[8M]

5.(d) Let $\overline{r}=\overline{r}(s)$ represent a space curve. Find ${\displaystyle \frac{d^3\overline{r}}{d{s}^3}}$ in terms of $\overline{T},\overline{N}$ and $\overline{B}$ where $\overline{T}$, $\overline{N}$ and $\overline{B}$ represent tangent, principal normal and binormal respectively. Compute ${\displaystyle \frac{d\overline{r}}{ds}}\cdot ({\displaystyle \frac{d^2\overline{r}}{d{s}^2}}\times {\displaystyle \frac{d^3\overline{r}}{d{s}^3}})$ in terms of radius of curvature and the torsion.

[8M]

5.(e) Evaluate ${\int }_{(0,0)}^{(2,1)}(10x^4-2x{y}^3)dx-3x^2{y}^{2}dy$ along the path $x^4-6x{y}^3=4y^2$.

[8M]

6.(a) Solve by the method of variation of parameters the differential equation

[15M]

6.(b) Find the law of force for the orbit $r^2=a^2\cos 2\theta$ (the pole being the centre of the force).

[15M]

6.(c) Verify Stokes’ theorem for $\overline{V}=(2x-y)\hat{i}-y{z}^2\hat{j}-y^{2}z\hat{k},$ where $S$ is the upper half surface of the sphere $x^2+y^2+z^2=1$ and $C$ is its boundary.

[10M]

7.(a) Find the general solution of the differential equation

Hence find the particular solution satisfying the conditions

[15M]

7.(b) A vessel is in the shape of a hollow hemisphere surmounted by a cone held with the axis vertical and vertex uppermost. If it is filled with a liquid so as to submerge half the axis of the cone in the liquid and height of the cone be double the radius $(r)$ of its base, find the resultant downward thrust of the liquid on the vessel in terms of the radius of the hemisphere and density (p) of the liquid.

[15M]

7.(c) Derive the Frenet-Serret formulae. Verify the same for the space curve $x=3\cos t$, $y=3\sin t$, $z=4t$

[10M]

8.(a) Find the general solution of the differential equation

[10M]

8.(b) A shot projected with a velocity $u$ can just reach a certain point on the horizontal plane through the point of projection. So in order to hit a mark $h$ metres above the ground at the same point, if the shot is projected at the same elevation, find increase in the velocity of projection.

[15M]

8.(c) Derive ${\mathrm{\nabla }}^2={\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}}$ in $$ spherical coordinates and compute ${\mathrm{\nabla }}^2\left({\displaystyle \frac{x}{{(x^2+y^2+z^2)}^{{\displaystyle \frac{3}{2}}}}}\right)$ in spherical coordinates.

[15M]