Paper I PYQs-2011
Section A
1.(a) Let \(V\) be the vector space of \(2 \times 2\) matrices over the field of real numbers \(\mathrm{R}\). Let \(\mathrm{W}=\{ \mathrm{A} \in \mathrm{V} \vert\) Trace \(\mathrm{A}=0\}\). Show that \(\mathrm{W}\) is a subspace of \(\mathrm{V}\). Find a basis of \(\mathrm{W}\) and dimension of \(\mathrm{W}\).
[10M]
1.(b) Find the linear transformation from \(\mathbf{R}^{3}\) into \(\mathbf{R}^{3}\) which has its range the subspace spanned by \((1,0,-1)\), \((1,2,2)\).
[10M]
1.(c) Show that the function defined by
\[f(x, y)=\left\{\begin{array}{cc} \dfrac{x^{3}+y^{3}}{x-y}, & x \neq y \\ 0, & x=y \end{array}\right.\]is discontinuous at the origin but possesses partial derivatives \(f_{x}\) and \(f_{y}\) thereat.
[10M]
1.(d) Let the function \(\mathrm{f}\) be defined by
\[f(t)=\left\{\begin{array}{ll} 0, & \text { for } t<0 \\ t, & \text { for } \quad 0 \leq t \leq 1 \\ 4, & \text { for } t>1 \end{array}\right.\](i) Determine the function \(\mathrm{F}(\mathrm{x})=\int_{0}^{\mathrm{x}} \mathrm{f}(\mathrm{t}) \mathrm{dt}\).
(ii) Where is \(F\) non-differentiable? Justify your answer.
[10M]
1.(e) A variable plane is at a constant distance p from the origin and meets the axes at \(\mathrm{A}, \mathrm{B}, \mathrm{C}\). Prove that the locus of the centroid of the tetrahedron \(OABC\) is \(\dfrac{1}{x^{2}}+\dfrac{1}{y^{2}}+\dfrac{1}{z^{2}}=\dfrac{16}{p^{2}}\).
[10M]
2.(a) Let \(\mathrm{V}=\left\{(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{u}) \in \mathrm{R}^{4}: \mathrm{y}+\mathrm{z}+\mathrm{u}=0\right\}\) \(\cdot \mathrm{W}=\left\{(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{u}) \in \mathrm{R}^{4}: \mathrm{x}+\mathrm{y}=0, \mathrm{z}=2 \mathrm{u}\right\}\) be two subspaces of \(\mathbf{R}^{4}\). Find bases for \(\mathrm{V}, \mathrm{W}\), \(\mathrm{V}+\mathrm{W}\) and \(\mathrm{V} \cap \mathrm{W}\).
[10M]
2.(b) Find the characteristic polynomial of the matrix
\[A=\left(\begin{array}{ccc} 3 & 1 & 1 \\ 2 & 4 & 2 \\ -1 & -1 & 1 \end{array}\right)\]and hence compute \(\mathrm{A}^{10}\).
[10M]
2.(c) Let \(A=\left(\begin{array}{ccc}1 & -3 & 3 \\ 0 & -5 & 6 \\ 0 & -3 & 4\end{array}\right)\). Find an invertible matrix \(P\) such that \(P^{-1} A P\) is a diagonal matrix.
[10M]
2.(d) Find an orthogonal transformation to reduce the quadratic form \(5 x^{2}+2 y^{2}+4 x y\) to a canonical form.
[10M]
3.(a) Show that the equation \(3^{x}+4^{x}=5^{x}\) has exactly one root.
[8M]
3.(b) Test for convergence the integral \(\int_{0}^{\infty} \sqrt{\mathrm{x} \mathrm{e}^{-\mathrm{x}}} \mathrm{dx}\).
[8M]
3.(c) Show that thé area of the surface of the sphere \(x^{2}+y^{2}+z^{2}=a^{2} \quad\) cut off by \(x^{2}+y^{2}=a x\) is \(2(\pi-2) \mathrm{a}^{2}\).
[12M]
3.(d) Show that the function defined by \(f(x, y, z)=3 \log \left(x^{2}+y^{2}+z^{2}\right)-2 x^{2}-2 y^{3}-2 z^{3}\), \((x, y, z) \neq(0,0,0)\)
has only one extreme value, \(\log \left(\dfrac{3}{\mathrm{e}^{2}}\right)\)
[12M]
4.(a) Find the equation of the right circular cylinder of radius \(2\) whose axis is the line \(\dfrac{x-1}{2}=\dfrac{y-2}{1}=\dfrac{z-3}{2}\).$$.
[10M]
4.(b) Find the tangent planes to the ellipsoid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\) which are parallel to the plane \(l x+m y+n z=0\).
[10M]
4.(c) Prove that the semi-latus rectum of any conic is a harmonic mean between the segments of any focal chord.
[8M]
4.(d) Tangent planes at two points \(P\) and \(Q\) of a paraboloid meet in the line RS. Show that the plane through \(\mathrm{RS}\) and middle point of \(\mathrm{PQ}\) is parallel to the axis of the paraboloid.
[12M]
Section B
5.(a) Find the family of curves whose tangents form an angle \(\pi / 4\) with hyperbolas \(x y=c\).
[10M]
5.(b) Solve:
\(\dfrac{d^{2} y}{d x^{2}}-2 \tan x \dfrac{d y}{d x}+5 y=\sec x \cdot e^{x}\)
[10M]
5.(c) The apses of a satellite of the Earth are at distances \(r_{1}\) and \(r_{2}\) from the centre of the Earth. Find the velocities at the apses in terms of \(r_{1}\) and \(\mathrm{r}_{2}\).
[10M]
5.(d) A cable of length 160 meters and weighing \(2 \mathrm{~kg}\) per meter is suspended from two points in the same horizontal plane. The ension at the points of support is \(200 \mathrm{~kg}\). Show that the span of the cable is \(120 \cosh ^{-1}\left(\dfrac{5}{3}\right)\) and also find the sag.
[10M]
5.(e) Evaluate the line integral
\(\oint_{\mathrm{C}}\left(\sin \mathrm{x} \mathrm{d} \mathrm{x}+\mathrm{y}^{2} \mathrm{dy}-\mathrm{dz}\right)\), where \(\mathrm{C}\) is the circle
\(\mathrm{x}^{2}+\mathrm{x}^{2}=16, \mathrm{z}=3\), by using Stokes’ theorem.
[10M]
6.(a) Solve:
\(p^{2}+2\) py \(\cot x=y^{2}\)
where \(p=\dfrac{d y}{d x}\)
[10M]
6.(b) Solve:
\((x^{4} D^{4}+6 x^{3} D^{3}+9 x^{2} D^{2}+3 x D+1) y=(1+\log x)^{2}\),
where \(D \equiv \dfrac{d}{d x}\)
[15M]
6.(c) Solve:
\(\left(\mathrm{D}^{4}+\mathrm{D}^{2}+1\right) \mathrm{y}=\mathrm{ax}^{2}+\mathrm{be}^{-\mathrm{x}} \sin 2 \mathrm{x}\),
where \(D=\dfrac{d}{d x}\)
[15M]
7.(a) One end of a uniform rod \(\mathrm{AB},\) of length \(2 \mathrm{a}\) and weight \(W,\) is attached by a frictionless joint to asmooth wall and the other end \(\mathrm{B}\) is smoothly hinged to an equal rod BC. The middle points of the rods are connected by an elastic cord of natural length a and modulus of elasticity \(4 \mathrm{W}\). Prove that the system can rest in equilibrium in a vertical plane with \(\mathrm{C}\) in contact with the wall below \(A\), and the angle between the rod is \(2 \sin ^{-1}\left(\dfrac{3}{4}\right)\).
[13M]
7.(b) AB is a uniform rod, of length 8 a, which can turn freely about the end \(\mathrm{A},\) which is fixed. \(\mathrm{C}\) is a smooth ring, whose weight is twice that of the rod, which can slide on the rod, and is attachéd by a’string CD to a point \(\mathrm{D}\) in the same horizontal plane as the point A. If AD and CD are each of length a, fix the position of the ring and the tension of the string when the system is in equilibrium.
Show also that the action on the rod at the fixed end A is a horizontal force equal to \(\sqrt{3} \mathrm{W}\), where \(W\) is the weight of the rod.
[14M]
7.(c) A stream is rushing from a boiler through a conical pipe, the diameter of the ends of which are \(\mathrm{D}\) and \(\mathrm{d} ;\) if \(\mathrm{V}\) and \(v\) be the corresponding velocities of the stream and if the motion is supposed to be that of the divergence from the vertex of cone, prove that
\[\dfrac{v}{\mathrm{V}}=\dfrac{\mathrm{D}^{2}}{\mathrm{d}^{2}} \mathrm{e}^{\left(v^{2}-\mathrm{V}^{2}\right) / 2 \mathrm{~K}}\]where \(\mathrm{K}\) is the pressure divided by the density and supposed constant.
[13M]
8.(a) Find the curvature, torsion and the relation between the arc length \(\mathrm{S}\) and parameter \(\mathrm{u}\) for the curve:
\(\overrightarrow{\mathbf{r}}=\overrightarrow{\mathbf{r}}(\mathrm{u})=2 \log _{\mathbf{e}} \mathbf{u} \hat{\mathbf{i}}+4 \mathbf{u} \hat{\mathbf{j}}+\left(2 \mathbf{u}^{2}+1\right) \hat{\mathbf{k}}\)
[10M]
8.(b) Prove the vector identity:
\(\operatorname{curl}(\vec{f} \times \overrightarrow{\mathrm{g}})=\overrightarrow{\mathrm{f}} \operatorname{div} \overrightarrow{\mathrm{g}}-\overrightarrow{\mathrm{g}} \operatorname{div} \overrightarrow{\mathrm{f}}+\) \((\overrightarrow{\mathrm{g}} \cdot \nabla) \overrightarrow{\mathrm{f}}-(\overrightarrow{\mathrm{f}} \cdot \nabla) \overrightarrow{\mathrm{g}}\)
and verify it for the vectors \(\overrightarrow{\mathrm{f}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{z} \hat{\mathrm{j}}+\mathrm{y} \hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{g}}=\mathrm{y} \hat{\mathrm{i}}+\mathrm{z} \hat{\mathrm{k}}\).
[10M]
8.(c) Verify Green’s theorem in the plane for
\[\oint_{C}\left[\left(3 x^{2}-8 y^{2}\right) d x+(4 y-6 x y) d y\right]\]where \(\mathrm{C}\) is the boundary of the region enclosed by the curves \(y=\sqrt{x}\) and \(y=x^{2}\).
[10M]
8.(d) The position vector \(\overrightarrow{\mathbf{r}}\) of a particle of mass 2 units at any time t, referred to fixed origin and axes, is
\[\vec{r}=\left(t^{2}-2 t\right) \hat{i}+\left(\dfrac{1}{2} t^{2}+1\right) \hat{\jmath} \Delta+\dfrac{1}{2} t^{2} \hat{k}\]At time \(t=1\), find its kinetic energy, angular momentum, time rate of change of angular momentum and the moment of the resultant force, acting at the particle, about the origin.
[10M]