Paper I PYQs-2017

1.(a) Let \(A= \begin{bmatrix}{2} & {2} \\ {1} & {3}\end{bmatrix}\). Find a non-singular matrix \(P\) such that \(P^{-1} A P\) is diagonal matrix.

[10M]

1.(b) Show that similar matrices have the same characteristic polynomial.

[10M]

1.(c) Integrate the function \(f(x, y)=x y\left(x^{2}+y^{2}\right)\) over the domain \(R:\left\{-3 \leq x^{2}-y^{2} \leq 3,1 \leq x y \leq 4\right\}\)

[10M]

1.(d) Find the equation of the tangent at the point (1,1,1) to the conicoid \(3 x^{2}-y^{2}=2 z\).

[10M]

1.(e) Reduce the following equation to the standard form and hence determine the conicoid:

\[x^{2}+y^{2}+z^{2}-y z-z x-x y-3 x-6 y-9 z+21=0\]

[15M]

2.(a) Find the volume of the solid above the \(xy-plane\) and directly below the portion of the elliptic paraboloid \(x^{2}+\dfrac{y^{2}}{4}=z\) which is cut off by the plane \(z=9\).

[15M]

2.(b) A plane passes through a fixed point \((a, b, c)\) and cuts the axes at the points \(A\), \(B\), \(C\) respectively. Find the locus of the center of the sphere which passes through the origin \(O\) and \(A\), \(B\), \(C\).

[15M]

2.(c) Show that the plane \(2 x-2 y+z+12=0\) touches the sphere \(x^{2}+y^{2}+z^{2}-2 x-4 y+2 z-3=0\). Find the point of contact.

[10M]

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

[10M]

3.(a) Consider the matrix mapping \(A: R^{4} \rightarrow R^{3}\), where \(A= \begin{bmatrix}{1} & {2} & {3} & {1} \\ {1} & {3} & {5} & {-2} \\ {3} & {8} & {13} & {-3}\end{bmatrix}\). Find a basis and dimension of the image of \(A\) and those of the kernel \(A\).

[15M]

3.(b) Prove that distance non-zero eigenvectors of a matrix are linearly independent.

[10M]

3.(c) If \(f(x, y)=\left\{\begin{array}{ll}{\dfrac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}},(x, y) \neq(0,0)} \\ {0, (x, y)=(0,0)}\end{array}\right.\)
Calculate \(\dfrac{\partial^{2} f}{\partial x \partial y}\) and \(\dfrac{\partial^{2} f}{\partial y \partial x}\) at (0,0).

[15M]

3.(d) Find the locus of the points of intersection of three mutually perpendicular tangent planes to \(a x^{2}+b y^{2}+c z^{2}=1\).

[10M]

4.(a) Reduce the following equation to the standard form and hence determine the nature of the conicoid: \(x^2 + y^2 + z^2 - yz-zx-xy-3x-6y-9z+21=0\).

[15M]

4.(b) Consider the following system of equation in \(x\), \(y\), \(z\):
\(\quad x+2 y+2 z=1\)
\(\quad x+a y+3 z=3\)
\(\quad x+11 y+a z=b\)

i) For which values of \(a\) does the system have a unique solution?
ii) For which of values \((a, b)\) does the system have more than one solution?

[15M]

4.(c) Examine if the improper integral \(\int_{0}^{3} \dfrac{2 x d x}{\left(1-x^{2}\right)^{2 / 3}}\) exists.

[10M]

4.(d) Prove that \(\dfrac{\pi}{3} \leq \iint \dfrac{d x d y}{\sqrt{x^{2}+(y-2)^{2}}} \leq \pi\), where \(D\) is the unit disc.

[10M]

5.(a) Find the differential equation representing the entire circle in the \(xy-plane\).

[10M]

5.(b) Suppose that the streamlines of the fluid are given by a family of curves \(xy=c\). Find the equipotential lines, that is, the orthogonal trajectories of the family of curves representing the streamlines.

[10M]

5.(c) A fixed wire is in the shape of the cardiod \(r=a(1+\cos \theta)\), the initial line being the downward vertical. A small ring of mass \(m\) can slide on the wire and is attached to point \(r=0\) of the cardiod by an elastic string of natural length \(a\) and modulus of elasticity 4 \(\mathrm{mg}\). The string is released from rest when the string is horizontal. Show by laws of conservation of energy that \(a \theta^{2}(1+\cos \theta)-g \cos \theta(1-\cos \theta)=0\), \(g\) being the acceleration due to gravity.

[10M]

5.(d) For what values of the constant \(a\), \(b\) and \(c\) the vector \(\overline{V}=(x+y+a z) \hat{i}+(b x+2 y-z) j+(-x+c y+2 z) k\) is irrotational. Find the divergence in cylindrical coordinates of the vector with these values.

[10M]

5.(e) The position vector of a moving point at time \(t\) is \(\overline{r}=\sin t \hat{i}+\cos 2 t j+\left(t^{2}+2 t\right) k\). Find the components of acceleration \(\overline{a}\) in the direction parallel to the velocity vector \(\overline{v}\) and perpendicular to the plane of \(\overline{r}\) and \(\overline{v}\) at time \(t=0\).

[10M]

6.(a)(i) Solve the following simultaneous liner differential equations:
\((D+1) y=z+e^{x}\) and \((D+1) z=y+e^{x}\)
where \(y\) and \(z\) are functions of independent variable \(x\) and \(D \equiv \dfrac{d}{d x}\).

[8M]

6.(a)(ii) If the growth rate of the population of bacteria at time \(t\) is proportional to the amount present at the time \(t\) and population doubles in one week, then how much bacteria can be expected after 4 weeks?

[8M]

6.(b)(i) Consider the differential equation \(x y p^{2}-\left(x^{2}+y^{2}-1\right) p+x y=0\) where \(p=\dfrac{d y}{d x}\) substituting \(u=x^{2}\) and \(v=y^{2}\) reduce the equation to Clairaut’s form in terms of \(u\), \(v\) and \(p^{\prime}=\dfrac{d v}{d u}\) hence or otherwise solve the equation.

[10M]

6.(b)(ii) Solve the following initial value problem using Laplace transform: \(\dfrac{d^{2} y}{d x^{2}}+9 y=r(x)\), \(y(0)=0\), \(y^{\prime}(0)=4\) where \(r(x)=\left\{\begin{array}{ll}{8 \sin x} & {\text { if } 0< x < \pi} \\ {0} & {\text { if } x \geq \pi}\end{array}\right.\).

[17M]

6.(c) A uniform solid hemisphere rests on a rough plane inclined to the horizon at an angle \(\phi\) with its curved surface touching the plane. Find the greatest admissible value of the inclination \(\phi\) for equilibrium. If \(\phi\) be less than this value, is the equilibrium stable?

[17M]

7.(a) Find the curvature vector and its magnitude at any point \(\overline{r}=(\theta)\) of the curve \(\overline{r}=(a \cos \theta, a \sin \theta, a \theta)\). Show that the locus of the feet of the perpendicular from the origin to the tangent is a curve that completely lies on the hyperboloid \(x^{2}+y^{2}-z^{2}=a^{2}\).

[16M]

7.(b)(i) Solve the differential equation: \(x \dfrac{d^{2} y}{d x^{2}}-\dfrac{d y}{d x}-4 x^{3} y=8 x^{3} \sin \left(x^{2}\right)\)

[8M]

7.(b)(ii) Solve the following initial value differential equations: \(20 y^{\prime \prime}+4 y^{\prime}+y=0\), \(y(0)=3.2\), \(y^{\prime}(0)=0\).

[7M]

7.(c) A particle is free to move on a smooth vertical circular wire of radius \(a\). At time \(t=0\) it is projected along the circle from its lowest point \(A\) with velocity just sufficient to carry it to the highest point \(B\). Find the time \(T\) at which the reaction between the particle and the wire is zero.

[17M]

8.(a) A spherical shot of \(W\) gm weight and radius \(r\) cm, lies at the bottom of cylindrical bucket of radius \(R \mathrm{cm}\). The bucket is filled with water up to a depth of \(\mathrm{h} \mathrm{cm}(\mathrm{h}>2 \mathrm{r})\). Show that the minimum amount of work done in lifting the shot just clear of the water must be \(\left[W\left(h-\dfrac{4 r^{3}}{3 R^{2}}\right)\right]+W^{\prime}\left(r-h+\dfrac{2 r^{3}}{3 R^{2}}\right) \mathrm{cm} \mathrm{gm}\). \(\mathrm{W}^{\prime} \mathrm{gm}\) is the weight of water displaced by the shot.

[17M]

8.(b) Solve the following initial value problem using Laplace transform:

\[\dfrac{d^{2} y}{d x^{2}}+9 y=r(x), y(0)=0, y^{\prime}(0)=4\]

where

\[r(x)=\left\{\begin{array}{ll}{8 \sin x} & {\text { if } 0< x < \pi} \\ {0} & {\text { if } x \geq \pi}\end{array}\right.\]

[17M]

8.(c)(i) Evaluate the integral \(\iint_{S} F nds\), where \(\overline{F}=3 x y^{2} \hat{i}+\left(y x^{2}-y^{3}\right) j+3 z x^{2} K\) and \(S\) is a surface of the cylinder \(y^{2}+z^{2} \leq 4,-3 \leq x \leq 3\) using divergence theorem.

[9M]

8.(c)(ii) Using Green theorem, evaluate the \(\int_{C} F(\vec{r}) . d \vec{r}\) counterclockwise where \(F(r)=\left(x^{2}+y^{2}\right) \hat{i}+\left(x^{2}-y^{2}\right) j\) and \(d \vec{r}=x \hat{i}+d y j\) and the curve \(C\) is the boundary off the region \(R=\left\{(x, y) \vert 1 \leq y \leq 2-x^{2}\right\}\).

[8M]