Paper I PYQs-2014
Section A
1.(a) Show that u1=(1,−1,0),u2=(1,1,0) and u3=(0,1,1) form a basis for R3. Express (5,3,4) in terms of u1,u2 and u3.
[8M]
1.(b) For the matrix A=[100101010]. Prove that An=An−2+A2−I,n≥3.
[8M]
1.(c) Show that the function given by
f(x)={x(e1/x−1)(e1/x+1),x≠00,x=0is continuous but not differentiable at x=0.
[8M]
1.(d) Evaluate ∬ 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=-cis 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=0represents 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]