Paper I PYQs-2015
Section A
1.(a) The vectors \(V_{1}=(1,1,2,4)\), \(V_{2}=(2,-1,-5,2)\), \(V_{3}=(1,-1,-4,0)\) and \(V_{4}=(2,1,1,6)\) are linearly independent. Is it true? Justify your answer.
[10M]
1.(b) Reduce the following matrix to row echelon form and hence find its rank: \(\begin{bmatrix}{1} & {2} & {3} & {4} \\ {2} & {1} & {4} & {5} \\ {1} & {5} & {5} & {7} \\ {8} & {1} & {14} & {17}\end{bmatrix}\).
[10M]
1.(c) Evaluate the following limit
\[\lim _{x \rightarrow a}\left(2-\dfrac{x}{a}\right)^{\tan \left(\dfrac{\pi x}{2 a}\right)}\][10M]
1.(d) Evaluate the following integral:
\[\int_{\pi / 6}^{\pi / 3} \dfrac{\sqrt[3]{\sin x}}{\sqrt[3]{\sin x}+\sqrt[3]{\cos x}} d x\][10M]
1.(e) Find the positive value of \(a\) for which the plane \(a x-2 y+z+12=0\) touches the sphere \(x^{2}+y^{2}+z^{2}-2 x-4 y+2 z-3=0\). Also find the point of contact.
[10M]
2.(a) If matrix \(A= \begin{bmatrix}{1} & {0} & {0} \\ {1} & {0} & {1} \\ {0} & {1} & {0}\end{bmatrix}\) then find \(A^{30}\).
[12M]
2.(b) A conical tent is of given capacity. For the least amount of Canvas required for it, find the ratio of its height to the radius of its base.
[13M]
2.(c) Find the eigen values and eigen vectors of the matrix \(\begin{bmatrix}{1} & {1} & {3} \\ {1} & {5} & {1} \\ {3} & {1} & {1} \end{bmatrix}\).
[12M]
2.(d) If \(6 x=3 y=2 z\) represents one of the mutually perpendicular generators of the cone \(5 y z-8 z x-3 x y=0\), then obtain the equations of the other two generators.
[13M]
3.(a) Let \(V=R^{3}\) and \(T \in A(V)\), for all \(a_{i} \in A(V)\), be defined by \(T\left(a_{1}, a_{2}, a_{3}\right)=\left(2 a_{1}+5 a_{2}+a_{3},-3 a_{1}+a_{2}-a_{3}, a_{1}+2 a_{2}+3 a_{3}\right)\). What is the matrix \(T\) relative to the basis \(V_{1}=(1,0,1)\), \(V_{2}=(-1,2,1)\), \(V_{3}=(3,-1,1)\)?
[12M]
3.(b) Which point of the sphere \(x^{2}+y^{2}+z^{2}=1\) is at the maximum distance from the point (2,1,3).
[13M]
3.(c)(i) Obtain the equation of the plane passing through the points \((2,3,1)\) and \((4,-5,3)\) parallel to \(x- axis\).
[6M]
3.(c)(ii) Verify if the lines: \(\dfrac{x-a+d}{\alpha-\delta}=\dfrac{y-a}{\alpha}=\dfrac{z-a-d}{\alpha+\delta}\) and \(\dfrac{x-b+c}{\beta-\gamma}=\dfrac{y-b}{\beta}=\dfrac{z-b-c}{\beta+\gamma}\) are coplanar. If yes, find the equation of the plane in which they lie.
[7M]
3.(d) Evaluate the integral \(\iint(x-y)^{2} \cos ^{2}(x+y) d x d y\) where \(R\) is the rhombus with successive vertices as \((\pi, 0)\), \((2 \pi, \pi)\), \((\pi, 2 \pi)\), \((0, \pi)\).
[12M]
4.(a) Evaluate \(\iint_{R} \sqrt{\vert y-x^{2} \vert } d x d y\) where \(R=[-1,1 ; 0,2]\).
[13M]
4.(b) Find the dimension of the subspace of \(R^{4}\), spanned by the set \(\{(1,0,0,0),(0,1,0,0),(1,2,0,1),(0,0,0,1)\}\). Hence find its basis.
[12M]
4.(c) Two perpendicular tangent planes to the paraboloid \(x^{2}+y^{2}=2 z\) intersect in a straight line in the plane \(x=0\). Obtain the curve to which this straight line touches.
[13M]
4.(d) For the function
\[f(x, y)=\left\{\begin{array}{cc}{\dfrac{x^{2}-x \sqrt{y}}{x^{2}+y}} & {,(x, y) \neq(0,0)} \\ {0} & {,(x, y)=(0,0)}\end{array}\right.\]Examine the continuity and differentiability.
[12M]
Section B
5.(a) Solve the differential equation: \(x \cos x \dfrac{d y}{d x}+y(x \sin x+\cos x)=1\).
[10M]
5.(b) Solve the differential equation:
\(\left(2 x y^{4} e^{y}+2 x y^{3}+y\right) d x+\left(x^{2} y^{4} e^{y}-x^{2} y^{2}-3 x\right) d y=0\).
[10M]
5.(c) A body moving under SHM has an amplitude \('a'\) and time period \('T'\). If the velocity is trebled, when the distance from mean position is \(\dfrac{2}{3} a\), the period being unaltered, find the new amplitude.
[10M]
5.(d) A rod of 8 kg movable in a vertical plane about a hinge at one end, another end is fastened aweight equal to half of the rod, this end is fastened by a string of length \(l\) to a point at a height \(b\) above the hinge vertically. Obtain the tension in the string.
[10M]
5.(e) Find the angle between the surfaces \(x^{2}+y^{2}+z^{2}-9=0\) and \(z=x^{2}+y^{2}-3\) at \((2,-1,2)\).
[10M]
6.(a) Find the constant \(a\) so that \((x+y)^{a}\) is the integrating factor of \(\left(4 x^{2}+2 x y+6 y\right) d x+\left(2 x^{2}+9 y+3 x\right) d y=0\) and hence solve the differential equation.
[12M]
6.(b) Two equal ladders of weight 4 \(\mathrm{kg}\) each are placed so as to lean at \(A\) against each other with their ends resting on a rough floor, given the coefficient of friction is \(\mu\). The ladders at \(A\) make an angle \(60^{\circ}\) with each other. Find what weight on the top would cause them to slip.
[13M]
6.(c) Find the value of \(\lambda\) and \(\mu\) so that the surfaces \(( \lambda x^2 -\mu yz)\) = \((\lambda+2)x\) and \(4x^2y+z^3=4\) may interesect orthogonally at \((1, -1, 2)\)
[12M]
6.(d) A mass starts from rest at a distance \('a'\) from the centre of force which attracts inversely as the distance. Find the time of arriving at the centre.
[13M]
7.(a)(i) Obtain inverse Laplace transform of \(\left\{\ln \left(1+\dfrac{1}{s^{2}}\right)+\dfrac{s}{s^{2}+25} e^{-5 s}\right\}\).
[6M]
7.(a)(ii) Using Laplace transform, solve \(y^{\prime \prime}+y=t\), \(y(0)=1\), \(y^{\prime}(0)=-2\).
[6M]
7.(b) A particle is projected from the base of a hill whose slope is that of a right circular cone, whose axis is vertical. The projectile gazes the vertex and strikes the hill again at a point on the base. If the semi vertical angle of the cone is \(30^{\circ}\), \(h\) is height, determine the initial velocity on \(u\) of the projection and its angle of projection.
[13M]
7.(c) A vector field is given by \(\vec{F}=\left(x^{2}+x y^{2}\right) \hat{i}+\left(y^{2}+x^{2} y\right) \hat{j}\). Verify that the field is irrotational or not. Find the scalar potential.
[12M]
7.(d) Solve the differential equation \(x=p y-p^{2}\) where \(p=\dfrac{d y}{d x}\).
[13M]
8.(a) Find the length of an endless chain which will hang over a circular pulley of radius \('a'\) so as to be in contact with the two-thirds of the circumference of the pulley.
[12M]
8.(b) A particle moves in a plane under a force, towards a fixed centre, proportional to the distance. If the path of the particle has two apsidal distances \(a\), \(b\) (\(a>b\)), find the equation of the path.
[13M]
8.(c) Evaluate \(\int_{C} e^{-x}(\sin y d x+\cos y d y)\), where \(C\) is the rectangle with vertices \((0,0)\), \((\pi, 0)\), \(\left(\pi, \dfrac{\pi}{2}\right)\), \(\left(0, \dfrac{\pi}{2}\right)\).
[12M]
8.(d) Solve \(x^{4} \dfrac{d^{4} y}{d x^{4}}+6 x^{3} \dfrac{d^{3} y}{d x^{3}}+4 x^{2} \dfrac{d^{2} y}{d x^{2}}-2 x \dfrac{d y}{d x}-4 y=x^{2}+2 \cos \left(\log _{e} x\right)\).
[13M]