78 questions from the UPSC 2022 examination.
78 questions
A cable of weight w per unit length and length 2l hangs from two points P and Q in the same horizontal line. Show that the span of the cable is 2l * [ 1 - (2h^2) / (3l^2) ], where h is the sag in the middle of the tightly stretched position.
Prove that any set of n linearly independent vectors in a vector space V of dimension n constitutes a basis for V.
Evaluate lim (ex + x)^(1/x) as x -> infinity.
A particle at a distance r from the centre of force moves under the influence of the central force F = -k / r^2, where k is a constant. Obtain the Lagrangian and derive the equations of motion.
Using Gauss' divergence theorem, evaluate surface integral over S of F dot n ds, where F = xi - yj + (z^2 - 1)k and S is the cylinder formed by the surfaces z = 0, z = 1, x^2 + y^2 = 4.
Evaluate contour integral over C ((z + 4) / (z^2 + 2z + 5)) dz, where C is |z + 1 - i| = 2.
A chain of n equal uniform rods is smoothly jointed together and suspended from its one end A1. A horizontal force P is applied to the other end An+1 of the chain. Find the inclinations of the rods to the downward vertical line in the equilibrium configuration.
Find the maximum and minimum values of (x^2 / a^4) + (y^2 / b^4) + (z^2 / c^4), when lx + my + nz = 0 and (x^2 / a^2) + (y^2 / b^2) + (z^2 / c^2) = 1. Interpret the result geometrically.
Find a combinatorial circuit corresponding to the Boolean function f(x, y, z) = [x . (y' + z)] + y and write the input/output table for the circuit.
Let R be a field of real numbers and S, the field of all those polynomials f(x) in R[x] such that f(0) = f(1). Prove that S is an ideal of R[x]. Is the residue class ring R[x]/S an integral domain? Give justification for your answer.
Show that A = (6xy + z^3)i + (3x^2 - z)j + (3xz^2 - y)k is irrotational. Also find phi such that A = grad phi.
Evaluate contour integral over C of ((z + 4) / (z^2 + 2z + 5)) dz, where C is |z + 1 - i| = 2.
Let T : R^2 -> R^3 be a linear transformation such that T(1, 0) = (1, 2, 3) and T(1, 1) = (-3, 2, 8). Find T(2, 4).
Use two-phase method to solve the following linear programming problem: Minimize Z = x1 + x2 subject to 2x1 + x2 >= 4, x1 + 7x2 >= 7, x1, x2 >= 0.
Verify Green's theorem in the plane for line integral over C of (3x^2 - 8y^2) dx + (4y - 6xy) dy, where C is the boundary curve of the region defined by x = 0, y = 0, x + y = 1.
Use two-phase method to solve the following linear programming problem : Minimize Z = x_1 + x_2 subject to 2x_1 + x_2 >= 4, x_1 + 7x_2 >= 7, x_1, x_2 >= 0
Apply the calculus of residues to evaluate integral from -infinity to infinity (cos x dx / ((x^2 + a^2)(x^2 + b^2))), a > b > 0.
Evaluate lim (e^x + x)^(1/x) as x -> inf.
Find the general and singular solutions of the differential equation : (x^2 - a^2) p^2 - 2xyp + y^2 + a^2 = 0, where p = dy/dx. Also give the geometric relation between the general and singular solutions.
Expand f(z) = 1 / ((z - 1)^2 (z - 3)) in a Laurent series valid for the regions (i) 0 < |z - 1| < 2 and (ii) 0 < |z - 3| < 2.
Using Gauss' divergence theorem, evaluate surface integral over S of F_vector dot n_vector dS, where F_vector = x i_cap - y j_cap + (z^2 - 1) k_cap and S is the cylinder formed by the surfaces z = 0, z = 1, x^2 + y^2 = 4.
Using Runge-Kutta method of fourth order, solve the differential equation dy / dx = x + y^2 with y(0) = 1, at x = 0.2. Use four decimal places for calculation and step length 0.1.
Reduce the following partial differential equation to a canonical form and hence solve it: y u_xx + (x + y) u_xy + x u_yy = 0
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: Destination A B C D, Source S1 (21, 16, 25, 13, Availability 11), Source S2 (17, 18, 14, 23, Availability 13), Source S3 (32, 27, 18, 41, Availability 19), Requirement (6, 10, 12, 15, 43).
Show that the general solution of the differential equation dy/dx + Py = Q can be written in the form y = Q/P - e^(- \int P dx) { C + \int e^\int P dx d(Q/P) }, where P, Q are non-zero functions of x and C, an arbitrary constant.
Solve the heat equation (del u)/(del t) = (del^2 u)/(del x^2), 0 < x < l, t > 0 subject to the conditions u(0, t) = u(l, t) = 0, u(x, 0) = x(l - x), 0 <= x <= l
If the plane ux + vy + wz = 0 cuts the cone ax^2 + by^2 + cz^2 = 0 in perpendicular generators, then prove that (b + c)u^2 + (c + a)v^2 + (a + b)w^2 = 0.
A wire of length l is cut into two parts which are bent in the form of a square and a circle respectively. Using Lagrange's method of undetermined multipliers, find the least value of the sum of the areas so formed.
Find a combinatorial circuit corresponding to the Boolean function f(x, y, z) = [x . (not y + z)] + y and write the input/output table for the circuit.
It is given that the equation of any cone with vertex at (a, b, c) is f((x - a)/(z - c), (y - b)/(z - c)) = 0. Find the differential equation of the cone.
A body of weight w rests on a rough inclined plane of inclination theta, the coefficient of friction, mu, being greater than tan theta. Find the work done in slowly dragging the body a distance 'b' up the plane and then dragging it back to the starting point, the applied force being in each case parallel to the plane.
Let the set P = { (x, y, z) | x - y - z = 0 and 2x - y + z = 0 } be the collection of vectors of a vector space R^3(R). Then (i) prove that P is a subspace of R^3. (ii) find a basis and dimension of P.
Show that the multiplicative group G = {1, -1, i, -i}, where i = sqrt(-1), is isomorphic to the group G' = ({0, 1, 2, 3}, +_4).
Solve the following initial value problem by Laplace's transformation (d^2y/dt^2) - 3(dy/dt) + 2y = h(t), where h(t) = 2 for 0 < t < 4, h(t) = 0 for t > 4, y(0) = 0, y'(0) = 0.
Solve the following differential equation : (3x + 2)^2 * (d^2y/dx^2) + 5(3x + 2) * (dy/dx) - 3y = x^2 + x + 1
Test for convergence or divergence of the series x + ((2^2 x^2) / 2!) + ((3^3 x^3) / 3!) + ((4^4 x^4) / 4!) + ((5^5 x^5) / 5!) + ... (x > 0)
Find all solutions to the following system of equations by row-reduced method: x1 + 2x2 - x3 = 2, 2x1 + 3x2 + 5x3 = 5, -x1 - 3x2 + 8x3 = -1.
The velocity components of an incompressible fluid in spherical polar coordinates (r, theta, psi) are (2Mr^(-3) cos theta, Mr^(-2) sin theta, 0), where M is a constant. Show that the velocity is of the potential kind. Find the velocity potential and the equations of the streamlines.
Reduce the following partial differential equation to a canonical form and hence solve it : u_xx + (x + y)u_xy + x u_yy = 0
Solve the heat equation del u / del t = del^2 u / del x^2, 0 < x < 1, t > 0 subject to the conditions u(0, t) = u(1, t) = 0, u(x, 0) = x(1 - x), 0 <= x <= 1.
Solve the following linear programming problem by the simplex method. Write its dual. Also, write the optimal solution of the dual from the optimal table of the given problem : Maximize Z = x_1 + x_2 + x_3 subject to 2x_1 + x_2 + x_3 <= 2, 4x_1 + 2x_2 + x_3 <= 2, x_1, x_2, x_3 >= 0
Solve the following linear programming problem by the simplex method. Write its dual. Also, write the optimal solution of the dual from the optimal table of the given problem: Maximize Z = x1 + x2 + x3 subject to 2x1 + x2 + x3 <= 2, 4x1 + 2x2 + x3 <= 2, x1, x2, x3 >= 0.
The velocity of a train which starts from rest is given by the following table, the time being reckoned in minutes from the start and the velocity in km/hour: t (minutes) = [2, 4, 6, 8, 10, 12, 14, 16, 18, 20], v (km/hour) = [16, 28.8, 40, 46.4, 51.2, 32, 17.6, 8, 3.2, 0]. Using Simpson's (1/3)rd rule, estimate approximately in km the total distance run in 20 minutes.
Solve the following initial value problem by using Laplace's transformation: (d^2y/dt^2) - 3(dy/dt) + 2y = h(t), where h(t) = 2 for 0 < t < 4 and 0 for t > 4, y(0) = 0, y'(0) = 0.
Find the equation of the sphere of smallest possible radius which touches the straight lines: (x - 3)/3 = (y - 8)/(-1) = (z - 3)/1 and (x + 3)/(-3) = (y + 7)/2 = (z - 6)/4.
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 : Destination A B C D Availability S_1 21 16 25 13 11 Source S_2 17 18 14 23 13 S_3 32 27 18 41 19 Requirement 6 10 12 15 43
If P, Q, R; P', Q', R' are feet of the six normals drawn from a point to the ellipsoid (x^2)/a^2 + (y^2)/b^2 + (z^2)/c^2 = 1, and the plane PQR is represented by lx + my + nz = p, show that the plane P'Q'R' is given by (x / a^2 l) + (y / b^2 m) + (z / c^2 n) + 1/p = 0.
Find a linear map T : R^2 -> R^2 which rotates each vector of R^2 by an angle theta. Also, prove that for theta = pi/2, T has no eigenvalue in R.
Suppose a cylinder of any cross-section is balanced on another fixed cylinder, the contact of curved surfaces being rough and the common tangent line horizontal. Let rho and rho' be the radii of curvature of the two cylinders at the point of contact and h be the height of centre of gravity of the upper cylinder above the point of contact. Show that the upper cylinder is balanced in stable equilibrium if h < (rho * rho') / (rho + rho').
Convert the number (1093.21875)_10 into octal and the number (1693.0628)_10 into hexadecimal systems.
Verify Stokes' theorem for F = xi + z^2j + y^2k over the plane surface : x + y + z = 1 lying in the first octant.
Trace the curve y^2x^2 = x^2 - a^2, where a is a real constant.
Let f(x) = x^2 on [0, k], k > 0. Show that f is Riemann integrable on the closed interval [0, k] and integral from 0 to k of f dx = k^3 / 3.
Verify that w = i k log {(z - i a) / (z + i a)} is the complex potential of a steady flow of fluid about a circular cylinder, where the plane y = 0 is a rigid boundary. Find also the force exerted by the fluid on unit length of the cylinder.
Show that the orthogonal trajectories of the system of parabolas : x^2 = 4a (y + a) belong to the same system.
Show that A_vector = (6xy + z^3)i_cap + (3x^2 - z)j_cap + (3xz^2 - y)k_cap is irrotational. Also find phi such that A_vector = grad phi.
If f(z) = u + iv is an analytic function of z, and u - v = (cos x + sin x - e^(-y)) / (2 cos x - e^y - e^(-y)), then find f(z) subject to the condition f(pi / 2) = 0.
Express the Boolean function F(x, y, z) = xy + x'z in a product of maxterms form.
Two point vortices each of strength k are situated at (+/-a, 0) and a point vortex of strength -k/2 is situated at the origin. Show that the fluid motion is stationary and also find the equations of streamlines. If the streamlines, which pass through the stagnation points, meet the x-axis at (+/-b, 0), then show that 3*sqrt(3)*(b^2 - a^2)^2 = 16a^3 b.
Solve, by Gauss elimination method, the system of equations 2x + 2y + 4z = 18, x + 3y + 2z = 13, 3x + y + 3z = 14.
Verify Stokes' theorem for F_vector = x i_cap + z^2 j_cap + y^2 k_cap over the plane surface : x + y + z = 1 lying in the first octant.
Find the general solution of the partial differential equation (D^2 + DD' - 6D'^2)z = x^2 sin(x + y) where D = del / del x and D' = del / del y.
A projectile is fired from a point O with velocity sqrt(2gh) and hits a tangent at the point P(x, y) in the plane, the axes OX and OY being horizontal and vertically downward lines through the point O, respectively. Show that if the two possible directions of projection be at right angles, then x^2 = 2hy and then one of the possible directions of projection bisects the angle POX.
Find the moment of inertia of a right circular solid cone about one of its slant sides (generator) in terms of its mass M, height h and the radius of base as a.
Examine the convergence of the integral from 0 to 2 of dx / (2x - x^2).
Test the convergence of integral from 0 to infinity (cos x / (1 + x^2)) dx.
If P, Q, R; P', Q', R' are feet of the six normals drawn from a point to the ellipsoid x^2/a^2 + y^2/b^2 + z^2/c^2 = 1, and the plane PQR is represented by lx + my + nz = p, show that the plane P'Q'R' is given by (x/a^2l) + (y/b^2m) + (z/c^2n) + 1/p = 0.
Let R be a field of real numbers and S, the field of all those polynomials f(x) in R[x] such that f(0) = 0 = f(1). Prove that S is an ideal of R[x]. Is the residue class ring R[x]/S an integral domain? Give justification for your answer.
Examine the convergence of integral from 0 to 2 of dx / (2x - x^2).
Use double integration to calculate the area common to the circle x^2 + y^2 = 4 and the parabola y^2 = 3x.
Trace the curve y^2 x^2 = x^2 - a^2, where a is a real constant.
Solve the following differential equation by using the method of variation of parameters : (x^2 - 1) * (d^2y/dx^2) - 2x * (dy/dx) + 2y = (x^2 - 1)^2, given that y = x is one solution of the reduced equation.
Verify that w = ik log {(z - ia) / (z + ia)} is the complex potential of a steady flow of fluid about a circular cylinder, where the plane y = 0 is a rigid boundary. Find also the force exerted by the fluid on unit length of the cylinder.
Prove that every homomorphic image of a group G is isomorphic to some quotient group of G.
A variable plane passes through a fixed point (a, b, c) and meets the axes at points A, B and C respectively. Find the locus of the centre of the sphere passing through the points O, A, B and C, O being the origin.
A chain of n uniform rods is smoothly jointed together and suspended from its one end A1. A horizontal force P_vector is applied to the other end A_(n+1) of the chain. Find the inclinations of the rods to the downward vertical line in the equilibrium configuration.
(i) Convert the number (1093.21875)_10 into octal and the number (1693.0628)_10 into hexadecimal systems. (ii) Express the Boolean function F(x, y, z) = xy + x'z in a product of maxterms form.
Verify Green's theorem in the plane for the closed line integral over C of (3x^2 - 8y^2) dx + (4y - 6xy) dy, where C is the boundary curve of the region defined by x = 0, y = 0, x + y = 1.