Mathematics UPSC PYQ 2017

Consider the following system of equations in x, y, z: x + 2y + 2z = 1, x + ay + 3z = 3, x + 11y + az = b. (i) For which values of a does the system have a unique solution? (ii) For which pair of values (a, b) does the system have more than one solution?

A stream is rushing from a boiler through a conical pipe, the diameters of the ends of which are D and d. If V and v be the corresponding velocities of the stream and if the motion is assumed to be steady and diverging from the vertex of the cone, then prove that \frac{V}{v} = \frac{D^2}{d^2} e^{(v^2 - V^2)/2K}, where K is the pressure divided by the density and is constant.

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 centre of the sphere which passes through the origin O and A, B, C.

Write an algorithm in the form of a flow chart for Newton-Raphson method. Describe the cases of failure of this method.

Find the differential equation representing all the circles in the x-y plane.

Using graphical method, find the maximum value of 2x + y subject to 4x + 3y \le 12, 4x + y \le 8, 4x - y \le 8, x, y \ge 0.

Solve the following linear programming problem by simplex method : Maximize z = 3x_1 + 5x_2 + 4x_3 subject to 2x_1 + 3x_2 \le 8, 2x_2 + 5x_3 \le 10, 3x_1 + 2x_2 + 4x_3 \le 15, x_1, x_2, x_3 \ge 0.

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

Explain the main steps of the Gauss-Jordan method and apply this method to find the inverse of the matrix \begin{pmatrix} 2 & 6 & 6 \\ 2 & 8 & 6 \\ 2 & 6 & 8 \end{pmatrix}.

For what values of the constants a, b and c the vector vec{V} = (x + y + az)hat{i} + (bx + 2y - z)hat{j} + (-x + cy + 2z)hat{k} is irrotational. Find the divergence in cylindrical coordinates of this vector with these values.

Examine if the improper integral int_0^3 (2xdx)/(1 - x^2)^(2/3) exists.

Show that the groups \mathbb{Z}_5 \times \mathbb{Z}_7 and \mathbb{Z}_{35} are isomorphic.

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

Solve (D^2 - 2DD' + D'^2)z = e^{x+2y} + x^3 + \sin 2x, where D = \frac{\partial}{\partial x}, D' = \frac{\partial}{\partial y}, D^2 = \frac{\partial^2}{\partial x^2}, D'^2 = \frac{\partial^2}{\partial y^2}.

Solve the following simultaneous linear 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 = d/dx.

Let G be a group of order n. Show that G is isomorphic to a subgroup of the permutation group S_n.

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.

Let \sum_{n=1}^{\infty} x_n be a conditionally convergent series of real numbers. Show that there is a rearrangement \sum_{n=1}^{\infty} x_{\pi(n)} of the series \sum_{n=1}^{\infty} x_n that converges to 100.

Two uniform rods AB, AC, each of mass m and length 2a, are smoothly hinged together at A and move on a horizontal plane. At time t, the mass centre of the rods is at the point (\xi, \eta) referred to fixed perpendicular axes Ox, Oy in the plane, and the rods make angles \theta \pm \phi with Ox. Prove that the kinetic energy of the system is m \left[ \dot{\xi}^2 + \dot{\eta}^2 + \left(\frac{1}{3} + \sin^2 \phi\right) a^2 \dot{\theta}^2 + \left(\frac{1}{3} + \cos^2 \phi\right) a^2 \dot{\phi}^2 \right]. Also derive Lagrange's equations of motion for the system if an external force with components [X, Y] along the axes acts at A.

(ii) Solve the following initial value differential equations: 20y'' + 4y' + y = 0, y(0) = 3.2 and y'(0) = 0.

Integrate the function f(x, y) = xy(x^2 + y^2) over the domain R: {-3 <= x^2 - y^2 <= 3, 1 <= xy <= 4}.

Prove that (pi)/3 <= iint_D (dxdy)/(sqrt(x^2 + (y - 2)^2)) <= pi where D is the unit disc.

(i) Solve the differential equation: x (d^2 y)/(dx^2) - (dy)/(dx) - 4x^3 y = 8x^3 sin(x^2).

Let x_1 = 2 and x_{n+1} = \sqrt{x_n} + 20, n = 1, 2, 3, .... Show that the sequence x_1, x_2, x_3, ... is convergent.

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

Suppose that the streamlines of the fluid flow 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.

Solve the following initial value problem using Laplace transform: (d^2 y)/(dx^2) + 9y = r(x), y(0) = 0, y'(0) = 4 where r(x) = {8 sin x if 0 < x < pi, 0 if x >= pi}

Determine all entire functions f(z) such that 0 is a removable singularity of f\left(\frac{1}{z}\right).

(i) Consider the differential equation xy p^2 - (x^2 + y^2 - 1)p + xy = 0 where p = dy/dx. Substituting u = x^2 and v = y^2 reduce the equation to Clairaut's form in terms of u, v and p' = dv/du. Hence, or otherwise solve the equation.

Let f = u + iv be an analytic function on the unit disc D = \{z \in \mathbb{C} : |z| < 1\}. Show that \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} at all points of D.

Show that the moment of inertia of an elliptic area of mass M and semi-axis a and b about a semi-diameter of length r is \frac{1}{4} M \frac{a^2 b^2}{r^2}. Further, prove that the moment of inertia about a tangent is \frac{5M}{4} p^2, where p is the perpendicular distance from the centre of the ellipse to the tangent.

(ii) Using Green's theorem, evaluate the int_C vec{F} dot dvec{r} counterclockwise where vec{F} = (x^2 + y^2)hat{i} + (x^2 - y^2)hat{j} and dvec{r} = dx hat{i} + dy hat{j} and the curve C is the boundary of the region R = {(x, y) | 1 <= y <= 2 - x^2}.

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 intersection W.

Find a complete integral of the partial differential equation 2(pq + yp + qx) + x^2 + y^2 = 0.

Find the locus of the point of intersection of three mutually perpendicular tangent planes to ax^2 + by^2 + cz^2 = 1.

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

Find the equation of the tangent plane at point (1, 1, 1) to the conicooid 3x^2 - y^2 = 2z.

Derive the formula \int_{a}^{b} y dx = \frac{3h}{8} [(y_0 + y_n) + 3(y_1 + y_2 + y_4 + y_5 + ... + y_{n-1}) + 2(y_3 + y_6 + ... + y_{n-3})]. Is there any restriction on n? State that condition. What is the error bound in the case of Simpson's \frac{3}{8} rule?

Let f(t) = \int_{0}^{t} [x] dx, where [x] denotes the largest integer less than or equal to x. (i) Determine all the real numbers t at which f is differentiable. (ii) Determine all the real numbers t at which f is continuous but not differentiable.

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.

If the velocity of an incompressible fluid at the point (x, y, z) is given by \left(\frac{3xz}{r^5}, \frac{3yz}{r^5}, \frac{3z^2 - r^2}{r^5}\right), r^2 = x^2 + y^2 + z^2, then prove that the liquid motion is possible and that the velocity potential is \frac{z}{r^3}. Further, determine the streamlines.

Show that similar matrices have the same characteristic polynomial.

For given equidistant values u_{-1}, u_0, u_1 and u_2, a value is interpolated by Lagrange's formula. Show that it may be written in the form u_x = yu_0 + xu_1 + \frac{y(y^2 - 1)}{3!} \Delta^2 u_{-1} + \frac{x(x^2 - 1)}{3!} \Delta^2 u_0 where x + y = 1.

Using contour integral method, prove that \int_{0}^{\infty} \frac{x \sin mx}{a^2 + x^2} dx = \frac{\pi}{2} e^{-ma}

Find the initial basic feasible solution of the following transportation problem using Vogel's approximation method and find the cost. Destinations D_1, D_2, D_3, D_4, D_5. Origins O_1 (4, 7, 0, 3, 6, 14), O_2 (1, 2, -3, 3, 8, 9), O_3 (3, -1, 4, 0, 5, 17). Demand (8, 3, 8, 13, 8).

Find the shortest distance between the skew lines: (x - 3)/3 = (8 - y)/1 = (z - 3)/1 and (x + 3)/-3 = (y + 7)/2 = (z - 6)/4

If f(x, y) = {(xy(x^2 - y^2)/(x^2 + y^2) if (x, y) != (0, 0)), (0 if (x, y) = (0, 0))}, calculate (del^2 f)/(del x del y) and (del^2 f)/(del y del x) at (0, 0).

Given the one-dimensional wave equation \frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2}, t > 0, where c^2 = \frac{T}{m}, T is the constant tension in the string and m is the mass per unit length of the string. (i) Find the appropriate solution of the above wave equation. (ii) Find also the solution under the conditions y(0, t) = 0, y(l, t) = 0 for all t and \left[\frac{\partial y}{\partial t}\right]_{t=0} = 0, y(x, 0) = a \sin \frac{\pi x}{l}, 0 < x < l, a > 0.

Consider the matrix mapping A: R^4 -> R^3, where A = [[1, 2, 3, 1], [1, 3, 5, -2], [3, 8, 13, -3]]. Find a basis and dimension of the image of A and those of the kernel A.

Find the supremum and the infimum of \frac{x}{\sin x} on the interval \left(0, \frac{\pi}{2}\right].

Reduce the equation y^2 \frac{\partial^2 z}{\partial x^2} - 2xy \frac{\partial^2 z}{\partial x \partial y} + x^2 \frac{\partial^2 z}{\partial y^2} = \frac{y^2}{x} \frac{\partial z}{\partial x} - \frac{x^2}{y} \frac{\partial z}{\partial y} to canonical form and hence solve it.

For a function f : \mathbb{C} \to \mathbb{C} and n \ge 1, let f^{(n)} denote the n^{th} derivative of f and f^{(0)} = f. Let f be an entire function such that for some n \ge 1, f^{(n)}\left(\frac{1}{k}\right) = 0 for all k = 1, 2, 3, .... Show that f is a polynomial.

Write the Boolean expression z(y + z)(x + y + z) in its simplest form using Boolean postulate rules. Mention the rules used during simplification. Verify your result by constructing the truth table for the given expression and for its simplest form.

Let \Gamma be a closed curve in xy-plane and let S denote the region bounded by the curve \Gamma. Let \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} = f(x, y) \forall (x, y) \in S. If f is prescribed at each point (x, y) of S and w is prescribed on the boundary \Gamma of S, then prove that any solution w = w(x, y), satisfying these conditions, is unique.

A fixed wire is in the shape of the cardioid 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 the point r = 0 of the cardioid by an elastic string of natural length a and modulus of elasticity 4 mg. The string is released from rest when the string is horizontal. Show by using the laws of conservation of energy that a dot{theta}^2 (1 + cos theta) - g cos theta (1 - cos theta) = 0, g being the acceleration due to gravity.

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?

Prove that distinct non-zero eigenvectors of a matrix are linearly independent.

Show that the plane 2x - 2y + z = 0 touches the sphere x^2 + y^2 + z^2 - 2x - 4y + 2z - 3 = 0. Find the point of contact.

Let A = [[2, 2], [1, 3]]. Find a non-singular matrix P such that P^{-1}AP is a diagonal matrix.

(i) Evaluate the integral: iiint_S vec{F} dot hat{n} ds where vec{F} = 3xy^2 i + (yx^2 - y^3)hat{j} + 3zx^2 hat{k} and S is a surface of the cylinder y^2 + z^2 <= 4, -3 <= x <= 3, using divergence theorem.

(ii) Solve the following differential equation using method of variation of parameters: (d^2 y)/(dx^2) - (dy)/(dx) - 2y = 44 - 76x - 48x^2.

Let F be a field and F[X] denote the ring of polynomials over F in a single variable X. For f(x), g(x) \in F[X] with g(X) \neq 0, show that there exist q(X), r(X) \in F[X] such that degree(r(X)) < degree(g(X)) and f(X) = q(X) \cdot g(X) + r(X).

Find the curvature vector and its magnitude at any point vec{r} = vec{r}(theta) of the curve vec{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.