Paper I PYQs-2013

1.(a) Find the dimension and a basis of the solution space \(\mathrm{W}\) of the system \(x+2 y+2 z-s+3 t=0\), \(x+2 y+3 z+s+t=0\), \(3 x+6 y+8 z+s+5 t=0\).

[8M]

1.(b) Find the characteristic equation of the matrix \(A=\left[\begin{array}{lll}2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 2\end{array}\right]\) and hence find the matrix represented by \(A^{8}-5 A^{7}+7 A^{6}-3 A^{5}+A^{4}-5 A^{3}+8 A^{2}-2 A+I\).

[8M]

1.(c) Evaluate the integral \(\int_{0}^{\infty} \int_{0}^{x} x \mathrm{e}^{-\mathrm{x}^{2} / \mathrm{y}} \mathrm{dydx}\) by changing the order of integration.

[8M]

1.(d) Find the surface generated by the straight line which intersects the lines \(y=z=a\) and \(x+3 z=a=y+z\) and is parallel to the plane \(x+y=0\).

[8M]

1.(e) Find \(C\) of the mean value theorem, if \(f(x)=x(x-1)(x-2), a=0, b=\dfrac{1}{2}\) and \(C\) has usual meaning.

[8M]

2.(a) Let \(\mathrm{V}\) be the vector space of \(2 \times 2\) matrices over \(\mathbb{R}\) and let \(\mathrm{M}=\left[\begin{array}{rr}1 & -1 \\ -2 & 2\end{array}\right]\) Let \(\mathrm{F}: \mathrm{V} \rightarrow \mathrm{V}\) be the linear map defined by \(\mathrm{F}(\mathrm{A})=\mathrm{MA}\). Find a basis and the dimension of
(i) the kernel of W of \(\mathrm{F}\) (ii) the image U of F.

[10M]

2.(b) Locate the stationary points of the function \(x^{4}+y^{4}-2 x^{2}+4 x y-2 y^{2}\) and determine their nature.

[10M]

2.(c) Find an orthogonal transformation of co-ordinates which diagonalizes the quadratic form

\[q(x, y)=2 x^{2}-4 x y+5 y^{2}\]

[10M]

2.(d) Discuss the consistency and the solutions of the equations

\[x+a y+a z=1, a x+y+2 a z=-4, a x-a y+4 z=2\]

for different values of \(a\).

[10M]

3.(a) Prove that if \(a_{0}, a_{1}, a_{2}, \ldots \ldots, a_{n}\) are the real numbers such that

\[\dfrac{a_{0}}{n+1}+\dfrac{a_{1}}{n}+\dfrac{a_{2}}{n-1}+\ldots \ldots \ldots+\dfrac{a_{n-1}}{2}+a_{n}=0\]

then there exists at least one real number \(x\) between 0 and 1 such that

\[a_{0} x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\ldots+f_{n-1} x+a_{n}=0\]

[10M]

3.(b) Reduce the following equation to its canonical form and determine the nature of the conic \(4 x^{2}+4 x y+y^{2}-12 x-6 y+5=0\).

[10M]

3.(c) Let \(\mathrm{F}\) be a subfield of complex numbers and \(\mathrm{T}\) a function from \(\mathrm{F}^{3} \rightarrow \mathrm{F}^{3}\) defined by \(\mathrm{T}\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}\right)=\left(\mathrm{x}_{1}+\mathrm{x}_{2}+3 \mathrm{x}_{3}, 2 \mathrm{x}_{1}-\mathrm{x}_{2},-3 \mathrm{x}_{1}+\mathrm{x}_{2}-\mathrm{x}_{3}\right) .\) What are the conditions on \((a, b, c)\) such that \((a, b, c)\) be in the null space of \(T\)? Find the nullity of \(T\).

[10M]

3.(d) Find the equations to the tangent planes to the surface \(7 x^{2}-3 y^{2}-z^{2}+21=0\), which pass through the line \(7 x-6 y+9=0\), \(z=3\).

[10M]

4.(a) Evaluate

\[\int_{0}^{\pi / 2} \dfrac{x \sin x \cos x d x}{\sin ^{4} x+\cos ^{4} x}\]

[10M]

4.(b) Let \(H=\left[\begin{array}{ccc}1 & \text { i } & 2+i \\ -i & 2 & 1-i \\ 2-i & 1+i & 2\end{array}\right]\) be a Hermitian matrix. Find a non-singular matrix \(P\) such that \(\mathrm{P}^{\mathrm{t}} \mathrm{H} \overline{\mathrm{P}}\) is diagonal and also find its signature.

[10M]

4.(c) Find the magnitude and the equations of the line of shortest distance between the lines

\[\begin{array}{l} \dfrac{x-8}{3}=\dfrac{y+9}{-16}=\dfrac{z-10}{7} \\ \dfrac{x-15}{3}=\dfrac{y-29}{8}=\dfrac{z-5}{-5} \end{array}\]

[10M]

4.(d) Find all the asymptotes of the curve

\[x^{4}-y^{4}+3 x^{2} y+3 x y^{2}+x y=0\]

[10M]

5.(a) Solve:

\[\dfrac{d y}{d x}+x \sin 2 y=x^{3} \cos ^{2} y\]

[8M]

5.(b) A particle is performing a simple harmopiç motion of period \(\mathrm{T}\) about centre \(\mathrm{O}\) and it passes through a point \(P\), where \(O P=(b\) with velocity \(v\) in the direction of \(OP\). Find the time which elapses before it returps to \(\mathrm{P}\).

[8M]

5.(c) \(\overrightarrow{\mathrm{F}}\) being a vector, prove that
curl curl \(\overrightarrow{\mathrm{F}}=\) grad \(\operatorname{div} \overrightarrow{\mathrm{F}}-\nabla^{2} \overrightarrow{\mathrm{F}}\)
where \(\nabla^{2}=\dfrac{\partial^{2}}{\partial x^{2}}+\dfrac{\partial^{2}}{\partial y^{2}}+\dfrac{\partial^{2}}{\partial z^{2}}\)

[8M]

5.(d) A triangular lamina \(ABC\) of density \(\rho\) floats in a liquid of density \(\sigma\) with its plane vertical, the angle \(\mathrm{B}\) being in the surface of the liquid, and the angle \(\mathrm{A}\) not immersed. Find \(p/\sigma\) in terms of the lengths of the sides of the triangle.

[8M]

5.(e) A heavy uniform rod rests with one end against a smooth vertical wall and with a point in its length resting on a smooth peg. Find the position of equilibrium and discuss the nature of equilibrium.

[8M]

6.(a) Solve the differential equation

\[\dfrac{d^{2} y}{d x^{2}}-4 x \dfrac{d y}{d x}+\left(4 x^{2}-1\right) y=-3 e^{x^{2}} \sin ^{4} 2 x\]

by changing the dependent variable.

[13M]

6.(b) Evaluate \(\int_{\mathrm{S}} \overrightarrow{\mathrm{F}} \cdot \mathrm{d} \overrightarrow{\mathrm{s}},\) where \(\overrightarrow{\mathrm{F}}=4 \mathrm{xi}-2 \mathrm{y}^{2} \mathrm{j}+z^{2} \overrightarrow{\mathrm{k}}\) and \(\mathrm{s}\) is the surface bounding the region \(x^{2}+y^{2}=4, z=0\) and \(z=3\).

[13M]

6.(c) Two bodies of weights \(w_{1}\) and \(w_{2}\) are placed on an inclined plane and are connected by a light string which coincides with a line of greatest slope of the plane; if the coefficient of friction between the bodies and the plane are respectively \(\mu_{1}\) and \(\mu_{2}\), find the inclination of the plane to the horizontal when both bodies are on the point of motion, it being assumed that smoother body is below the other.

[14M]

7.(a) Solve

\[\begin{array}{l} \left(D^{3}+1\right) y=e^{\dfrac{x}{2}} \sin \left(\dfrac{\sqrt{3}}{2} x\right) \\ \text { where } D=\dfrac{d}{d x} \end{array}\]

[13M]

7.(b) A body floating in water has volumes \(v_{1}, v_{2}\) and \(v_{3}\) above the surface, when the densities of the surrounding air are respectively \(\rho_{1}, \rho_{2}, \rho_{3}\). Find the value of:

\[\dfrac{\rho_{2}-\rho_{3}}{v_{1}}+\dfrac{\rho_{3}-\rho_{1}}{v_{2}}+\dfrac{\rho_{1}-\rho_{2}}{v_{3}}\]

[13M]

7.(c) A particle is projected vertically upwards with a velocity \(\mathrm{u},\) in a resisting medium which produces a retardation \(\mathrm{k} v^{2}\) when the velocity is \(v\). Find the height when the particle comes to rest above the point of projection.

[14M]

8.(a) Apply the method of variation of parameters to solve

\[\dfrac{d^{2} y}{d x^{2}}-y=2\left(1+e^{x}\right)^{-1}\]

[13M]

8.(b) Verify the Divergence theorem for the vector function

\[\overrightarrow{\mathrm{F}}=\left(x^{2}-y z\right) \vec{i}+\left(y^{2}-x z\right) \vec{j}+\left(z^{2}-x y\right) \bar{k}\]

taken over the rectangular parallelopiped

\[0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq c\]

[14M]

8.(c) A particle is projected with a velocity \(v\) along a smooth horizontal plane in a medium whose resistance per unit mass is double the cube of the velocity. Find the distance it will describe in time \(t\).

[13M]