Mathematics UPSC PYQ 2018

For the system of linear equations x + 3y - 2z = -1 5y + 3z = 8 x - 2y - 5z = 7 determine which of the following statements are true and which are false : (i) The system has no solution. (ii) The system has a unique solution. (iii) The system has infinitely many solutions.

For an incompressible fluid flow, two components of velocity (u, v, w) are given by u = -x^{2} - 2yz - 3z^{2}, v = x^{2} - y^{2} + zx. Determine the third component w so that they satisfy the equation of continuity. Also, find the z-component of acceleration.

Prove that the function f(x, y) = x^{3} - 3xy^{2} - 3y^{2} is harmonic and find its harmonic conjugate and the corresponding analytic function f(z) in terms of z.

An agricultural farm has 180 tons of nitrogen fertilizer, 250 tons of phosphate and 220 tons of potash. It will be able to sell a mixture of these substances in their respective ratio 3 : 3 : 4 at a profit of Rs. 1500 per ton and a mixture in the ratio 2 : 4 : 2 at a profit of Rs. 1200 per ton. Pose a linear programming problem to show how many tons of these two mixtures should be prepared to obtain the maximum profit.

Let A be a 3 x 2 matrix and B a 2 x 3 matrix. Show that C = A B is a singular matrix.

Find the maximum and the minimum values of x^4 - 5x^2 + 4 on the interval [2, 3].

Find the angle between the tangent at a general point of the curve whose equations are x = 3t, y = 3t^2, z = 3t^3 and the line y - x = 0, z = 0.

Show that if a function f defined on an open interval (a, b) of R is convex, then f is continuous. Show, by example, if the condition of open interval is dropped, then the convex function need not be continuous.

Find f(y) such that (2xe^x + 3y^2)dx - (3x^2 + f(y))dx = 0 is exact and hence solve.

How many basic solutions are there in the following linearly independent set of equations ? Find all of them. 2x_{1} - x_{2} + 3x_{3} + x_{4} = 6, 4x_{1} - 2x_{2} + x_{3} + 2x_{4} = 10.

Find the projection of the straight line (x-1)/2 = (y-1)/3 = (z-1)/(-2) on the plane x - y - 2z = 6.

A particle moving with simple harmonic motion in a straight line has velocities v1 and v2 at distances x1 and x2 respectively from the centre of its path. Find the period of its motion.

Using Newton's forward difference formula find the lowest degree polynomial u_{x} when it is given that u_{1} = 1, u_{2} = 9, u_{3} = 25, u_{4} = 55 and u_{5} = 105.

For a two-dimensional potential flow, the velocity potential is given by \phi = x^{2}y - y^{3} + \frac{1}{3}(x^{3} - xy^{2}). Determine the velocity components along the directions x and y. Also, determine the stream function \psi and check whether \phi represents a possible case of flow or not.

Evaluate the integral int_0^a int_0^x (xdy dx)/(x^2 + y^2).

Find the Laplace transform of f(t) = |sin t|.

Solve : (1 - x^2) y'' + x y' + y = 4 cos(log(1 + x))

Find all the proper subgroups of the multiplicative group of the field (Z_{13}, +_{13}, \times_{13}), where +_{13} and \times_{13} represent addition modulo 13 and multiplication modulo 13 respectively.

If S is the surface of the sphere x^2 + y^2 + z^2 = a^2, then evaluate iiint_S (x(z - z)dydz + (y + z)dzdx + (x + y)dxdy) using Gauss' divergence theorem.

Write down the basic algorithm for solving the equation x e^{x} - 1 = 0 by bisection method, correct to 4 decimal places.

Find the equivalent of numbers given in a specified number system to the system mentioned against them : (i) (11011.101)_{2} to decimal system (ii) (100011110000.00101100)_{2} to hexadecimal system (iii) (C4F2)_{16} to decimal system (iv) (418)_{10} to binary system.

Determine which of (partial f)/(partial x) (0, 0) and (partial f)/(partial y) (0, 0) exists and which does not exist, for f(x, y) = xy^2 / (x^2 + y^4), if y != 0 = 0, if y = 0

Suppose R be the set of all real numbers and f : R \rightarrow R is a function such that the following equations hold for all a, y \in R : (i) f(x \cdot y) = f(x) + f(y) (ii) f(xy) = f(x) f(y). Show that \forall x \in R either f(x) = 0, or f(x) = x.

Solve : y'' + 16y = 32 sec 2x

Find the inverse Laplace transform of (3s^2 + 3s - 16) / ((s - 1)(s - 2)(s + 3)).

The Hamiltonian of a mechanical system is given by H = p_{1}q_{1} - aq_{1}^{2} + bg_{2}^{2} - p_{2}q_{2}, where a, b are the constants. Solve the Hamiltonian equations and show that \frac{p_{2}}{q_{1}} = \text{constant}.

Evaluate the line integral int_C (-y^3 dx - x^3 dy + z^3 dz) using Stokes' theorem. Here C is the intersection of the cylinder x^2 + y^2 = 1 and the plane x + y + z = 1. The orientation on C corresponds to counterclockwise motion in the xy plane.

Solve the partial differential equation : (2D^{2} - 5DD' + 2D'^{2})z = 5\sin(2x + y) + 24(y - x) e^{x-1}, where D = \frac{\partial}{\partial x}, D' = \frac{\partial}{\partial y}.

Find the equation of the plane parallel to 3x - y + 3z = 8 and passing through the point (1, 1, 1).

Show that the quotient group of (R, +) modulo Z is isomorphic to the multiplicative group of complex numbers on the unit circle in the complex plane. Here R is the set of real numbers and Z is the set of integers.

Find the Laurent's series which represent the function \frac{1}{(1+z^{2})(2-z)} when (i) |z| < 1 (ii) 1 < |z| < 2 (iii) |z| > 2.

Show by applying the residue theorem that \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+a^{2})^{2}} = \frac{\pi}{4a^{3}}, a>0.

Solve : y'' - 6y' + 12y' - 6y = 120 e^(x + 27 e^(-x))

Solve : ((dy)/(dx))^2 + y^2 = x y (dy)/(dx)

A thin annulus occupies the region 0 < a \leq r \leq b, 0 \leq \theta \leq 2\pi. The faces are insulated. Along the inner edge the temperature is maintained at 0^{\circ}, while along the outer edge the temperature is held at T = K \cos \frac{\theta}{2}, where K is a constant. Determine the temperature distribution in the annulus.

Find alpha and beta such that x^alpha y^beta is an integrating factor of (y^2 + 2xy)dx - (2xy - 2x^2)dy = 0 and solve the equation.

Solve the following linear programming problem by Big M-method and show that the problem has finite optimal solutions. Also find the value of the objective function : Minimize : Z = 3x_{1} + 5x_{2} subject to x_{1} + 2x_{2} \geq 8, 3x_{1} - 2x_{2} \leq 12, 5x_{1} + 6x_{2} \leq 60, x_{1}, x_{2} \geq 0.

Find the shortest distance from the point (1, 0) to the parabola y^2 = 4x.

In a factory there are five operators O_{1}, O_{2}, O_{3}, O_{4}, O_{5} and five machines M_{1}, M_{2}, M_{3}, M_{4}, M_{5}. The operating costs are given when the O_{i} operator operates the M_{i} machine (i, j = 1, 2, ..., 5). But there is a restriction that O_{3} cannot be allowed to operate the third machine M_{3} and O_{2} cannot be allowed to operate the fifth machine M_{5}. The cost matrix is given above. Find the optimal assignment and the optimal assignment cost also.

Find the shortest distance between the lines (a1x + b1y + c1z + d1 = 0, a2x + b2y + c2z + d2 = 0) and the z-axis.

Let F = xy^2 i + y x^2 j. Integrate (v x F) . k over the region in the first quadrant bounded by the curves y = x^2 and y = x using Green's Theorem.

Suppose the Lagrangian of a mechanical system is given by L = \frac{1}{2}m(\dot{x}^{2} + 2b\dot{x}\dot{y} + \dot{y}^{2}) - \frac{1}{2}k(a x^{2} + 2bxy - cy^{2}), where a, b, c, m(>0), k(>0) are constants and b^{2} \neq ac. Write down the Lagrangian equations of motion and identify the system.

Find the equation of the cone with (0, 0, 1) as the vertex and 2x^2 - y^2 = 4, z = 0 as the guiding curve.

Show that if A and B are similar n x n matrices, then they have the same eigenvalues.

Find the general solution of the partial differential equation : (y^{2}z - 2x^{2}p)p + (2y^{2}q - x^{2}z)q = 9x^{2}y^{2}, where p = \frac{\partial z}{\partial x}, q = \frac{\partial z}{\partial y}, and find its integral surface that passes through the curve x, y, z^{2} = 1.

Solve the initial value problem y'' - 5y' + 4y = e^(4t) y(0) = 19/12, y'(0) = 8/3

Determine if lim (x->0) (1 - 2tan^2(x))^(1/x^2) exists or not. If the limit exists, then find its value.

Find the range of p (>0) for which the series \sum_{n=1}^{\infty} \frac{1}{(1+n^{p})(2+n^{p})(3+n^{p})} is (i) absolutely convergent and (ii) conditionally convergent.

Find the limit lim (n->infinity) (1/n) sum_{r=1}^n e^(r^2/n^2).

Prove the inequality : \int_{0}^{\pi} \frac{x^{2}}{\sin x} dx < \frac{2\pi^{2}}{9}

Solve : y'' - 9y' = x^2 e^(2x)

The ellipse x^2/a^2 + y^2/b^2 = 1 revolves about the x-axis. Find the volume of the solid of revolution.

Let R be an integral domain with unit element. Show that any unit in R[x] is a unit in R.

Simplify the boolean expression : (a+b) \cdot (\overline{b} - c) + (b - \overline{c}) \cdot (b + \overline{a} + c) by using the laws of boolean algebra. From its truth table write it in an interim normal form.

Find the partial differential equation of the family of all tangent planes to the ellipsoid : x^{2} + 4y^{2} - 4z^{2} = 4, which are not perpendicular to the xy plane.

Find the values of the constants a, b, c such that the quadrature formula \int_{a}^{b} f(x) dx = h[af(a) + bf(\frac{h}{3}) + cf(h)] is exact for polynomials of as high degree as possible, and hence find the order of the truncation error.

Starting from rest in the beginning, the speed (in Km/h) of a train at different times (in minutes) is given by the above table : Using Simpson's \frac{1}{3}rd rule, find the approximate distance travelled (in Km) in 20 minutes from the beginning.

A particle projected from a given point on the ground just clears a wall of height h at a distance d from the point of projection. If the particle moves in a vertical plane and if the horizontal range is R, find the elevation of the projection.

Let v = v1i + v2j + v3k. Show that curl(curl v) = grad(div v) - v^2 v.

Find the equations to the generating lines of the paraboloid x^2 + y^2 + 2xoy + 2xyz - 5z = 0 which pass through the point (1, 1, 1).

Find the curvature and torsion of the curve r(t) = a(t - sin t)i + a(1 - cos t)j + bt k.

Express basis vectors e1 = (1, 0) and e2 = (0, 1) as linear combinations of alpha1 = (2, -1) and alpha2 = (1, 3).

Find the equation of the sphere in yz-plane passing through the points (0, 0, 0), (0, 1, 1), (1, 2, 0) and (1, 2, 3).