Paper I PYQs-2014
Section A
1.(a) Show that \(\mathrm{u}_{1}=(1,-1,0), \mathrm{u}_{2}=(1,1,0)\) and \(\mathrm{u}_{3}=(0,1,1)\) form a basis for \(\mathbb{R}^{3} .\) Express (5,3,4) in terms of \(\mathrm{u}_{1}, \mathrm{u}_{2}\) and \(\mathrm{u}_{3}\).
[8M]
1.(b) For the matrix \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right] .\) Prove that \(A^{n}=A^{n-2}+A^{2}-I, n \geq 3\).
[8M]
1.(c) Show that the function given by
\[f(x)=\left\{\begin{array}{cl} \dfrac{x\left(e^{1 / x}-1\right)}{\left(e^{1 / x}+1\right)}, & x \neq 0 \\ 0 & , x=0 \end{array}\right.\]is continuous but not differentiable at \(x=0\).
[8M]
1.(d) Evaluate \(\iint_{\mathbf{R}} \mathrm{y} \dfrac{\sin \mathrm{x}}{\mathrm{x}} \mathrm{dx} \mathrm{dy}\) over \(\mathrm{R}\) where \(\mathrm{R}=\{(\mathrm{x}, \mathrm{y}): \mathrm{y} \leq \mathrm{x} \leq \pi / 2, \quad 0 \leq \mathrm{y} \leq \pi / 2\}\).
[8M]
1.(e) Prove that the locus of a variable line which intersects the three lines:
\[y=m x, z=c ; \quad y=-m x, \quad z=-c ; \quad y=z, m x=-c\]is the surface \(y^{2}-m^{2} x^{2}=z^{2}-c^{2}\).
[8M]
2.(a) Let \(B=\left[\begin{array}{rr}1 & -1 \\ 2 & -1\end{array}\right]\). Find all eigen values and corresponding eigen vectors of \(\mathrm{B}\) viewed as a matrix over: (i) the real field \(R\) (ii) the complex field \(C\).
[10M]
2.(b) If \(\mathrm{xyz}=\mathrm{a}^{3}\) then show that the minimum value of \(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}\) is \(3 \mathrm{a}^{2}\).
[10M]
2.(c) Prove that every sphere passing through the circle \(x^{2}+y^{2}+2 a x+r^{2}=0\), \(z=0\) cut orthogonally every sphere through the circle \(x^{2}+z^{2}=r^{2}\), \(y=0\).
[10M]
2.(d) Show that the mapping \(\mathrm{T}: \mathrm{V}_{2}(\overline{\mathrm{R}}) \rightarrow \mathrm{V}_{3}(\overline{\mathrm{R}})\) defined as \(\mathrm{T}(\mathrm{a}, \mathrm{b})=(\mathrm{a}+\mathrm{b}, \mathrm{a}-\mathrm{b}, \mathrm{b})\) is a linear transformation. Find the range, rank and nullity of \(\overline{T}\).
[10M]
3.(a) Examine whether the matrix \(A=\left[\begin{array}{rrr}-2 & 2 & -3 \\ 2 & 1 & 6 \\ -1 & -2 & 0\end{array}\right]\) is diagonalizable. Find all eigen values. Then obtain a matrix \(P\) such that \(P^{-1}AP\) is a digonal matrix.
[10M]
3.(b) A moving plane passes throughla fixed point \((2,2,2)\) and meets the coordinate axes at the points \(\mathrm{A}\), \(\mathrm{B}\), \(\mathrm{C}\), all away from the origin \(\mathrm{O}\). Find the locus of the centre of the sphere passing through the points \(\mathrm{O}\), \(\mathrm{A},\)\mathrm{B}, \(\mathrm{C}\)
[10M]
3.(c) Evaluate the integral \(\) I=\int_{0}^{\infty} 2^{-a x^{2}} d x \(\) using Gamma function.
[10M]
3.(d) Prove that the equation:
\[4 x^{2}-y^{2}+z^{2}-3 y z+2 x y+12 x-11 y+6 z+4=0\]represents a cone with vertex at \((-1,-2,-3)\).
[10M]
4.(a) Let \(f\) be a real valued function defined on [0,1] as follows:
\[f(x)=\left\{\begin{array}{cc} \dfrac{1}{a^{r-1}}, & \dfrac{1}{a^{r}}<x \leq \dfrac{1}{a^{r-1}}, r=1,2,3 \ldots \ldots \\ 0 & x=0 \end{array}\right.\]where \(a\) is an integer greater than \(2\). Show that \(\int_{0}^{1} f(x) d x\) exists and is equal to \(\dfrac{a}{a+1}\).
[10M]
4.(b) Prove that the plane \(a x+b y+c z=0\) cuts the cone \(y z+z x+x y=0\) in perpendicular lines if \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=0\).
[10M]
4.(c) Evaluate the integral \(\iint_{R} \dfrac{y}{\sqrt{x^{2}+y^{2}+1}} d x d y\) over the region \(R\) bounded between \(0 \leq x \leq \dfrac{y^{2}}{2}\) and \(0 \leq y \leq 2\).
[10M]
4.(d) Consider the linear mapping \(\mathrm{F}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) given as \(\mathrm{F}(\mathrm{x}, \mathrm{y})=(3 \mathrm{x}+4 \mathrm{y}, 2 \mathrm{x}-5 \mathrm{y})\) with usual basis.
Find the matrix associated with the linear transformation relative to the basis \(\mathrm{S}=\left\{\mathrm{u}_{1}, \mathrm{u}_{2}\right\}\) where \(u_{1}=(1,2), u_{2}=(2,3)\).
[10M]
Section B
5.(a) Solve the differential equation : \(y=2 p x+p^{2} y, p=\dfrac{d y}{d x}\)
and obtain the non-singular solution.
[8M]
5.(b) Solve :
\[\dfrac{d^{4} y}{d x^{4}}-16 y=x^{4}+\sin x\][8M]
5.(c) A particle whose mass is \(m\), is acted upon by a force \(m \mu\left(x+\dfrac{a^{4}}{x^{3}}\right)\) towards the origin. If it starts from rest at a distance ‘a’ from the origin, prove that it will arrive at the origin in time \(\dfrac{\pi}{4 \sqrt{\mu}}\).
[8M]
5.(d) A hollow weightless hemisphere filled with liquid is suspended from a point on the rim of its base. Show that the ratio of the thrust on the plane base to the weight of the contained liquid is \(12: \sqrt{73}\).
[8M]
5.(e) For three vectors show that:
\[\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+\overline{\mathrm{b}} \times(\overline{\mathrm{c}} \times \overline{\mathrm{a}})+\overline{\mathrm{c}} \times(\overline{\mathrm{a}} \times \overline{\mathrm{b}})=0\][8M]
6.(a) Solve the following differential equation:
\[\dfrac{d y}{d x}=\dfrac{2 y}{x}+\dfrac{x^{3}}{y}+x \tan \dfrac{y}{x^{2}}\][10M]
6.(b) An engine, working at a constant rate \(\mathrm{H},\) draws a load \(\mathrm{M}\) against a resistance \(\mathrm{R} .\) Show that the maximum speed is \(\mathrm{H} / \mathrm{R}\) and the time taken to attain half of this speed is \(\dfrac{\mathrm{MH}}{\mathrm{R}^{2}}\left(\log 2-\dfrac{1}{2}\right)\)
[10M]
6.(c) Solve by the method of variation of parameters:
\[y^{\prime \prime}+3 y^{\prime}+2 y=x+\cos x\][10M]
6.(d) For the vector \(\bar{A}=\dfrac{x \hat{i}+y \hat{j}+2 \hat{k}}{x^{2}+y^{2}+z^{2}}\) examine if \(\bar{A}\) is an irrotational vector. Then determine \(\phi\) such that \(\overline{\mathrm{A}}=\nabla \phi\).
[10M]
7.(a) A solid consisting of a cone and a hemisphere on the same base rests on a rough horizontal table with the hemisphere in contact with the table. Shoy that the largest height of the cone so that the equilibrium is stable is \(\sqrt{3} \times\) radius of hemisphere.
[15M]
7.(b) Evaluate \(\iint_{\mathrm{S}} \nabla \times \overline{\mathrm{A}} \cdot \overline{\mathrm{n}} \mathrm{dS}\) for \(\overline{\mathrm{A}}=\left(\mathrm{x}^{2}+\mathrm{y}-4\right) \hat{\mathrm{i}}+3 \mathrm{x} \mathrm{y} \hat{\mathrm{j}}+\left(2 \mathrm{x} \mathrm{z}+\mathrm{z}^{2}\right) \hat{\mathrm{k}}\) and \(\mathrm{S}\) is the surface of hemisphere \(x^{2}+y^{2}+z^{2}=16\) above \(xy\) plane.
[15M]
7.(c) Solve the D.E.:
\[\dfrac{d^{3} y}{d x^{3}}-3 \dfrac{d^{2} y}{d x^{2}}+4 \dfrac{d y}{d x}-2 y=e^{x}+\cos x\][10M]
8.(a) A semi circular disc rests in a vertical plane with its curved edge on a rough horizontal and equally rough yertical plane. If the coeff. of friction is \(\mu,\) prove that the greatest angle that the bounding diameter can make with the horizontal plane is:
\[\sin ^{-1}\left(\dfrac{3 \pi}{4} \dfrac{\mu+\mu^{2}}{1+\mu^{2}}\right)\][10M]
8.(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 \(\rho_{1}, \rho_{2}, \rho_{3}\) respectively. Prove that:
\[\dfrac{\rho_{2}-\rho_{3}}{\mathrm{V}_{1}}+\dfrac{\rho_{3}-\rho_{1}}{\mathrm{V}_{2}}+\dfrac{\rho_{1}-\rho_{2}}{\mathrm{V}_{3}}=0\][15M]
8.(c) Verify the divergence theorem for \(\bar{A}=4 x \hat{i}-2 y^{2} \hat{j}+z^{2} \hat{k}\) over the region \(x^{2}+y^{2}=4\), \(z=0\), \(z=3\).
[10M]