Mathematics UPSC PYQ 2020

Using Laplace transform, solve the initial value problem ty'' + 2ty' + 2y = 2; y(0) = 1 and y'(0) is arbitrary. Does this problem have a unique solution?

Find the equation of the cylinder whose generators are parallel to the line \frac{x}{1} = \frac{y}{-2} = \frac{z}{3} and whose guiding curve is x^2 + y^2 = 4, z = 2.

Let S3 and Z3 be permutation group on 3 symbols and group of residue classes module 3 respectively. Show that there is no homomorphism of S3 in Z3 except the trivial homomorphism.

If v(r, \theta) = (r-\frac{1}{r}) \sin \theta, r \neq 0, then find an analytic function f(z) = u(r, \theta) + iv(r, \theta)

Let F be a subfield of complex numbers and T a function from F^3 \to F^3 defined by T(x_1, x_2, x_3) = (x_1 + x_2 + 3x_3, 2x_1 - x_2, -3x_1 + x_2 - x_3). What are the conditions on a, b, c such that (a, b, c) be in the null space of T? Find the nullity of T.

Let G be a finite cyclic group of order n. Then prove that G has \phi(n) generators (where \phi is Euler's \phi-function).

Find all the asymptotes of the curve (2x + 3)y = (x - 1)^2.

Find the orthogonal trajectories of the family of circles passing through the points (0, 2) and (0, -2).

Find the equations of the tangent plane to the ellipsoid 2x^2 + 6y^2 + 3z^2 = 27 which passes through the line x - y - z = 0 = x - y + 2z - 9.

UPSC maintenance section has purchased sufficient number of curtain cloth pieces to meet the curtain requirement of its building. The length of each piece is 17 feet. The requirement according to curtain length is as follows : Curtain length (in feet) | Number required 5 | 700 9 | 400 7 | 300 The width of all curtains is same as that of available pieces. Form a linear programming problem in standard form that decides the number of pieces cut in different ways so that the total trim loss is minimum. Also give a basic feasible solution to it.

A light rigid rod ABC has three particles each of mass m attached to it at A, B and C. The rod is struck by a blow P at right angles to it at a point distant from A equal to BC. Prove that the kinetic energy set up is \frac{1}{2} \frac{P^2}{m} \frac{a^2 - ab + b^2}{a^2 + ab + b^2}, where AB = a and BC = b.

Using the method of variation of parameters, solve the differential equation y'' + (1 - \cot x)y' - y \cot x = \sin^2 x, if y = e^{-x} is one solution of CF.

By writing down the Hamiltonian, find the equations of motion of a particle of mass m constrained to move on the surface of a cylinder defined by x^2+y^2=R^2, R is a constant. The particle is subject to a force directed towards the origin and proportional to the distance r of the particle from the origin given by \vec{F} = -k\vec{r}, k is a constant.

Evaluate \int \tan^{-1}\left(1 - \frac{1}{x}\right) dx.

Using contour integration, evaluate the integral \int_0^{2\pi} \frac{1}{3+2\sin \theta} d\theta.

Find the integral surface of the partial differential equation : (x-y)y^2 \frac{\partial z}{\partial x} + (y-x)x^2 \frac{\partial z}{\partial y} = (x^2+y^2)z that contains the curve : xz = a^3, y = 0 on it.

Write the three point Lagrangian interpolating polynomial relative to the points x_0, x_0+\epsilon and x_1. Then by taking the limit \epsilon \to 0, establish the relation f(x) = \frac{(x_1-x)(x+x_1-2x_0)}{(x_1-x_0)^2} f(x_0) + \frac{(x_0-x)(x_1-x)}{(x_1-x_0)} f'(x_0) + \frac{(x-x_0)^2}{(x_1-x_0)^2} f(x_1) + E(x) where E(x) = \frac{1}{6}(x-x_0)^2(x-x_1)f'''(\xi) is the error function and \min.(x_0, x_0+\epsilon, x_1) < \xi < \max.(x_0, x_0+\epsilon, x_1)

A uniform rod, in vertical position, can freely about one of its ends and is pulled aside from the vertical by a horizontal force acting at the other end of the rod and equal to half its weight. At what inclination to the vertical will the rod rest?

Let M_2(R) be the vector space of all 2 x 2 real matrices. Let B = \begin{bmatrix} 1 & -1 \\ -4 & 4 \end{bmatrix}. Suppose T: M_2(R) \to M_2(R) is a linear transformation defined by T(A) = BA. Find the rank and nullity of T. Find a matrix A which maps to the null matrix.

Evaluate the integral \int_C (z^2+3z)dz counterclockwise from (2, 0) to (0, 2) along the curve C, where C is the circle |z|=2.

Two sources of strength \frac{m}{2} are placed at the points (\pm a, 0). Show that at any point on the circle x^2+y^2=a^2, the velocity is parallel to the y-axis and is inversely proportional to y.

A velocity potential in a two-dimensional fluid flow is given by \phi(x,y) = xy+x^2-y^2. Find the stream function for this flow.

A four-wheeled railway truck has a total mass M, the mass and radius of gyration of each pair of wheels and axle are m and k respectively, and the radius of each wheel is r. Prove that if the truck is propelled along a level track by a force P, the acceleration is \frac{P}{M + \frac{2mk^2}{r^2}}, and find the horizontal force exerted on each axle by the truck. The axle friction and wind resistance are to be neglected.

Find an extreme value of the function u = x^2 + y^2 + z^2, subject to the condition 2x + 3y + 5z = 30, by using Lagrange's method of undetermined multiplier.

Evaluate \lim_{x \to \frac{\pi}{4}} (\tan x)^{\tan 2x}.

Find the locus of the point of intersection of the perpendicular generators of the hyperbolic paraboloid \frac{x^2}{a^2} - \frac{y^2}{b^2} = 2z.

Find the initial basic feasible solution of the following transportation problem by Vogel's approximation method and use it to find the optimal solution and the transportation cost of the problem.

Show that \int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} dx = \frac{1}{\sqrt{2}} \log_e (1+\sqrt{2})

Solve the following differential equation: x \cos\left(\frac{y}{x}\right)(y \, dx + x \, dy) = y \sin\left(\frac{y}{x}\right)(x \, dy - y \, dx)

Show that the equation f(x) = \cos \frac{\pi(x+1)}{8} + 0.148x - 0.9062 = 0 has one root in the interval (-1, 0) and one in (0, 1). Calculate the negative root correct to four decimal places using Newton-Raphson method.

For the solution of the system of equations : 4x + y + 2z = 4 3x + 5y + z = 7 x + y + 3z = 3 set up the Gauss-Seidel iterative scheme and iterate three times starting with the initial vector X^{(0)} = 0. Also find the exact solutions and compare with the iterated solutions.

Prove that the moment of inertia of a triangular lamina ABC about any axis through A in its plane is \frac{M}{6} (\beta^2 + \beta\gamma + \gamma^2) where M is the mass of the lamina and \beta, \gamma are respectively the length of perpendiculars from B and C on the axis.

Let A = \begin{bmatrix} 1 & 0 & 2 \\ 2 & -1 & 3 \\ 4 & 1 & 8 \end{bmatrix} and B = \begin{bmatrix} -11 & 2 & 2 \\ -4 & 0 & 1 \\ 6 & -1 & -1 \end{bmatrix}. (i) Find AB. (ii) Find \det(A) and \det(B). (iii) Solve the following system of linear equations: x + 2z = 3, 2x - y + 3z = 3, 4x + y + 8z = 14

Solve the linear programming problem using simplex method : Minimize z = -6x_1 - 2x_2 - 5x_3 subject to 2x_1 - 3x_2 + x_3 <= 14 -4x_1 + 4x_2 + 10x_3 <= 46 2x_1 + 2x_2 - 4x_3 <= 37 x_1 >= 2, x_2 >= 1, x_3 >= 3

If u = \tan^{-1} \frac{x^3+y^3}{x-y}, x \neq y then show that x^2 \frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2} = (1-4 \sin^2 u) \sin 2u

(i) A square framework formed of uniform heavy rods of equal weight W jointed together, is hung up by one corner. A weight W is suspended from each of the three lower corners, and the shape of the square is preserved by a light rod along the horizontal diagonal. Find the thrust of the light rod. (ii) A particle starts at a great distance with velocity V. Let p be the length of the perpendicular from the centre of a star on the tangent to the initial path of the particle. Show that the least distance of the particle from the centre of the star is \lambda, where V^2\lambda^2 = \mu^2 + p^2 V^4 - \mu. Here \mu is a constant.

Solve the partial differential equation : (D^3 - 2D^2 D' - DD'^2 + 2D'^3)z = e^{2x+y} + \sin(x-2y), D = \frac{\partial}{\partial x}, D' = \frac{\partial}{\partial y}

Let R be a finite field of characteristic p(>0). Show that the mapping f: R \to R defined by f(a) = a^p, \forall a \in R is an isomorphism.

Prove that the function f(x) = \sin x^2 is not uniformly continuous on the interval [0, \infty[.

Evaluate the surface integral \iint_{S} (\nabla \times \vec{F}) \cdot \hat{n} \, ds for \vec{F} = y\hat{i} + (x - 2xz)\hat{j} - xy\hat{k} and S is the surface of the sphere x^2 + y^2 + z^2 = a^2 above the xy-plane.

For what value of a, b, c is the vector field \vec{V} = (-4x - 3y + az)\hat{i} + (bx + 3y + 5z)\hat{j} + (4x + cy + 3z)\hat{k} irrotational? Hence, express V as the gradient of a scalar function \phi. Determine \phi.

A beam AD rests on two supports B and C, where AB = BC = CD. It is found that the beam will tilt when a weight of p kg is hung from A or when a weight of q kg is hung from D. Find the weight of the beam.

Define an n x n matrix as A = I - 2u \cdot u^T, where u is a unit column vector. (i) Examine if A is symmetric. (ii) Examine if A is orthogonal. (iii) Show that \text{trace}(A) = n - 2. (iv) Find A_{3,3} when u = \begin{bmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}.

Find the solution of the partial differential equation : z = \frac{1}{2}(p^2+q^2) + (p-x)(q-y); p = \frac{\partial z}{\partial x}, q = \frac{\partial z}{\partial y} which passes through the x-axis.

Verify the Stokes' theorem for the vector field \vec{F} = xy\hat{i} + yz\hat{j} + xz\hat{k} on the surface S which is the part of the cylinder z = 1 - x^2 for 0 \le x \le 1, -2 \le y \le 2; S is oriented upwards.

Prove that the sequence (a_n) satisfying the condition |a_{n+1}-a_n| <= alpha |a_n - a_{n-1}|, 0<alpha<1 for all natural numbers n>=2, is a Cauchy sequence.

Consider the set V of all n x n real magic squares. Show that V is a vector space over R. Give examples of two distinct 2 x 2 magic squares.

Find a quadrature formula \int_0^1 f(x) \frac{dx}{\sqrt{x(1-x)}} = \alpha_1 f(0) + \alpha_2 f(\frac{1}{2}) + \alpha_3 f(1) which is exact for polynomials of highest possible degree. Then use the formula to evaluate \int_0^1 \frac{dx}{\sqrt{x-x^3}} (correct up to three decimal places).

If the straight line \frac{x}{1} = \frac{y}{2} = \frac{z}{3} represents one of a set of three mutually perpendicular generators of the cone 5yz - 8zx - 3xy = 0, then find the equations of the other two generators.

Form a partial differential equation by eliminating the arbitrary functions f(x) and g(y) from z=yf(x)+xg(y) and specify its nature (elliptic, hyperbolic or parabolic) in the region x>0, y>0.

(i) Solve the following differential equation: (x + 1)^2 y'' - 4(x + 1)y' + 6y = (x + 1)^2 + \sin \log(x + 1) (ii) Find the general and singular solutions of the differential equation 9p^2(2 - y)^2 = 4(3 - y), where p = \frac{dy}{dx}.

Let R be a principal ideal domain. Show that every ideal of a quotient ring of R is principal ideal and R/P is a principal ideal domain for a prime ideal P of R.

Consider the function f(x) = \int_{0}^{x} (t^2 - 5t + 4)(t^2 - 5t + 6) dt. (i) Find the critical points of the function f(x). (ii) Find the points at which local minimum occurs. (iii) Find the points at which local maximum occurs. (iv) Find the number of zeros of the function f(x) in [0, 5].

For the vector function \vec{A}, where \vec{A} = (3x^2 + 6y)\hat{i} - 14yz\hat{j} + 20xz^2 \hat{k}, calculate \int_{C} \vec{A} \cdot d\vec{r} from (0, 0, 0) to (1, 1, 1) along the following paths: (i) x = t, y = t^2, z = t^3 (ii) Straight lines joining (0, 0, 0) to (1, 0, 0) and then to (1, 1, 0) and then to (1, 1, 1) (iii) Straight line joining (0, 0, 0) to (1, 1, 1) Is the result same in all the cases? Explain the reason.

One end of a tightly stretched flexible thin string of length l is fixed at the origin and the other at x = l. It is plucked at x = \frac{l}{3} so that it assumes initially the shape of a triangle of height h in the x-y plane. Find the displacement y at any distance x and at any time t after the string is released from rest. Take, \frac{\text{horizontal tension}}{\text{mass per unit length}} = c^2.

Let g(w, x, y, z) = (w+x+y)(x+\bar{y}+z)(w+\bar{y}) be a Boolean function. Obtain the conjunctive normal form for g(w, x, y, z). Also express g(w, x, y, z) as a product of maxterms.