Paper I PYQs-2016

1.(a) Let \(\mathrm{T}: \mathrm{I} \mathrm{B}^{3} \rightarrow \mathrm{B}^{4}\) be given by
\(T(x, y, z)=(2 x-y, 2 x+z,(z)+2 z, x+y+z)\).

Find the matrix of \(T\) with respect to standard basis of \(\mathrm{EB}^{3}\) and \(\mathrm{R}^{4}\) (i.e., \((1,0,0)\), \((0,1,0)\), etc. Examine if \(\mathrm{T}\) is a linear map.

[8M]

1.(b) Show that \(\frac{x}{(1)+x)}<\log (1+x)<x$ for $x>0\).

[8M]

1.(c) Examine if the function \(f(x, y)=\dfrac{x y}{x^{2}+y^{2}},(x, y) \neq(0,0)\) and \(f(0,0)=0\) is continuous at \((0,0)\). Find \(\dfrac{\partial f}{\partial x}\) and \(\dfrac{\partial f}{\partial y}\) at points other than origin.

[8M]

1.(d) If the point (2,3) is the mid-point of a chord of the parabola \(y^{2}=4 x\), then obtain the equation of the chord.

[8M]

1.(e) For the matrix \(A=\left[\begin{array}{rrr}-1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1\end{array}\right],\) obtain the eigen value and get the value of \(A^{4}+3 A^{3}-9 A^{2}\).

[8M]

2.(a) After changing the order of integration of \(\int_{0}^{\infty} \int_{0}^{\infty} e^{-x y} \sin n x d x d y\), show that \(\int_{0}^{\infty} \dfrac{\sin n x}{x} d x=\dfrac{\pi}{2}\).

[10M]

2.(b) A perpendicular is drawn from the centre of ellipse \(Q \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\) to any tangent. Prove that the locus of the foot of the perpendicular is given-by \(\left(x^{2}+y^{2}\right)^{2}=a^{2} x^{2}+b^{2} y^{2}\).

[10M]

2.(c) Using mean value theorem, find a point on the curve \(y=\sqrt{x-2},\) defined on \([2,3],\) where the tangent \((\mathrm{is},\) parallel to the chord joining the end points of the curve.

[10M]

2.(d) Let \(\mathrm{T}\) be a linear map such that \(\mathrm{T}: \mathrm{v}_{3} \rightarrow \mathrm{v}_{2}\) defined by \(\mathrm{T}\left(\mathrm{e}_{1}\right)=2 \mathrm{f}_{1}-\mathrm{f}_{2}\), \(\mathrm{T}\left(\mathrm{e}_{2}\right)=\mathrm{f}_{1}+2 \mathrm{f}_{2} \mathrm{T} \mathrm{T}\left(\mathrm{e}_{3}\right)=0 \mathrm{f}_{1}+0 \mathrm{f}_{2},\) where \(\mathrm{e}_{1}, \mathrm{e}_{2}, \mathrm{e}_{3}\) and \(\mathrm{f}_{1}, \mathrm{f}_{2}\) are standard basis in \(\mathrm{v}_{3}\) and \(\mathrm{V}_{2}\). Find the matrix of \(\mathrm{T}\) relative to these basis. Further take two other basis \(\mathrm{B}_{1}[(1,1,0)(1,0,1)(0,1,1)]\) and \(\mathrm{B}_{2}[(1,1)(1,-1)]\). Obtain the matrix \(\mathrm{T}_{1}\) relative to \(\mathrm{B}_{1}\) and \(\mathrm{B}_{2}\).

[10M]

3.(a) For the matrix \(A=\left[\begin{array}{ccc}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]\), find two non-singular matrices \(P\) and \(Q\) such that \(\mathrm{PAQ}=\mathrm{I}\). Hence find \(A^{-1}\).

[10M]

3.(b) Using Lagrange’s method of multipliers, find the point on the plane \(2 x+3 y+4 z=5\) which is closest to the point \((1,0,0)\).

[10M]

3.(c) Obtain the area between the curve \(x=3(\sec \theta+\cos \theta)\) and its asymptote \(x=3\).

[10M]

3.(d) Obtain the equation of the sphere on which the intersection of the plane \(5 x-2 y+4 z+7=0\) with the sphere which has \((0,1,0)\) and \((3,-5,2)\) as the end points of its diameter is a great circle.

[10M]

4.(a) Examine whether the real quadratic form \(4 x^{2}-y^{2}+2 z^{2}>2 x y-2 y z-4 x z\) is a positive definite or not. Reduce it to its Aiagonal form and determine its signature.

[10M]

4.(b) Show that the integral \(\int_{0}^{\infty} \mathrm{e}^{-x} \mathrm{x}^{\alpha - 1} \mathrm{dx}, \alpha>0\) exists, by separately taking the cases for \(\alpha \geq 1\) and \(0<\alpha<\mathbf{1}\).

[10M]

4.(c) Prove that \(\Gamma (2z)\) = \(\dfrac{2^{2z-1}}{\sqrt{\pi}} \Gamma (z) \Gamma \left(z + \dfrac{1}{2} \right)\)

[10M]

4.(d) A plane \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=a_{2}\) cuts the coordinate plane at \(A\), \(B\), \(C\). Find the equation of the cone with vertex at origin and guiding curve as the circle passing through \(\mathrm{A}\), \(\mathrm{B},\)\mathrm{C}$$.

[10M]

5.(a) Obtain the curve which passes through (1,2) and has a slope \(=\dfrac{-2 x y}{x^{2}+1}\) Dbtain one asymptote to the curve.

[8M]

5.(b) Solve the de to get the particular integral of \(\dfrac{\mathrm{d}^{4} \mathrm{y}}{\mathrm{dx}^{4}}+2 \dfrac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+\mathrm{y}=\mathrm{x}^{2} \cos \mathrm{x}\).

[8M]

5.(c) A weight \(W\) is hanging with the help of two strings of length \(l\) and \(2 l\) in such a way that the other ends A and \(B\) of those strings lie on a horizontal line at a distance \(2 l\). Obtain the tension in the two strings.

[8M]

5.(d) From a point in a smooth horizontal plane, a particle is projected with velocity u at angle \(\alpha\) to the horizontal from the foot of a plane, inclined at an angle \(\beta\) with respect to the horizon. Show that if will strike the plane at right angles, if cot \(\beta=2 \tan (\alpha-\beta)\).

[8M]

5.(e) If \(\mathrm{E}\) be the solid bounded by the xy plane and the paraboloid the volume \(E\) and \(\bar{F}\)=\(\left(z x \sin y z+x^{3}\right) \hat{i}\)+ \(\cos y z \hat{j}\) + \(\left(3 z y^{2}-e^{\lambda^{2}+y^{2}}\right) \hat{k}\).

[8M]

6.(a) A stone is thrown vertically with the velocity which would just carry it to a height of \(40 \mathrm{~m}\). Two seconds later another stone is projected vertically from the same place with the same velocity. When and where will they meet?

[10M]

6.(b) Using the method of variation of parameters, solve

[10M]

6.(c) Water is flowing through a pipe of \(80 \mathrm{~mm}\) diameter under a gauge pressure of \(60 \mathrm{kPa}\), with a mean velocity of 2 \(m/s\). Find the total head, if the pipe is 7 \(m\) above the datum line.

[10M]

6.(d) Evaluate \(\iint_{\mathrm{S}}(\nabla \times \bar{f}), \hat{\mathrm{n}} \mathrm{dS}\) for \(\overline{\mathrm{f}}=(2 \mathrm{x}-\mathrm{y}) \hat{\mathrm{i}}-\mathrm{yz}^{2} \hat{\mathrm{j}}-\mathrm{y}^{2} \mathrm{z} \hat{\mathrm{k}}\) where \(\mathrm{S}\) is the upper half surface of the sphere \(x^{2}+y^{2}+z^{2}=1\) bounded by its projection on the \(xy\) plane.

[10M]

7.(a) State Stokes’ theorem. Verify the Stokes’ theorem for the function \(\overline{\mathrm{f}}=x \hat{i}+z \hat{j}+2 y \hat{k}\), where \(\mathrm{c}\) is the curve obtained by the intersection of the plane \(z=x\) and the cylinder \(x^{2}+y^{2}=1\) and \(S\) is the surface inside the interesected cone.

[15M]

7.(b) A uniform rod of weight \(\mathrm{W}\) is resting against an equally rough horizon and a wall, at an angle \(\alpha\) with the wall. At this condition, a horizontal force \(P\) is stopping them from sliding, implemented at the mid-point of the rod. Prove that \(P=W \tan (\alpha-2 \lambda),\) where \(\lambda\) is the angle of friction. Is there any condition on \(\lambda\) and \(\alpha\)?

[15M]

7.(c) Obtain the singular solution of the differential equation

[10M]

8.(a) A body immersed in a liquid is balanced by a weight \(\mathrm{P}\) to which it is attached by a thread passing over a fixed pulley and when half immersed, is balanced in the same manner by weight \(2 \mathrm{P}\). Prove that the density of the body and the liquid are in the ratio 3:2?

[10M]

8.(b) Solve the differential equation

[10M]

8.(c) Prove that \(\bar{a} \times(\bar{b} \times \bar{c})=(\bar{a} \times \vec{b}) \times \vec{c}\), if and only if either \(\bar{b}=\overline{0}\) or \(\bar{c}\) is collinear with \(\bar{a}\) or \(\bar{b}\) is perpendicular to both \(\bar{a}\) and \(\bar{c}\).

[10M]

8.(d) A particle is acted on a force parallel to the axis of \(y\) whose acceleration is \(\lambda y\), initially projected with a velocity \(a \sqrt{\lambda}\) parallel to \(x\) -axis at the point where \(y=a\). Prove that it will describe a catenary.

[10M]