Mathematics UPSC PYQ 2019

Using Hamilton's equation, find the acceleration for a sphere rolling down a rough inclined plane, if x be the distance of the point of contact of the sphere from a fixed point on the plane.

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 normals coincide.

Is f(x) = |cos x| + |sin x|, x = pi/2 differentiable at x = pi/2? If yes, then find its derivative at x = pi/2. If no, then give a proof of it.

Find the equivalent numbers given in a specified number to the system mentioned against them : (i) Integer 524 in binary system. (ii) 101010110101-101101011 to octal system. (iii) decimal number 5280 to hexadecimal system. (iv) Find the unknown number (1101.101)_{8} \rightarrow (?)_{10}.

Using differentials, find an approximate value of f(4.1, 4.9) where f(x, y) = (x^{3} + x^{2}y)^{\frac{1}{2}}.

Evaluate \int_{0}^{\infty} \frac{\tan^{-1}(ax)}{x(1+x^{2})} dx, a > 0, a \neq 1.

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 mu / (distance)^2, is a conic section. Find the conditions under which the path becomes (i) ellipse, (ii) parabola and (iii) hyperbola.

Consider the following LPP, Maximize Z = 2x_1 + 4x_2 + 4x_3 - 3x_4 subject to x_1 + x_2 + x_3 = 4 x_1 + 4x_2 + x_4 = 8 and x_1, x_2, x_3, x_4 \ge 0 Use the dual problem to verify that the basic solution (x_1, x_2) is not optimal.

Obtain the singular solution of the differential equation ((dy)/(dx))^2 (y/x)^2 cot^2 alpha - 2 ((dy)/(dx)) (y/x) + ((y/x))^2 cosec^2 alpha = 1. Also find the complete primitive of the given differential equation. Give the geometrical interpretations of the complete primitive and singular solution.

Let f : [0, pi/2] -> R be a continuous function such that f(x) = (cos^2 x) / (4x^2 - pi^2), 0 <= x < pi/2. Find the value of f(pi/2).

A circular cylinder of radius a and radius of gyration k rolls without slipping inside a fixed hollow cylinder of radius b. Show that the plane through axes moves in a circular pendulum of length (b - a)\left(1 + \frac{k^2}{a^2}\right).

Find the circulation of F vector round the curve C, where F vector = (2x + y^2) i + (3y - 4x) 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).

Show that an isolated singular point z_0 of a function f(z) is a pole of order m if and only if f(z) can be written in the form f(z) = \frac{\phi(z)}{(z - z_0)^m} where \phi(z) is analytic and non zero at z_0. Moreover \text{Res}_{z = z_0} f(z) = \frac{\phi^{(m-1)}(z_0)}{(m-1)!} if m \ge 1.

One end of a heavy uniform rod AB can slide along a rough horizontal rod AC, to which it is attached 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 prove that the coefficient of friction is cot(theta) / (2 + cot^2(theta)).

Find the dimension of the subspace V = {(x_1, x_2, x_3, x_4) in R^4 | A [[x_1], [x_2], [x_3], [x_4]] = 0} where A = [[5, 7, 2, 1], [1, 1, -8, 1], [2, 3, 5, 0], [3, 4, -3, 1]].

Find the rank of matrix A.

Obtain the singular solution of the differential equation ((dy/dx) * (y/x))^2 cot^2 alpha - 2 ((dy/dx) * (y/x)) + ((dy/dx) * (y/x))^2 cosec^2 alpha = 1. Also find the complete primitive of the given differential equation. Give the geometrical interpretations of the complete primitive and singular solution.

Solve the differential equation (d^2 y)/(dx^2) + (3 sin x - cot x) (dy)/(dx) + 2y sin^2 x = e^(-cos x) sin^2 x

If u = sin^-1 ((x^(1/3) + y^(1/3)) / (x^(1/2) + y^(1/2))), then show that sin^2 u is a homogeneous function of x and y of degree -1/6. Hence show that x^2 (d^2 u / dx^2) + 2xy (d^2 u / dx dy) + y^2 (d^2 u / dy^2) = (tan u / 12) (13/12 + (tan^2 u)/12).

Draw a flow chart and write a basic algorithm (in FORTRAN/C/C++) for evaluating y = \int_{0}^{6} \frac{dx}{1 + x^2} using Trapezoidal rule.

Discuss the convergence of \int_{1}^{2} \frac{\sqrt{x}}{\ln x} dx.

Find the circulation of F vector round the curve C, where F vector = (2x + y^2) i_hat + (3y - 4x) j_hat 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).

Use graphical method to solve the linear programming problem. Maximize Z = 3x_1 + 2x_2 subject to x_1 - x_2 \ge 1, x_1 + x_3 \ge 3 and x_1, x_2, x_3 \ge 0

Suppose f(z) is analytic function on a domain D in \mathbb{C} and satisfies the equation Im f(z) = (Re f(z))^{2}, z \in D. Show that f(z) is constant in D.

Evaluate the integral \int_{C} Re(z^{2})dz from 0 to 2 + 4i along the curve C where C is a parabola y = x^{2}.

Consider the following LPP, Maximize Z = 2x_{1} + 4x_{2} + 4x_{3} - 3x_{4} subject to x_{1} + x_{2} + x_{3} = 4, x_{1} + 4x_{2} + x_{4} = 8 and x_{1}, x_{2}, x_{3}, x_{4} \geq 0. Use the dual problem to verify that the basic solution (x_{1}, x_{2}) is not optimal.

Find the Laplace transforms of t^(-1/2) and t^(1/2). Prove that the Laplace transform of t^(n + 1/2) is (Gamma(n + 1/2)) / (s^(n + 1/2)), where n in N.

Reduce the following second order partial differential equation to canonical form and find the general solution : \frac{\partial^{2}u}{\partial x^{2}} - 2x \frac{\partial^{2}u}{\partial x \partial y} + x^{2} \frac{\partial^{2}u}{\partial y^{2}} = \frac{\partial u}{\partial y} + 12x.

If A = [[1, 2, 1], [1, -4, 1], [3, 0, -3]] and B = [[2, 1, 1], [1, -1, 0], [2, 1, -1]], then show that AB = 6I3. Use this result to solve the following system of equations : 2x + y + z = 5, x - y = 0, 2x + y - z = 1

State Gauss divergence theorem. Verify this theorem for F vector = 4xi - 2y^2 j + z^2 k, taken over the region bounded by x^2 + y^2 = 4, z = 0 and z = 3.

Two sources, each of strength m, are placed at the points (-a, 0), (a, 0) and a sink of strength 2m at origin. Show that the stream lines are the curves (x^{2}+y^{2})^{2} = a^{2}(x^{2}-y^{2}+\lambda xy), where \lambda is a variable parameter. Show also that the fluid speed at any point is (2ma^{2})/(r_{1}r_{2}r_{3}), where r_{1}, r_{2} and r_{3} are the distances of the points from the sources and the sink, respectively.

The force of attraction of a particle by the earth is 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 3/4 hW, where h is the radius of the earth.

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.

Find the radius of curvature and radius of torsion of the helix x = a cos u, y = a sin u, z = a u tan alpha.

Use graphical method to solve the linear programming problem. Maximize Z = 3x_{1} + 2x_{2} subject to x_{1} - x_{2} \geq 1, x_{1} + x_{3} \geq 3 and x_{1}, x_{2}, x_{3} \geq 0

Find the maximum value of f(x, y, z) = x^{2}y^{2}z^{2} subject to the subsidiary condition x^{2} + y^{2} + z^{2} = c^{2}, (x, y, z > 0).

Draw a flow chart and write a basic algorithm (in FORTRAN/C/C++) for evaluating \int_{0}^{6} \frac{dx}{1+x^{2}} using Trapezoidal rule.

Show that the function f(x,y) = \begin{cases} \frac{x^2 - y^2}{x - y}, & (x, y) \neq (1, -1), (1, 1) \\ 0, & (x, y) = (1, 1), (1, -1) \end{cases} is continuous and differentiable at (1, -1).

Obtain the first three terms of the Laurent series expansion of the function f(z) = \frac{1}{(e^{z}-1)} about the point z = 0 valid in the region 0 < |z| < 2\pi.

Given the Boolean expression X = AB + ABC + \bar{A}\overline{C} + A\overline{C} (i) Draw the logical diagram for the expression. (ii) Minimize the expression. (iii) Draw the logical diagram for the reduced expression.

If G and H are finite groups whose orders are relatively prime, then prove that there is only one homomorphism from G to H, the trivial one.

Let f : D (subset of R^2) -> R be a function and (a, b) in D. If f(x, y) is continuous at (a, b), then show that the functions f(x, b) and f(a, y) are continuous at x = a and at y = b respectively.

Find the length of the normal chord through a point P of the ellipsoid (x^2)/(a^2) + (y^2)/(b^2) + (z^2)/(c^2) = 1 and prove that it is equal to 4PG3, where G3 is the point where the normal chord through P meets the xy-plane, then P lies on the cone (x^2)/(a^6) (2c^2 - a^2) + (y^2)/(b^6) (2c^2 - b^2) + (z^2)/(c^4) = 0.

Solve the differential equation (2y sin x + 3y^4 sin x cos x) dx - (4y^3 cos^2 x + cos x) dy = 0.

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.

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 (Gamma(n + 1 + 1/2)) / (s^(n + 1 + 1/2)).

Show that the lines (x + 1)/(-3) = (y - 3)/2 = (z + 2)/1 and x/1 = (y - 7)/(-3) = (z + 7)/2 intersect. Find the coordinates of the point of intersection and the equation of the plane containing them.

Solve the linear programming problem using Simplex method. Minimize Z = x_{1} + 2x_{2} - 3x_{3} - 2x_{4} subject to x_{1} + 2x_{2} - 3x_{3} + x_{4} = 4, x_{1} + 2x_{2} + x_{3} + 2x_{4} = 4 and x_{1}, x_{2}, x_{3}, x_{4} \geq 0

Let T : R^2 -> R^2 be a linear map such that T(2, 1) = (5, 7) and T(1, 2) = (3, 3). If A is the matrix corresponding to T with respect to the standard bases e1, e2, then find Rank(A).

Using Runge-Kutta method of fourth order, solve \frac{dy}{dx} = \frac{y^{2}-x^{2}}{y^{2}+x^{2}} with y(0) = 1 at x = 0.2. Use four decimal places for calculation and step length 0.2.

State Gauss divergence theorem. Verify this theorem for F vector = 4x i_hat - 2y^2 j_hat + z^2 k_hat, taken over the region bounded by x^2 + y^2 = 4, z = 0 and z = 3.

Discuss the convergence of \int_{1}^{2} \frac{\sqrt{x}}{l_{n}x} dx.

Solve the differential equation d^2 y / dx^2 + (3 sin x - cot x) (dy / dx) + 2 sin^2 x y = e^(- cos x) sin^2 x.

State the Cayley-Hamilton theorem. Use this theorem to find A^100, where A = [[1, 0, 0], [1, 0, 1], [0, 1, 0]].

Find the length of the normal chord through a point P of the ellipsoid x^2/a^2 + y^2/b^2 + 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 x^2/a^6 (2c^2 - a^2) + y^2/b^6 (2c^2 - b^2) + z^2/c^4 = 0.

Evaluate by Stokes' theorem integral over C (e^x dx + 2dy - dz), where C is the curve x^2 + y^2 = 4, z = 2.

Let f : D (subset of R^2) -> R be a function and (a, b) in D. If f(x, y) is continuous at (a, b), then show that the functions f(x, b) and f(a, y) are continuous at x = a and y = b respectively.

Find the equivalent numbers given in a specified number to the system mentioned against them: (i) Integer 524 in binary system. (ii) 101010110101-10110101 to octal system. (iii) decimal number 5280 to hexadecimal system. (iv) Find the unknown number (1101.101)_8 \rightarrow (?)_{10}.

Let T : R^2 -> R^2 be a linear map such that T(2, 1) = (5, 7) and T(1, 2) = (3, 3). If A is the matrix corresponding to T with respect to the standard bases e_1, e_2, then find Rank(A).

Find the maximum and the minimum value of the function f(x) = 2x^3 - 9x^2 + 12x + 6 on the interval [2, 3].

A uniform rod OA, of length 2a, free to turn about its end O, revolves with angular velocity \omega about the vertical OZ through O, and is inclined at a constant angle \alpha to OZ; find the value of \alpha.

A particle moving along the y-axis has an acceleration Fy towards the origin, where F is a positive and even function of y. The periodic time, when the particle vibrates between y = -a and y = a, is T. Show that (2pi)/sqrt(F_1) < T < (2pi)/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 deg on either side of the vertical line, T lies between 2pi sqrt(l/g) and 2pi sqrt(l/g) * (pi / sqrt(3)).

Let a be an irreducible element of the Euclidean ring R, then prove that R/(a) is a field.

Discuss the uniform convergence of f_{n}(x) = \frac{nx}{1+n^{2}x^{2}}, \forall x \in \mathbb{R}(-\infty, \infty), n = 1, 2, 3, ....

Using the Jacobian method, show that if f'(x) = 1 / (1 + x^2) and f(0) = 0, then f(x) + f(y) = f((x + y) / (1 - xy)).

Form a partial differential equation of the family of surfaces given by the following expression : \psi(x^{2} + y^{2} + 2z^{2}, y^{2} - 2zx) = 0.

Apply Gauss-Seidel iteration method to solve the following system of equations : 2x + y - 2z = 17, 3x + 20y - z = -18, 2x - 3y + 20z = 25, correct to three decimal places.

Let G be a finite group, H and K subgroups of G such that K \subset H. Show that (G : K) = (G : H)(H : K).

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 root(3) a.

Given the Boolean expression X = AB + ABC + \bar{A}\bar{B}C + A\bar{C} (i) Draw the logical diagram for the expression. (ii) Minimize the expression. (iii) Draw the logical diagram for the reduced expression.

Determine the complete solution of the differential equation d^2 y / dx^2 - 4 (dy / dx) + 4y = 3 x^2 e^(2x) sin 2x.

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^3 at the point (1, 1, 1).

Find the linearly independent solutions of the corresponding homogeneous differential equation of the equation x^2 y'' - 2xy' + 2y = x^3 sin x and then find the general solution of the given equation by the method of variation of parameters.

Find the radius of curvature and radius of torsion of the helix x = a cos u, y = a sin u, z = au tan alpha.

Find the dimension of the subspace V = {(x1, x2, x3, x4) in R^4 | A [x1, x2, x3, x4]^T = 0} where A = [[5, 7, 2, 1], [1, 1, -8, 1], [2, 3, 5, 0], [3, 4, -3, 1]].

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 rectangular hyperbola.

Apply Newton-Raphson method, to find a real root of transcendental equation x \log_{10} x = 1.2, correct to three decimal places.

Show that an isolated singular point z_{0} of a function f(z) is a pole of order m if and only if f(z) can be written in the form f(z) = \frac{\phi(z)}{(z-z_{0})^{m}} where \phi(z) is analytic and non zero at z_{0}. Moreover Res_{z=z_{0}} f(z) = \frac{\phi^{(m-1)}(z_{0})}{(m-1)!} if m \geq 1.

Determine the complete solution of the differential equation (d^2 y)/(dx^2) - 4 (dy)/(dx) + 4y = 3x^2 e^(2x) sin 2x

Is f(x) = |cos x| + |sin x| differentiable at x = pi/2? If yes, then find its derivative at x = pi/2. If no, then give a proof of it.

Solve the first order quasilinear partial differential equation by the method of characteristics: x \frac{\partial u}{\partial x} + (u - x - y)\frac{\partial u}{\partial y} = x + 2y in x > 0, -\infty < y < \infty with u = 1 + y on x = 1.

Solve the linear programming problem using Simplex method. Minimize Z = x_1 + 2x_2 - 3x_3 - 2x_4 subject to x_1 + 2x_2 - 3x_3 + x_4 = 4 x_1 + 2x_2 + x_3 + 2x_4 = 4 and x_1, x_2, x_3, x_4 \ge 0

A particle moving along the y-axis has an acceleration Fy towards the origin, where F is a positive and even function of y. The periodic time, when the particle vibrates between y = -a and y = a, is T. Show that (2pi)/sqrt(F1) < T < (2pi)/sqrt(F2) where F1 and F2 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 degrees on either side of the vertical line, T lies between 2pi sqrt(l/g) and 2pi sqrt(l/g) * sqrt(pi/3).

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.

If A = [[1, 2, 1], [1, -4, 1], [3, 0, -3]] and B = [[2, 1, 1], [1, -1, 0], [2, 1, -1]], then show that AB = 6 I_3. Use this result to solve the following system of equations : 2x + y + z = 5, x - y = 0, 2x + y - z = 1.

A sphere of radius R, whose centre is at rest, vibrates radially in an infinite incompressible fluid of density \rho, which is at rest at infinity. If the pressure at infinity is \Pi, so that the pressure at the surface of the sphere at time t is \Pi + \frac{1}{2}\rho \left[ \frac{d^{2}R^{2}}{dt^{2}} + \left(\frac{dR}{dt}\right)^{2} \right].

Write down all quotient groups of the group Z_{12}.