Paper I PYQs-2015

1.(a) The vectors $V_1=(1,1,2,4)$, $V_2=(2,-1,-5,2)$, $V_3=(1,-1,-4,0)$ and $V_4=(2,1,1,6)$ are linearly independent. Is it true? Justify your answer.

[10M]

1.(b) Reduce the following matrix to row echelon form and hence find its rank: $\left[\begin{array}{cccc}1& 2& 3& 4\\ 2& 1& 4& 5\\ 1& 5& 5& 7\\ 8& 1& 14& 17\end{array}\right]$.

[10M]

1.(c) Evaluate the following limit

[10M]

1.(d) Evaluate the following integral:

[10M]

1.(e) Find the positive value of $a$ for which the plane $ax-2y+z+12=0$ touches the sphere $x^2+y^2+z^2-2x-4y+2z-3=0$. Also find the point of contact.

[10M]

2.(a) If matrix $A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ then find $A^{30}$.

[12M]

2.(b) A conical tent is of given capacity. For the least amount of Canvas required for it, find the ratio of its height to the radius of its base.

[13M]

2.(c) Find the eigen values and eigen vectors of the matrix $\left[\begin{array}{ccc}1& 1& 3\\ 1& 5& 1\\ 3& 1& 1\end{array}\right]$.

[12M]

2.(d) If $6x=3y=2z$ represents one of the mutually perpendicular generators of the cone $5yz-8zx-3xy=0$, then obtain the equations of the other two generators.

[13M]

3.(a) Let $V=R^3$ and $T\in A(V)$, for all $a_i\in A(V)$, be defined by $T(a_1,a_2,a_3)=(2a_1+5a_2+a_3,-3a_1+a_2-a_3,a_1+2a_2+3a_3)$. What is the matrix $T$ relative to the basis $V_1=(1,0,1)$, $V_2=(-1,2,1)$, $V_3=(3,-1,1)$?

[12M]

3.(b) Which point of the sphere $x^2+y^2+z^2=1$ is at the maximum distance from the point (2,1,3).

[13M]

3.(c)(i) Obtain the equation of the plane passing through the points $(2,3,1)$ and $(4,-5,3)$ parallel to $x-axis$.

[6M]

3.(c)(ii) Verify if the lines: ${\displaystyle \frac{x-a+d}{\alpha -\delta }}={\displaystyle \frac{y-a}{\alpha }}={\displaystyle \frac{z-a-d}{\alpha +\delta }}$ and ${\displaystyle \frac{x-b+c}{\beta -\gamma }}={\displaystyle \frac{y-b}{\beta }}={\displaystyle \frac{z-b-c}{\beta +\gamma }}$ are coplanar. If yes, find the equation of the plane in which they lie.

[7M]

3.(d) Evaluate the integral $\iint (x-y{)}^2{\cos }^2(x+y)dxdy$ where $R$ is the rhombus with successive vertices as $(\pi ,0)$, $(2\pi ,\pi )$, $(\pi ,2\pi )$, $(0,\pi )$.

[12M]

4.(a) Evaluate ${\iint }_R\sqrt{|y-x^2|}dxdy$ where $R=[-1,1;0,2]$.

[13M]

4.(b) Find the dimension of the subspace of $R^4$, spanned by the set $\{(1,0,0,0),(0,1,0,0),(1,2,0,1),(0,0,0,1)\}$. Hence find its basis.

[12M]

4.(c) Two perpendicular tangent planes to the paraboloid $x^2+y^2=2z$ intersect in a straight line in the plane $x=0$. Obtain the curve to which this straight line touches.

[13M]

4.(d) For the function

Examine the continuity and differentiability.

[12M]

5.(a) Solve the differential equation: $x\cos x{\displaystyle \frac{dy}{dx}}+y(x\sin x+\cos x)=1$.

[10M]

5.(b) Solve the differential equation:

$(2x{y}^4{e}^y+2x{y}^3+y)dx+(x^2{y}^4{e}^y-x^2{y}^2-3x)dy=0$.

[10M]

5.(c) A body moving under SHM has an amplitude ${}^{\prime }{a}^{\prime }$ and time period ${}^{\prime }{T}^{\prime }$. If the velocity is trebled, when the distance from mean position is ${\displaystyle \frac{2}{3}}a$, the period being unaltered, find the new amplitude.

[10M]

5.(d) A rod of 8 kg movable in a vertical plane about a hinge at one end, another end is fastened aweight equal to half of the rod, this end is fastened by a string of length $l$ to a point at a height $b$ above the hinge vertically. Obtain the tension in the string.

[10M]

5.(e) Find the angle between the surfaces $x^2+y^2+z^2-9=0$ and $z=x^2+y^2-3$ at $(2,-1,2)$.

[10M]

6.(a) Find the constant $a$ so that $(x+y{)}^a$ is the integrating factor of $(4x^2+2xy+6y)dx+(2x^2+9y+3x)dy=0$ and hence solve the differential equation.

[12M]

6.(b) Two equal ladders of weight 4 $\mathrm{k}\mathrm{g}$ each are placed so as to lean at $A$ against each other with their ends resting on a rough floor, given the coefficient of friction is $\mu$. The ladders at $A$ make an angle ${60}^{\circ }$ with each other. Find what weight on the top would cause them to slip.

[13M]

6.(c) Find the value of $\lambda$ and $\mu$ so that the surfaces $(\lambda x^2-\mu yz)$ = $(\lambda +2)x$ and $4x^{2}y+z^3=4$ may interesect orthogonally at $(1,-1,2)$

[12M]

6.(d) A mass starts from rest at a distance ${}^{\prime }{a}^{\prime }$ from the centre of force which attracts inversely as the distance. Find the time of arriving at the centre.

[13M]

7.(a)(i) Obtain inverse Laplace transform of $\{\ln (1+{\displaystyle \frac{1}{s^2}})+{\displaystyle \frac{s}{s^2+25}}{e}^{-5s}\}$.

[6M]

7.(a)(ii) Using Laplace transform, solve $y^{\mathrm{\prime }\mathrm{\prime }}+y=t$, $y(0)=1$, $y^{\mathrm{\prime }}(0)=-2$.

[6M]

7.(b) A particle is projected from the base of a hill whose slope is that of a right circular cone, whose axis is vertical. The projectile gazes the vertex and strikes the hill again at a point on the base. If the semi vertical angle of the cone is ${30}^{\circ }$, $h$ is height, determine the initial velocity on $u$ of the projection and its angle of projection.

[13M]

7.(c) A vector field is given by $\overrightarrow{F}=(x^2+x{y}^2)\hat{i}+(y^2+x^{2}y)\hat{j}$. Verify that the field is irrotational or not. Find the scalar potential.

[12M]

7.(d) Solve the differential equation $x=py-p^2$ where $p={\displaystyle \frac{dy}{dx}}$.

[13M]

8.(a) Find the length of an endless chain which will hang over a circular pulley of radius ${}^{\prime }{a}^{\prime }$ so as to be in contact with the two-thirds of the circumference of the pulley.

[12M]

8.(b) A particle moves in a plane under a force, towards a fixed centre, proportional to the distance. If the path of the particle has two apsidal distances $a$, $b$ ($a>b$), find the equation of the path.

[13M]

8.(c) Evaluate ${\int }_C{e}^{-x}(\sin ydx+\cos ydy)$, where $C$ is the rectangle with vertices $(0,0)$, $(\pi ,0)$, $(\pi ,{\displaystyle \frac{\pi }{2}})$, $(0,{\displaystyle \frac{\pi }{2}})$.

[12M]

8.(d) Solve $x^4{\displaystyle \frac{d^{4}y}{d{x}^4}}+6x^3{\displaystyle \frac{d^{3}y}{d{x}^3}}+4x^2{\displaystyle \frac{d^{2}y}{d{x}^2}}-2x{\displaystyle \frac{dy}{dx}}-4y=x^2+2\cos ({\log }_{e}x)$.

[13M]