Paper I PYQs-2010

1.(a) Show that the set

\[\mathrm{P}[\mathrm{t}]=\left\{\mathrm{at}^{2}+\mathrm{bt}+\mathrm{c} / \mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathbb{R}\right\}\]

forms a vector space over the field \(\mathbb{R}\). Find a basis for this vector space. What is the dimension of this vector space?

[8M]

1.(b) Determine whether the quadratic form

\[\mathrm{q}=\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{xz}+4 \mathrm{yz}+3 \mathrm{z}^{2}\]

is positive definite.

[8M]

1.(c) Prove that. between any two real roots of \(\mathrm{e}^{\mathrm{x}} \sin \mathrm{x}=1\), there is at least one real root of \(e^{x} \cos x+1=0\).

[8M]

1.(d) Let \(f\) be a function defined on \(\mathbb{R}\) such that

\[f(x+y)=f(x)+f(y), \quad x, y \in \mathbb{R}\]

If \(\mathrm{f}\) is differentiable-at one point of \(\mathbb{R}\), then prove that \(f\) is differentiable on \(\mathbb{R}\)

[8M]

1.(e) If a plane cuts the axes in \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and \((\mathrm{a}, \mathrm{b}, \mathrm{c})\) are the coordinates of the centroid of the triangle ABC, then show that the equation of the plane is \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=3\).

[8M]

1.(f) Find the equations of the spheres passing through the circle

\[x^{2}+y^{2}+z^{2}-6 x-2 z+5=0, y=0\]

and touching the plane \(3 y+4 z+5=0\).

[8M]

2.(a) Show that the vectors

\[\alpha_{1}=(1,0,-1), \quad \alpha_{2}=(1,2,1), \quad \alpha_{3}=(0,-3,2)\]

form a basis for \(\mathbf{R}^{3}\). Find the components of \((1,0,0)\) w.r.t. the basis \(\left\{\alpha_{1}, \alpha_{2}, \alpha_{3}\right\}.\)

[10M]

2.(b) Find the characteristic polynomial of \(\left(\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 2 \\ 0 & 1 & 3\end{array}\right)\). Verify Cayley-Hamilton theorem for this matrix and hence find its inverse.

[10M]

2.(c) Let \(\quad A=\left(\begin{array}{rrr}5 & -6 & -6 \\ -1 & .4 & 2 \\ 3 & -6 & -4\end{array}\right)\). Find an invertible matrix \(\mathrm{P}\) such that \(\mathrm{P}^{-1} \mathrm{~A} \mathrm{P}\) is a diagonal matrix.

[10M]

2.(d) Find the rank of the matrix
\(\left(\begin{array}{ccccc} 1 & 2 & 1 & 1 & 2 \\ 2 & 4 & 3 & 4 & 7 \\ -1 & -2 & 2 & 5 & 3 \\ 3 & 6 & 2 & 1 & 3 \\ 4 & 8 & 6 & 8 & 9 \end{array}\right)\)

[10M]

3.(a) Discuss the convergence of the integral \(\int_{0}^{\infty} \dfrac{d x}{1+x^{4} \sin ^{2} x}\)

[10M]

3.(b) Find the extreme value of \(x y z\) if \(x+y+z=a\).

[10M]

3.(c) Let \(\begin{aligned} f(x, y) &=\left\{\begin{array}{cl}\dfrac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}, & \text { if }(x, y) \neq(0,0) \\ 0, & \text { if }(x, y)=(0,0)\end{array}\right.\end{aligned}\)
Show that:
(i) \(f_{x y}(0,0) \neq f_{y x}(0,0)\) (ii) \(\mathrm{f}\) is differentiable at (0,0)

[10M]

3.(d) Evaluate \(\iint_{D}(x+2 y) dA\), where \(D\) is the region bounded by the parabolas \(y=2 x^{2}\) and \(y=1+x^{2}\).

[10M]

4.(a) Prove that the second degree equation

\[x^{2}-2 y^{2}+3 z^{2}+5 y z-6 z x-4 x y+ 8 x-19 y-2 z-20=0\]

represents a cone whose vertex is \((1,-2,3)\).

[10M]

4.(b) If the feet of three normals drawn from a point \(\mathrm{P}\) to the ellipsoid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\) lie in the plane \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1,\) prove that the feet of the other three normals lie in the plane \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}+1=0\).

[10M]

4.(c) If \(\dfrac{x}{1}=\dfrac{y}{2}=\dfrac{z}{3}\) represents one of the three mutually perpendicular generators of the cone \(5 \mathrm{yz}-8 \mathrm{zx}-3 \mathrm{xy}=0\), find the equations of the other two.

[10M]

4.(d) Prove that the locus of the point of intersection of three tangent planes to the ellipsoid \(\dfrac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\dfrac{\mathrm{y}^{2}}{\mathrm{b}^{2}}+\dfrac{\mathrm{z}^{2}}{\mathrm{c}^{2}}=1,\) which are parallel to the conjugate diametral planes of the ellipsoid \(\dfrac{x^{2}}{\alpha^{2}}+\dfrac{y^{2}}{\beta^{2}}+\dfrac{z^{2}}{\gamma^{2}}=1\) is \(\dfrac{x^{2}}{\alpha^{2}}+\dfrac{y^{2}}{\beta^{2}}+\dfrac{z^{2}}{\gamma^{2}}=\dfrac{a^{2}}{\alpha^{2}}+\dfrac{b^{2}}{\beta^{2}}+\dfrac{c^{2}}{\gamma^{2}}\).

[10M]

5.(a) Show that \(\cos (\mathrm{x}+\mathrm{y})\) is an integrating factor of

\[y d x+[y+\tan (x+y)] d y=0\]

Hence solve it.

[8M]

5.(b) Solve
\(\dfrac{d^{2} y}{d x^{2}}-2 \dfrac{d y}{d x}+y=x e^{x} \sin x\)

[8M]

5.(c) A uniform rod AB rests with one end on a smooth vertical wall and the other on a smooth inclined plane, making an angle \(\alpha\) with the horizon. Find the positions of equilibrium and discuss stability.

[8M]

5.(d) A particle is thrown over a triangle from one end of a horizontal base and grazing the vertex falls on che other end of the base. If \(\theta_{1}\) and \(\theta_{2}\) be the base angles and \(\theta\) be the angle of projection, prove that,

\[\tan \theta=\tan \theta_{1}+\tan \theta_{2}\]

[8M]

5.(e) Prove that the horizontal line through the centre of pressure of a rectangle immersed in a liquid with one side in the surface, divides the rectangle in two parts, the fluid pressure on which, are in the ratio, 4:5.

[8M]

5.(f) Find the directional derivation of \(\overrightarrow{\mathrm{V}}^{2}\), where, \(\overrightarrow{\mathrm{V}}=\mathrm{xy}^{2} \overrightarrow{\mathrm{i}}+\mathrm{zy}^{2} \overrightarrow{\mathrm{j}}+\mathrm{xz}^{2} \overrightarrow{\mathrm{k}}\) at the point \((2,0,3)\) in the direction of the outward normal to the surface \(x^{2}+y^{2}+z^{2}=14\) at the point \((3,2,1)\).

[8M]

6.(a) Solve the following differential equation

\[\dfrac{d y}{d x}=\sin ^{2}(x-y+6)\]

[8M]

6.(b) Find the general solution of

\[\dfrac{d^{2} y}{d x^{2}}+2 x \dfrac{d y}{d x}+\left(x^{2}+1\right) y=0\]

[12M]

6.(c) Solve

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

[10M]

6.(d) Solve by the method of variation of parameters the following equation

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

[10M]

7.(a) A aniform chain of length \(2 l\) and weight \(W,\) is suspended from two points \(A\) and \(B\) in the same horizontal line. A load \(P\) is now hung from the middle point \(\mathrm{D}\) of the chain and the depth of this point below \(\mathrm{AB}\) is found to be \(\mathrm{h}\). Show that each terminal tension is,

\[\dfrac{1}{2}\left[\mathbf{P} \cdot \dfrac{l}{\mathbf{h}}+\mathbf{W} \cdot \dfrac{\mathbf{h}^{2}+i^{2}}{2 \mathrm{h} l}\right]\]

[14M]

7.(b) A particle moves with a central acceleration \(\dfrac{\mu}{(\text { distance })^{2}},\) it is projected with velocity \(\mathrm{V}\) at \(\mathrm{a}\) distance \(\mathrm{R}\). Show that its path is a rectangular hyperbola if the angle of projection is,

\[\sin ^{-1}\left[\dfrac{\mu}{\mathrm{VR}\left(\mathrm{V}^{2}-\dfrac{2 \mu}{\mathrm{R}}\right)^{1 / 2}}\right]\]

[13M]

7.(c) A smooth wedge of mass \(M\) is placed on \(a\) smooth horizontal plane and a particle of mass \(\mathrm{m}\) slides down its slant face which is inclined at an angle \(\alpha\) to the horizontal plane, Prove that the acceleration of the wedge is,

\[\dfrac{m g \sin \alpha \cos \alpha}{M+m \sin ^{2} \alpha}\]

[13M]

8.(a)(i) Show that \(\overrightarrow{\mathrm{F}}=\left(2 \mathrm{x} y+\mathrm{z}^{3}\right) \overrightarrow{\mathrm{i}}+\mathrm{x}^{2} \overrightarrow{\mathrm{j}}+3 \mathrm{z}^{2} \mathrm{x} \overrightarrow{\mathrm{k}}\) is a conservative field. Find its scalar potential and also the work done in moving a particle from \((1,-2,1)\) to \((3,1,4)\).

[5M]

8.(a)(ii) Show that, \(\nabla^{2} f(r)=\left(\dfrac{2}{r}\right) f^{\prime}(r)+f^{\prime \prime}(r),\) where

\[r=\sqrt{x^{2}+y^{2}+z^{2}}\]

[5M]

8.(b) Use divergence theorem to evaluate,

\[\iint_{\mathrm{s}}\left(\mathrm{x}^{3} \mathrm{dy} \mathrm{dz}+\mathrm{x}^{2} \mathrm{y} \mathrm{dz} \mathrm{dx}+\mathrm{x}^{2} \mathrm{z} \mathrm{dy} \mathrm{d} \mathrm{x}\right)\]

where \(S\) is the sphere, \(x^{2}+y^{2}+z^{2}=1\).

[10M]

8.(c) If \(\vec{A}=2 y \vec{i}-z \vec{j}-x^{2} \vec{k}\) and \(S\) is the surface of the parabolic cylinder \(y^{2}=8 x\) in the first octant bounded by the planes \(y=4\), \(z=6\), evaluate the surface integral,

\[\iint_{\mathrm{S}} \overrightarrow{\mathrm{A}} \cdot \hat{\mathrm{n}} \overrightarrow{\mathrm{dS}}\]

[10M]

8.(d) Use Green’s theorem in a plane to evaluate the integral, \(\int_{\mathrm{C}}\left[\left(2 \mathrm{x}^{2}-\mathrm{y}^{2}\right) \mathrm{d} \mathrm{x}+\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right) \mathrm{dy}\right],\) where \(\mathrm{C}\) is the boundary of the surface in the xy-plane enclosed by \(y=0\) and the semi-circle, \(y=\sqrt{1-x^{2}}\).

[10M]