Paper I PYQs-2012

1.(a) Define a function \(f\) of two real variables in the plane by: \(f(x, y)=\left\{\begin{array}{c}{ \dfrac{x^{3} \cos \dfrac{1}{y}+y^{3} \cos \dfrac{1}{x}}{x^{2}+y^{2}}, \text { for } x, y \neq 0 } \\ {0, \text { otherwise }}\end{array}\right.\)

Check the continuity and differentiability of \(f\) at (0,0).

[12M]

1.(b) Let \(p\) and \(q\) be positive real numbers such that \(\dfrac{1}{p}\) + \(\dfrac{1}{q}\) =1. Show that for real numbers \(a\), \(b \geq0\),

[12M]

1.(c) Prove or disprove the following statement: If \(B=\left\{b_{1}, b_{2}, b_{3}, b_{4}, b_{5}\right\}\) is a basis for \(\mathbb{R}^{5}\) and \(V\) is a two-dimensional subspace of \(\mathbb{R}^{5}\), then \(V\) has a basis made of two members of \(B\).

[12M]

1.(d) Let \(T : \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) be the linear transformation defined by \(T(\alpha, \beta, \gamma)=(\alpha+2 \beta-3 \gamma, 2 \alpha+5 \beta-4 \gamma, \alpha+4 \beta+\gamma\). Find a basis and the dimension of the image of \(T\) and the kernel of \(T\).

[12M]

1.(e) Prove that two of the straight lines represented by the equation \(x^{3}+b x^{2} y+c x y^{2}+y^{3}=0\) will be at right angles, if \(b+c=-2\).

[12M]

2.(a)(i) Let \(V\) be the vector space of all \(2 \times 2\) matrices over the field of real numbers. Let \(W\) be the set consisting of all matrices with zero determinant. Is \(W\) a subspace of \(V\)? Justify your answer?

[8M]

2.(a)(ii) Find the dimension and a basis for the space \(W\) of all solutions of the following homogeneous system using matrix notation:

[12M]

2.(b)(i) Consider the linear mapping \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) by \(f(x, y)=(3 x+4 y, 2 x-5 y)\). Find the matrix \(A\) relative to the basis \(\{(1,0), (0,1)\}\) and the matrix \(B\) relative to the basis \(\{(1,2), (2,3)\}\).

[12M]

2.(b)(ii) If \(\lambda\) is a characteristic root of a non-singular matrix \(A\), then prove that \(\dfrac{\vert A \vert }{\lambda}\) is a characteristic root of \(\operatorname{Adj} A\).

[8M]

2.(c) Let \(H= \begin{bmatrix}{1} & {i} & {2+i} \\ {-i} & {2} & {1-i} \\ {2-i} & {1+i} & {2}\end{bmatrix}\) be a Hermitian matrix. Find a non-singular matrix \(P\) such that \(D=P^{T} H \overline{P}\) is diagonal.

[20M]

3.(a) Find the points of local extrema and saddle points of the function \(f\) for two variable defined by \(f(x, y)=x^{3}+y^{3}-63(x+y)+12 x y\).

[20M]

3.(b) Define a sequence \(s_{n}\) of real numbers by \(s_{n}=\sum_{i=1}^{n} \dfrac{(\log (n+i)-\log n)^{2}}{n+i}\). Does \(\lim _{n \rightarrow \infty} s_{n}\) exist? If so, compute the value of this limit and justify your answer.

[20M]

3.(c) Find all the real values of \(p\) and \(q\) so that the integral \(\int_{0}^{1} x^{p}\left(\log \dfrac{1}{x}\right)^{q} d x\) converges.

[20M]

4.(a) Compute the volume of the solid enclosed between the surfaces \(x^2+y^2=9\) and \(x^2+z^2=9\).

[20M]

4.(b) A variable plane is parallel to the plane \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=0\) and meets the axes in \(A\), \(B\), \(C\) respectively. Prove that circle \(A B C\) lies on the cone \(y z\left(\dfrac{b}{c}+\dfrac{c}{b}\right)+z x\left(\dfrac{c}{a}+\dfrac{a}{c}\right)+x y\left(\dfrac{a}{b}+\dfrac{b}{a}\right)=0\).

[20M]

4.(c) Show that locus of a point from which three mutually perpendicular tangent lines can be drawn to the paraboloid \(x^{2}+y^{2}+2 z^{2}=0\) is \(x^{2}+y^{2}+4 z=1\).

[20M]

5.(a) Solve \(\dfrac{d y}{d x}=\dfrac{2 x y e^{(x / y)^{2}}}{y^{2}\left(1+e^{(x / y)^{2}}\right)+2 x^{2} e^{(x / y)^{2}}}\).

[12M]

5.(b) Find the orthogonal trajectories of the family of curves \(x^{2}+y^{2}=a x\).

[12M]

5.(c) Using Laplace transforms, solve the initial value problem \(y^{\prime \prime}+2 y^{\prime}+y=e^{-t}\), \(y(0)=-1\), \(y^{\prime}(0)=1\).

[12M]

5.(d) A particle moves with an acceleration

towards the origin. It it starts from rest at a distance \(a\) from the origin, find its velocity when its distance from the origin is \(\dfrac{a}{2}\).

[12M]

5.(e) If \(\vec{A}=x^{2} y z \vec{i}-2 x z^{3} \vec{j}+x z^{2} \vec{k}\), \(\vec{B}=2 z \vec{i}+y \vec{j}-x^{2} \vec{k}\), find the value of \(\dfrac{\partial^{2}}{\partial x \partial y}(\vec{A}+\vec{B})\) at \((1,0,-2)\).

[12M]

6.(a) Show that the differential equation \((2 x y \log y) d x+\left(x^{2}+y^{2} \sqrt{y^{2}+1}\right) d y=0\) is not exact. Find an integrating factor and hence, the solution of the equation.

[20M]

6.(b) Find the general solution of the equation \(y^{\prime \prime \prime}-y^{\prime \prime}=12 x^{2}+6 x\).

[20M]

6.(c) Solve the ordinary differential equation \(x(x-1) y^{\prime \prime}-(2 x-1) y^{\prime}+2 y=x^{2}(2 x-3)\).

[20M]

7.(a) A heavy ring of mass \(m\), slides on a smooth vertical rod and is attached to a light string which passes over a small pulley distant \(a\) from the rod and has a mass \(M(>m)\) fastened to its other end. Show that if the ring be dropped from a point in the rod in the same horizontal plane as the pulley, it will descend a distance \(\dfrac{2 M m a}{M^{2}-m^{2}}\) before coming to rest.

[20M]

7.(b) A heavy hemispherical shell of radius \(a\) has a particle attached to a point on the rim, and rests with the curved surface in contact with a rough sphere of radius \(b\) at the highest point. Prove that if \(\dfrac{b}{a}>\sqrt{5}-1\), the equilibrium is stable, whatever be the weight of the particle.

[20M]

7.(c) The end links of a uniform chain slide along a fixed rough horizontal rod. Prove that the ratio of the maximum span to the length of the chain is \(\mu \log \left[\dfrac{1+\sqrt{1+\mu^{2}}}{\mu}\right]\), where \(\mu\) is the coefficient of friction.

[20M]

8.(a) Derive the Frenet-Serret formulae. Define the curvature and torsion for a space curve. Compute them for the space curve \(x=t\), \(y=t^{2}\), \(z=\dfrac{2}{3} t^{3}\). Show that the curvature and torsion are equal for this curve.

[20M]

8.(b) Verify Green’s theorem in the plane for \(\oint_{C}\left[x y+y^{2} d x+x^{2} d y\right]\) where \(C\) is the closed curve of the region bounded by \(y=x\) and \(y=x^{2}\).

[20M]

8.(c) If \(\vec{F}=y \vec{i}+(x-2 x z) \vec{j}-x y \vec{k}\), evaluate \(\iint_{S}(\vec{\nabla} \times \vec{F}) . \vec{n} d \vec{s}\), where \(S\) is the surface of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) above the \(xy-plane\).

[20M]