Paper I PYQs-2019
Section A
1.(a) Let \(f:[0,\dfrac{\pi}{2} \to R]\) be a continuous function such that
\[f(x)=\dfrac{cos^2 x}{4x^2-\pi^2}, 0\leq x\leq\dfrac{\pi}{2}\]Find the value of \(f\left( \dfrac{\pi}{2} \right)\).
[10M]
1.(b) Let \(f: D(\subseteq R^2)\to R\) be a function and \((a,b)\in D\). If \(f(x,y)\) is continuous at \((a,b)\), then show that the function \(f(x,b)\) and \(f(a,y)\) are continuous at \(x=a\) and at \(y=b\) respectively.
[10M]
1.(c) Let \(T:R^2 \to R^2\) be a linear map such that that \(T(2,1)=(5,7)\) and \(T(1,2)+(3,3)\). If \(A\) is the matrix corresponding to \(T\) with repect to the standard bases \(e_1\), \(e_2\), then find \(Rank(A)\).
[10M]
1.(d) If \(A=\begin{bmatrix} 1& 2& 1\\ 1& -4& 1\\ 3& 0& -3 \end{bmatrix}\) and \(B=\begin{bmatrix} 2& 1& 1\\1& -1& 0\\2& 1& -1 \end{bmatrix}\), then show that \(AB=6I_3\). Use this result to solve the following system of equations:
\(2x+y+z=5\)
\(x-y=0\)
\(2x+y-z=1\)
[10M]
1.(e) Show that the lines \(\dfrac{x+1}{-3}=\dfrac{y-3}{2}=\dfrac{z+2}{1}\) and \(\dfrac{x}{1}=\dfrac{y-7}{-3}=\dfrac{z+7}{2}\) intersect. Find the coordinates of the point of intersection and the equation of the plane containing them.
[10M]
2.(a) Is \(f(x)=\vert \cos x \vert + \vert \sin x \vert\) is differentiable at \(x=\dfrac{\pi}{2}\)? If yes, then find its derivative at \(x=\dfrac{\pi}{2}\). If no, then give a proof of it.
[15M]
2.(b) Let \(A\) and \(B\) be two orthogonal matrices of same order and \(det\) \(A\) + \(det\) \(B\)=0.Show that \(A+B\) is a singular matrix.
[15M]
2.(c)(i) The plane \(x+2y+3z=12\) cuts the axes of coordinates in \(A\), \(B\), \(C\). Find the equations of the circle circumscribing the triangle \(ABC\).
[10M]
2.(c)(ii) Prove that the plane \(z=0\) cuts the enveloping cone of the sphere \(x^2+y^2+z^2=11\) which has the vertex at \((2,4,1)\) in a rectangulat hyperbola.
[10M]
3.(a) Find the maximum and the minimum value of the function \(f(x)=2x^3-9x^2+12x+6\) on the interval \((2,3)\).
[15M]
3.(b) Prove that, in general, three normals can be drawn from a given point to the paraboloid \(x^2+y^2=2az\), but if the point lies on the surface
\[27a(x^2+y^2)+8(a-z)^3=0\]then two of the three normal coincide.
[10M]
3.(c) Let \(A=\begin{bmatrix} 5& 7& 2& 1\\ 1& 1& -8& 1\\ 2& 3& 5& 0\\ 3& 4& -3& 1 \end{bmatrix}\).
(i) Find the rank of matrix \(A\).
[15M]
(ii) Find the dimension of the subspace
\[V= \left\{ (x_1,x_2,x_3,x_4) \in R^4 \vert A\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}=0 \right\}\][5M]
4.(a) State the Cayley-Hamilton theorem. Use this theorem to find \(A^{100}\), where
\[A = \begin{bmatrix} 1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{bmatrix}\][15M]
4.(b) Find the length of the normal chord through a point P of the ellipsoid
\[\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1\]and prove that if it is equal to \(4PG_3\), where \(G_3\) is the point where the normal chord through \(P\) meets the \(xy-plane\), then \(P\) lies on the cone
\[\dfrac{x^6}{a^6}(2c^2-a^2)+\dfrac{y^6}{b^6}(2c^2-b^2)+\dfrac{z^4}{c^4}=0\][15M]
4.(c)(i) If
\[u=sin^{-1} \sqrt{\dfrac{x^{1/3}+y^{1/3}}{x^{1/2}+y^{1/2}}},\]then show that \(\sin^2 u\) is homogeneous function of \(x\) and \(y\) of degree \(-\dfrac{1}{6}\). Hence show that
\(x^2\dfrac{\partial^2 u}{\partial x^2}+2xy\dfrac{\partial^2 u}{\partial x\partial y}+y^2\dfrac{\partial^2 u}{\partial y^2}\)=\(\dfrac{\tan u}{12}(\dfrac{13}{12}+\dfrac{\tan^2 u}{12})\)
[12M]
4.(c)(ii) Using the Jacobian method, show that if \(f'(x)=\dfrac{1}{1+x^2}\) and \(f(0)=0\), then
\[f(x)+f(y)=f\left(\dfrac{x+y}{1-xy}\right)\][8M]
Section B
5.(a) Solve the differential equation
\[(2y \sin x+3y^4 \sin x \cos x)dx-(4y^3 \cos^2x+ \cos x)dy=0\][10M]
5.(b) Determine the complete solution of the differential equation
\[\dfrac{d^2y}{dx^2}-4\dfrac{dy}{dx}+4y=3x^2e^{2x} \sin x\][10M]
5.(c) One end of a heavy uniform rod AB can slide along a rough horizontal rod \(AC\), to which it is attracted by a ring. \(B\) and \(C\) are joined by a string. When the rod is on the point of sliding, then \(AC^2-AB^2=BC^2\). If \(\theta\) is the angle between \(AB\) and the horizontal line, then pove that the coefficient of friction is \(\dfrac{cot\theta}{2+cot^2\theta}\).
[10M]
5.(d) The force of attraction of a particle by the earth in inversely proportional to the square of its distance from the earth’s centre. A particle, whose weight on the surface of the earth is \(W\), falls to the surface of the earth from a height \(3h\) above it. Show that the magnitude of work done by the earth’s attraction force is \(\dfrac{3}{4}hW\), where \(h\) is the radius of the earth.
[10M]
5.(e) Find the directional derivative of the function \(xy^2+yz^2+zx^2\) along the tangent to the curve \(x=t\), \(y=t^2\), \(z=t^2\) at the point \((1,1,1)\).
[10M]
6.(a) A body consists of a cone and underlying hemisphere. The base of the cone and the top of the hemisphere have same radius \(a\). The whole body rests on a rough horizontal table with hemisphere in contact with the table. Show that the greatest height of the cone, so that the equilibrium may be stable, is \(\sqrt{3}a\).
[15M]
6.(b) Find the circulation of \(\vec{F}\) round the curve \(C\), where \(\vec{F}=(2x+y^2)\hat{i}+(3y-4x)\hat{j}\) and \(C\) is the curve \(y=x^2\) from \((0,0)\) to \((1,1)\) and the curve \(y^2=x\) from \((1,1)\) to \((0,0)\).
[15M]
6.(c)(i) Solve the differential equation
\[\dfrac{d^2y}{dx^2}+(3sin x-cot x)\dfrac{dy}{dx}+2ysin^2x=e^{-cos x}sin^2x\][10M]
6.(c)(ii) Find the Laplace transforms of \(t^{-1/2}\) and \(t^{-1/2}\). Prove that the Laplace transform of \(t^{n+1/2}\), where \(n\in N\), is
\[\dfrac{\Gamma{(n+1+\dfrac{1}{2})}}{s^{n+1+\dfrac{1}{2}}}\][10M]
7.(a) Find the linearly independent solutions of the corresponding homogeneous differential equation of the equation \(x^2y''-2xy'+2y=x^3 \sin x\) and then find the general solution of the given equation by the method of variation of parameters.
[15M]
7.(b) Find the radius of curvature and radius of torsion of the helix \(x=a \cos u\), \(y=a \sin u\), \(z=au \tan \alpha\).
[15M]
7.(c) A particle moving along the \(y-axis\) has an acceleration \(F_y\) towards the origin, where \(F\) is a psoitive and even function of \(y\). The periodic time, when the particle vibrates between \(y=-a\) and \(y=a\), is \(T\). Show that
\[\dfrac{2\pi}{\sqrt{F_1}}<T<\dfrac{2\pi}{\sqrt{F_2}}\]where \(F_1\) and \(F_2\) are the greatest and the least values of \(F\) within the range \([-a,a]\). Further, show that when a simple pendulum of length \(l\) oscillates through \(30^{\circ}\) on either side of the vertical line, \(T\) lies between \(2\pi\sqrt{l/g}\) and \(2\pi\sqrt{l/g}\sqrt{\pi/3}\).
[20M]
8.(a) Obtain the singular solution of the differential equation
\[(\dfrac{dy}{dx})^2(\dfrac{y}{x})^2cot^2\alpha-2(\dfrac{dy}{dx})(\dfrac{y}{x})+(\dfrac{dy}{dx})^2cosec^2\alpha=1\]Also find the complete primitive of the given differential equation. Give the geometrical interpretations of the complete primitive and singular solution.
[15M]
8.(b) Prove that the path of a planet, which is moving so that its acceleration is always directed to a fixed point (star) and is equal to \(\dfrac{\mu}{(distance)^2}\) is a conic section. Find the conditions under which the path becomes
(i) ellipse,
(ii) parabola and
(iii) hyperbola
[15M]
8.(c)(i) State Gauss divergence theorem. Verify this theorem for \(\vec{F}=4x\hat{i}-2y^2\hat{j}+z^2\hat{k}\), taken over the region bounded by \(x^2+y^2=4\), \(z=0\) and \(z=3\).
[15M]
8.(c)(ii) Evalute by Stokes’ theorem \(\oint_C e^xdx+2ydy-dz\), where \(C\) is the curve \(x^2+y^2=4\), \(z=2\).
[5M]