Paper I PYQs-2015

1.(a) Find an upper triangular matrix \(A\) such that \(A^{3}=\left[\begin{array}{cc}8 & -57 \\ 0 & 27\end{array}\right]\).

[8M]

1.(b) Let \(G\) be the linear operator on \(\mathbb{R}^{3}\) defined by

Find the matrix representation of \(G\) relative to the basis

[8M]

1.(c) Let \(f(x)\) be a real-valued function defined on the interval (-5,5) such that \(e^{-x} f(x)=2+\int_{0}^{x} \sqrt{t^{4}+1} d t\) for all \(x \in(-5,5) .\) Let \(f^{-1}(x)\) be the inverse function of \(f(x)\). Find \(\left(f^{-1}\right)^{\prime}(2)\).

[8M]

1.(d) For \(x>0,\) let \(f(x)=\int_{1}^{x} \dfrac{\ln t}{1+t} d t .\) Evaluate \(f(e)+f\left(\dfrac{1}{e}\right)\).

[8M]

1.(e) The tangent at \((a \cos \theta, b \sin \theta)\) on the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\) meets the auxiliary circle in two points. The chord joining them subtends a right angle at the centre. Find the eccentricity of the ellipse.

[8M]

2.(a) Suppose \(U\) and \(W\) are distinct four-dimensional subspaces of a vector space \(V\), where \(\operatorname{dim} V=6\). Find the possible dimensions of \(U \cap W\).

[10M]

2.(b) Find the condition on \(a, b\) and \(c\) so that the following system in unknowns \(x, y\) and \(z\) has a solution:

[10M]

2.(c) Consider the three-dimensional region \(R\) bounded by \(x+y+z=1, y=0, z=0\). Evaluate \(\iiint_{R}\left(x^{2}+y^{2}+z^{2}\right) dx dy dz\).

[10M]

2.(d) Find the area enclosed by the curve in which the plane \(z=2\) cuts the ellipsoid

[10M]

3.(a) Find the minimal polynomial of the matrix \(A=\left(\begin{array}{rrr}4 & -2 & 2 \\ 6 & -3 & 4 \\ 3 & -2 & 3\end{array}\right)\).

[10M]

3.(b) If \(\sqrt{x+y}+\sqrt{y-x}=c,\) find \(\dfrac{d^{2} y}{d x^{2}}\).

[10M]

3.(c) A rectangular box, open at the top, is said to have a volume of 32 cubic metres. Find the dimensions of the box so that the total surface is a minimum.

[10M]

3.(d) Find the equation of the plane containing the straight line \(y+z=1, x=0\) and parallel to the straight line \(x-z=1\), \(y=0\).

[10M]

4.(a) Find a \(3 \times 3\) orthogonal matrix whose first two rows are \(\left[\dfrac{1}{3}, \dfrac{2}{3}, \dfrac{2}{3}\right]\) and \(\left[0, \dfrac{1}{\sqrt{2}},-\dfrac{1}{\sqrt{2}}\right]\).

[10M]

4.(b) Find the locus of the variable straight line that always intersects \(x=1\), \(y=0\); \(y=1\), \(z=0\); \(z=1\), \(x=0\).

[10M]

4.(c) Find the locus of the poles of chords which are normal to the parabola \(y^{2}=4 a x\).

[10M]

4.(d) Evaluate \(\lim _{x \rightarrow 0}\left(\dfrac{2+\cos x}{x^{3} \sin x}-\dfrac{3}{x^{4}}\right)\).

[10M]

5.(a) Reduce the differential equation \(x^{2} p^{2}+y p(2 x+y)+y^{2}=0, \quad p=\dfrac{d y}{d x}\) to Clairaut’s form. Hence, find the singular solution of the equation.

[8M]

5.(b) A heavy particle is attached to one end of an elastic string, the other end of which is fixed. The modulus of elasticity of the string is equal to the weight of the particle. The string is drawn vertically down till it is four times its natural length \(a\) and then let go. Find the time taken by the particle to return to the starting point.

[8M]

5.(c) Find the curvature and torsion of the curve \(x=a \cos t, y=a \sin t, z=b t\).

[8M]

5.(d) A cylindrical vessel on a horizontal circular base of radius \(a\) is filled with a liquid of density \(w\) with a height \(h\). If a sphere of radius \(c\) and density greater than \(w\) is suspended by a thread so that it is completely immersed, determine the increase of the whole pressure on the curved surface.

[8M]

5.(e) Solve the differential equation \(x^{2} \dfrac{d^{2} y}{d x^{2}}+3 x \dfrac{d y}{d x}+y=\dfrac{1}{(1-x)^{2}}\).

[8M]

6.(a) Solve \(x \dfrac{d^{2} y}{d x^{2}}-\dfrac{d y}{d x}-4 x^{3} y=8 x^{3} \sin x^{2}\) by changing the independent variable.

[10M]

6.(b) The forces \(P\), \(Q\) and \(R\) act along three straight lines \(y=b\), \(z=-c\), \(z=c\), \(x=-a\) and \(x=a\), \(y=-b\) respectively. Find the condition for these forces to have a single resultant force. Also, determine the equations to its line of action.

[10M]

6.(c) Solve \(\left(D^{4}+D^{2}+1\right) y=e^{-x / 2} \cos \left(\dfrac{x \sqrt{3}}{2}\right),\) where \(D \equiv \dfrac{d}{d x}\).

[10M]

6.(d) Examine if the vector field defined by \(\vec{F}=2 x y z^{3} \hat{i}+x^{2} z^{3} \hat{j}+3 x^{2} y z^{2} \hat{k}\) is irrotational. If so, find the scalar potential \(\phi\) such that \(\vec{F}=\operatorname{grad} \phi\).

[10M]

7.(a) Determine the length of an endless chain which will hang over a circular pulley of radius \(a\) so as to be in contact with two-thirds of the circumference of the pulley.

[15M]

7.(b) Using divergence theorem, evaluate

where \(S\) is the surface of the sphere \(x^{2}+y^{2}+z^{2}=1\)

[15M]

7.(c) A particle of mass \(m\) is falling under the influence of gravity through a medium whose resistance equals \(\mu\) times the velocity. If the particle were released from rest, determine the distance fallen through in time \(t\).

[10M]

8.(a) An ellipse is just immersed in water with its major axis vertical. If the centre of pressure coincides with the focus, determine the eccentricity of the ellipse.

[15M]

8.(b) If \(\vec{F}=y \hat{i}+(x-2 x z) \hat{j}-x y \hat{k},\) evaluate \(\iint_{S}(\nabla \times \vec{F}) \cdot \hat{n} d S,\) where \(S\) is the surface of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) above the \(x y\) -plane.

[10M]

8.(c) A particle moves with a central acceleration which varies inversely as the cube of the distance. If it be projected from an apse at a distance \(a\) from the origin with a velocity which is \(\sqrt{2}\) times the velocity for a circle of radius \(a\), determine the equation to its path.

[15M]