Paper I PYQs-2018

1.(a) Let $A$ be a $3\times 2$ matrix and $B$ a $2\times 3$ matrix. Show that $C=A\cdot B$ is a singular matrix.

[10M]

1.(b) Express basis vectors $e_1=(1,0)$ and $e_2=(0,1)$ as linear combination of ${\alpha }_1=(2,-1)$ and ${\alpha }_2=(1,3)$.

[10M]

1.(c) Determine if $\underset{z\to 1}{lim}(1-z)\tan {\displaystyle \frac{\pi z}{2}}$ exists or not. If the limit exists, then find its value.

[10M]

1.(d) Find the limit $\underset{n\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{n^2}}\sum _{r=0}^{n-1}\sqrt{n^2-r^2}$.

[10M]

1.(e) Find the projection of the straight line ${\displaystyle \frac{x-1}{2}}={\displaystyle \frac{y-1}{3}}={\displaystyle \frac{z+1}{-1}}$ on the plane $x+y+2z=6$.

[10M]

2.(a) Show that if $A$ and $B$ are similar $n\times n$ matrices, then they have the same eigenvalues.

[12M]

2.(b) Find the shortest distance from the point $(1,0)$ to the parabola $y^2=4x$.

[13M]

2.(c) The ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$ revolves about the $x-axis$. Find the volume of the solid of revolution.

[13M]

2.(d) Find the shortest distance between the lines

and the $z-axis$.

[12M]

3.(a) For the system of linear equations $x+3y-2z=-1$
$5y+3z=-8$
$x-2y-5z-7$
determine which of the following statements are true and which are false:

(i) The system has no solution.
(ii) The system has a unique solution.
(iii) The system has infinitely many solutions.

[13M]

3.(b) Let
$f(x,y)=x{y}^2$, if $y>0$ and $f(x,y)=-x{y}^2$, if $y\le 0$

Determine which of ${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}}(0,1)$ and ${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }y}}(0,1)$ exists and which does not exist.

[12M]

3.(c) Find the equations to the generating lines of the paraboloid $(x+y+z)(2x+y-z)=6z$ which pass through the point $(1,1,1)$.

[13M]

3.(d) Find the equation of the sphere in $xyz-plane$ passing through the points $(0,0,0)$, $(0,1,-1)$, $(-1,2,0)$, $(1,2,3)$.

[12M]

4.(a) Find the maximum and the minimum value of $x^4-5x^2+4$ on the interval $[2,3]$.

[13M]

4.(b) Evaluate the integral ${\int }_0^a{\int }_{x/a}^x{\displaystyle \frac{xdydx}{x^2+y^2}}$.

[12M]

4.(c) Find the equation of the cone with $(0,0,1)$ as the vertex and $2x^2-y^2=4$, $z=0$ as the guiding curve.

[13M]

4.(d) Find the equation of the plane parallel to $3x-y+3z=8$ and passing through the point $(1,1,1)$.

[12M]

5.(a) Solve:

[10M]

5.(b) Find the angle between the tangent and a general point of the curve whose equations are $x=3t$, $y=3t^2$, $z=3t^3$ and the line $y=z-x=0$.

[10M]

5.(c) Solve:

[10M]

5.(d)(i) Find the Laplace transform of $f(t)={\displaystyle \frac{1}{\sqrt{t}}}$.

[5M]

5.(d)(ii) Find the inverse Laplace transform of ${\displaystyle \frac{5s^2+3s-16}{(s-1)(s-2)(s-3)}}$.

[5M]

5.(e) A particle projected from a given point on the ground just clears a wall of height $h$ at a distance from the point of projection. If the particle moves in a vertical plane and if the horizontal range is $R$, find the elevation of the projection.

[10M]

6.(a) Solve:

[13M]

6.(b) A particle moving with simple harmonic motion in a straight line has velocities $v_1$ and $v_2$ at the distances $x_1$ and $x_2$ respectively from the center of its path. Find the period of its motion.

[12M]

6.(c) Solve:

[13M]

6.(d) If $S$ is the surface of the sphere $x^2+y^2+z^2=a^2$, then evaluate

using Gauss’s divergence theorem.

[12M]

7.(a) Solve:

[13M]

7.(b) Find the curvature and torsion of the curve

[12M]

7.(c) Solve the initial value problem

$y^{″}-5y^{\prime }+4y=e^{2t}$
$y(0)={\displaystyle \frac{19}{20}},y^{\prime }(0)={\displaystyle \frac{8}{3}}$

[13M]

7.(d) Find $\alpha$ and $\beta$ such that $x^{\alpha }{y}^{\beta }$ is an integrating factor of $(4y^2+3xy)dx-(3xy+2x^2)dy=0$ and solve the equation.

[12M]

8.(a) Let $\overrightarrow{v}=v_1\hat{i}+v_2\hat{j}+v_3\hat{k}$. Show that $curl(curl\overrightarrow{v})=grad(div\overrightarrow{v})-{\mathrm{\nabla }}^2\overrightarrow{v}$

[12M]

8.(b) Evaluate the line integral ${\int }_C-y^{3}dx+x^{3}dy+z^{3}dz$ using Stokes’ theorem. Here $C$ is the intersection of the cylinder $x^2+y^2=1$ and the plane $x+y+z=1$. The orientation on $C$ corresponds to counterclockwise motion in the $xy-plane$.

[13M]

8.(c) Let $\overrightarrow{F}=x{y}^2\hat{i}+(y+x)\hat{j}$. Integrate $(\mathrm{\nabla }\times \overrightarrow{F})\cdot \overrightarrow{k}$ over the region in the first quadrant bounded by the curve $y=x^2$ and $y=x$ using Green’s theorem.

[12M]

8.(d) Find $f(y)$ such that $(2x{e}^y+3y^2)dy+(3x^2+f(y))dx=0$ is exact and hence solve.

[13M]