Paper I PYQs-2010

1.(a) Show that the set

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

is positive definite.

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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 $e^x\cos x+1=0$.

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

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

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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 ${\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}}+{\displaystyle \frac{z}{c}}=3$.

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

and touching the plane $3y+4z+5=0$.

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2.(a) Show that the vectors

form a basis for ${\mathbf{R}}^3$. Find the components of $(1,0,0)$ w.r.t. the basis $\{{\alpha }_1,{\alpha }_2,{\alpha }_3\}.$

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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.

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

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

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3.(a) Discuss the convergence of the integral ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{dx}{1+x^4{\sin }^{2}x}}$

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

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3.(c) Let $\begin{array}{rl}f(x,y)& =\{\begin{array}{cl}{\displaystyle \frac{xy(x^2-y^2)}{x^2+y^2}},& \text{ if }(x,y)\ne (0,0)\\ 0,& \text{ if }(x,y)=(0,0)\end{array}\end{array}$
Show that:
(i) $f_{xy}(0,0)\ne f_{yx}(0,0)$ (ii) $\mathrm{f}$ is differentiable at (0,0)

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

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4.(a) Prove that the second degree equation

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

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

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

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

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

Hence solve it.

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5.(b) Solve
${\displaystyle \frac{d^{2}y}{d{x}^2}}-2{\displaystyle \frac{dy}{dx}}+y=x{e}^x\sin x$

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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.

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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,

[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.

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5.(f) Find the directional derivation of ${\overrightarrow{\mathrm{V}}}^2$, where, $\overrightarrow{\mathrm{V}}={\mathrm{x}\mathrm{y}}^2\overrightarrow{\mathrm{i}}+{\mathrm{z}\mathrm{y}}^2\overrightarrow{\mathrm{j}}+{\mathrm{x}\mathrm{z}}^2\overrightarrow{\mathrm{k}}$ 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 $(3,2,1)$.

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6.(a) Solve the following differential equation

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6.(b) Find the general solution of

[12M]

6.(c) Solve

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

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7.(a) A aniform chain of length $2l$ 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 $\mathrm{A}\mathrm{B}$ is found to be $\mathrm{h}$. Show that each terminal tension is,

[14M]

7.(b) A particle moves with a central acceleration ${\displaystyle \frac{\mu }{(\text{ distance }{)}^2}},$ 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,

[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,

[13M]

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

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

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8.(b) Use divergence theorem to evaluate,

where $S$ is the sphere, $x^2+y^2+z^2=1$.

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

[10M]

8.(d) Use Green’s theorem in a plane to evaluate the integral, ${\int }_{\mathrm{C}}[(2{\mathrm{x}}^2-{\mathrm{y}}^2)\mathrm{d}\mathrm{x}+({\mathrm{x}}^2+{\mathrm{y}}^2)\mathrm{d}\mathrm{y}],$ 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]