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UPSC CSE Mains 2021 — Mathematics Questions with Answers

All 61 Mathematics previous-year questions from UPSC CSE Mains 2021, each with the correct answer and a full explanation. Practise them as a free, timed mock test with instant scoring.

1. Q1.Linear Algebra

   If A = [[1, -1, 1], [2, -1, 0], [1, 0, 0]], then show that A^2 = A^(-1) (without finding A^(-1)).

   Explanation: Use Cayley-Hamilton theorem to express A^2 as inverse of A.

2. Q2.Linear Algebra

   Find the matrix associated with the linear operator on V_3(R) defined by T(a, b, c) = (a + b, a - b, 2c) with respect to the ordered basis B = {(0, 1, 1), (1, 0, 1), (1, 1, 0)}.

   Explanation: Express images of basis vectors in terms of the given basis to form the matrix.

3. Q3.Calculus

   Given Δ(x) = | f(x+α) f(x+2α) f(x+3α) ; f(α) f(2α) f(3α) ; f'(α) f'(2α) f'(3α) |, where f is a real valued differentiable function and α is a constant. Find lim_{x→0} Δ(x)/x.

   Explanation: Differentiate the determinant and evaluate the limit at x=0.

4. Q4.Calculus

   Show that between any two roots of e^x cos x = 1, there exists at least one root of e^x sin x - 1 = 0.

   Explanation: Apply Rolle's theorem to a suitable auxiliary function.

5. Q5.Analytic Geometry

   Find the equation of the cylinder whose generators are parallel to the line x = -y/2 = z/3 and whose guiding curve is x^2 + 2y^2 = 1, z = 0.

   Explanation: Eliminate the parameter using direction ratios of the generator and the guiding curve.

6. Q6.Analytic Geometry

   Show that the planes, which cut the cone ax^2 + by^2 + cz^2 = 0 in perpendicular generators, touch the cone x^2/(b+c) + y^2/(c+a) + z^2/(a+b) = 0.

   Explanation: Use the perpendicularity condition of generators and the tangency condition of a plane to a cone.

7. Q7.Calculus

   Given that f(x, y) = |x^2 - y^2|. Find f_xy(0, 0) and f_yx(0, 0). Hence show that f_xy(0, 0) = f_yx(0, 0).

   Explanation: Compute mixed partial derivatives at the origin from first principles.

8. Q8.Linear Algebra

   Show that S = {(x, 2y, 3x) : x, y are real numbers} is a subspace of R^3(R). Find two bases of S. Also find the dimension of S.

   Explanation: Verify closure properties and identify spanning sets to determine bases and dimension.

9. Q9.Calculus

   If u = x^2 + y^2, v = x^2 - y^2, where x = r cosθ, y = r sinθ, then find ∂(u, v)/∂(r, θ).

   Explanation: Compute the Jacobian of (u,v) with respect to (r,θ).

10. Q10.Calculus

    If ∫_0^x f(t) dt = x + ∫_x^1 t f(t) dt, then find the value of f(1).

    Explanation: Differentiate both sides using Leibniz rule and evaluate at x=1.

11. Q11.Calculus

    Express ∫_a^b (x - a)^m (b - x)^n dx in terms of Beta function.

    Explanation: Substitute to reduce the integral to standard Beta function form.

12. Q12.Analytic Geometry

    A sphere of constant radius r passes through the origin O and cuts the axes at the points A, B and C. Find the locus of the foot of the perpendicular drawn from O to the plane ABC.

    Explanation: Find intercepts on axes, equation of plane ABC, and locus of foot of perpendicular from origin.

13. Q13.Linear Algebra

    Prove that the eigen vectors, corresponding to two distinct eigen values of a real symmetric matrix, are orthogonal.

    Explanation: Use symmetry of the matrix and the eigenvalue equation to show orthogonality.

14. Q14.Linear Algebra

    For two square matrices A and B of order 2, show that trace (AB) = trace (BA). Hence show that AB - BA ≠ I_2, where I_2 is an identity matrix of order 2.

    Explanation: Use trace properties to show trace of commutator is zero, contradicting trace of identity.

15. Q15.Linear Algebra

    Reduce the following matrix to a row-reduced echelon form and hence also, find its rank: A = [[1, 3, 2, 4, 1], [0, 0, 2, 2, 0], [2, 6, 2, 6, 2], [3, 9, 1, 10, 6]].

    Explanation: Apply elementary row operations to reach RREF and count non-zero rows for rank.

16. Q16.Linear Algebra

    Find the eigen values and the corresponding eigen vectors of the matrix A = [[0, -i], [i, 0]], over the complex-number field.

    Explanation: Solve the characteristic equation over C and find eigenvectors for each eigenvalue.

17. Q17.Calculus

    Show that the entire area of the Astroid: x^(2/3) + y^(2/3) = a^(2/3) is (3/8)πa^2.

    Explanation: Use parametric form and integration to compute the enclosed area.

18. Q18.Analytic Geometry

    Find equation of the plane containing the lines (x+1)/3 = (y+3)/5 = (z+5)/7 and (x-2)/1 = (y-4)/3 = (z-6)/5. Also find the point of intersection of the given lines.

    Explanation: Use direction ratios and a common point to derive the plane and the intersection point.

19. Q19.Ordinary Differential Equations

    Solve the differential equation: d^2y/dx^2 + 2y = x^2 e^(3x) + e^x cos 2x.

    Explanation: Find complementary function and particular integral using operator methods.

20. Q20.Ordinary Differential Equations

    Solve the initial value problem: d^2y/dx^2 + 4y = e^(-2x) sin 2x; y(0) = y'(0) = 0 using Laplace transform method.

    Explanation: Apply Laplace transform, solve algebraically, and invert to find y(x).

21. Q21.Dynamics and Statics

    Two rods LM and MN are joined rigidly at the point M such that (LM)^2 + (MN)^2 = (LN)^2 and they are hanged freely in equilibrium from a fixed point L. Let ω be the weight per unit length of both the rods which are uniform. Determine the angle, which the rod LM makes with the vertical direction, in terms of lengths of the rods.

    Explanation: Use centre of gravity and equilibrium of a hinged rod system to find the vertical angle.

22. Q22.Dynamics and Statics

    If a planet, which revolves around the Sun in a circular orbit, is suddenly stopped in its orbit, then find the time in which it would fall into the Sun. Also, find the ratio of its falling time to the period of revolution of the planet.

    Explanation: Treat the fall as a degenerate elliptic orbit and apply Kepler's third law.

23. Q23.Vector Analysis

    Show that ∇^2[∇·(r⃗/r)] = 2/r^4, where r⃗ = x î + y ĵ + z k̂.

    Explanation: Compute divergence then Laplacian using vector calculus identities for radial fields.

24. Q24.Dynamics and Statics

    A heavy string, which is not of uniform density, is hung up from two points. Let T_1, T_2, T_3 be the tensions at the intermediate points A, B, C of the catenary respectively where its inclinations to the horizontal are in arithmetic progression with common difference β. Let ω_1 and ω_2 be the weights of the parts AB and BC of the string respectively. Prove that (i) Harmonic mean of T_1, T_2 and T_3 = 3T_2/(1 + 2 cos β) (ii) T_1/T_3 = ω_1/ω_2.

    Explanation: Use catenary tension relations T = T_0 sec ψ and weight-tension geometry.

25. Q25.Ordinary Differential Equations

    Solve the equation: d^2y/dx^2 + (tan x - 3 cos x) dy/dx + 2y cos^2 x = cos^4 x completely by demonstrating all the steps involved.

    Explanation: Use a change of independent variable to reduce to constant-coefficient form and solve.

26. Q26.Vector Analysis

    Evaluate ∮_C F⃗·dr⃗, where C is an arbitrary closed curve in the xy-plane and F⃗ = (-y î + x ĵ)/(x^2 + y^2).

    Explanation: Analyze the line integral depending on whether C encloses the origin (singularity).

27. Q27.Vector Analysis

    Verify Gauss divergence theorem for F⃗ = 2x^2 y î - y^2 ĵ + 4xz^2 k̂ taken over the region in the first octant bounded by y^2 + z^2 = 9 and x = 2.

    Explanation: Compute volume integral of divergence and surface flux to verify equality.

28. Q28.Ordinary Differential Equations

    Find all possible solutions of the differential equation: y^2 log y = xy (dy/dx) + (dy/dx)^2.

    Explanation: Treat as a Clairaut-type / solvable equation in dy/dx to find general and singular solutions.

29. Q29.Dynamics and Statics

    A heavy particle hangs by an inextensible string of length a from a fixed point and is then projected horizontally with a velocity √(2gh). If 5a/2 > h > a, then prove that the circular motion ceases when the particle has reached the height (1/3)(a + 2h) from the point of projection. Also, prove that the greatest height ever reached by the particle above the point of projection is (4a - h)(a + 2h)^2 / (27a^2).

    Explanation: Apply energy conservation and the condition of string tension vanishing in vertical circular motion.

30. Q30.Ordinary Differential Equations

    Find the orthogonal trajectories of the family of confocal conics x^2/(a^2 + λ) + y^2/(b^2 + λ) = 1; a > b > 0 are constants and λ is a parameter. Show that the given family of curves is self orthogonal.

    Explanation: Form the differential equation of the family and show it is unchanged under dy/dx -> -dx/dy.

31. Q31.Ordinary Differential Equations

    Find the general solution of the differential equation: x^2 (d^2y/dx^2) - 2x(1 + x)(dy/dx) + 2(1 + x)y = 0. Hence, solve the differential equation: x^2 (d^2y/dx^2) - 2x(1 + x)(dy/dx) + 2(1 + x)y = x^3 by the method of variation of parameters.

    Explanation: Find independent solutions of the homogeneous equation then apply variation of parameters.

32. Q32.Dynamics and Statics

    Describe the motion and path of a particle of mass m which is projected in a vertical plane through a point of projection with velocity u in a direction making an angle θ with the horizontal direction. Further, if particles are projected from that point in the same vertical plane with velocity 4√g, then determine the locus of vertices of their paths.

    Explanation: Use projectile motion equations and eliminate the angle to find the locus of vertices.

33. Q33.Vector Analysis

    Using Stokes' theorem, evaluate ∬_S (∇ × F⃗)·n̂ dS, where F⃗ = (x^2 + y - 4) î + 3xy ĵ + (2xz + z^2) k̂ and S is the surface of the paraboloid z = 4 - (x^2 + y^2) above the xy-plane. Here, n̂ is the unit outward normal vector on S.

    Explanation: Convert the surface integral to a line integral over the boundary circle using Stokes' theorem.

34. Q34.Modern Algebra

    Let m_1, m_2, ..., m_k be positive integers and d > 0 the greatest common divisor of m_1, m_2, ..., m_k. Show that there exist integers x_1, x_2, ..., x_k such that d = x_1·m_1 + x_2·m_2 + ... + x_k·m_k.

    Explanation: Prove the GCD of several integers is expressible as an integer linear combination of them.

35. Q35.Real Analysis

    Test the uniform convergence of the series x^4 + x^4/(1+x^4) + x^4/(1+x^4)^2 + x^4/(1+x^4)^3 + ... on [0, 1].

    Explanation: Examine whether the given geometric-type series of functions converges uniformly on [0,1].

36. Q36.Real Analysis

    If a function f is monotonic in the interval [a, b], then prove that f is Riemann integrable in [a, b].

    Explanation: Prove that monotonicity on a closed interval implies Riemann integrability.

37. Q37.Complex Analysis

    Let c : [0, 1] → ℂ be the curve, where c(t) = e^(4πit), 0 ≤ t ≤ 1. Evaluate the contour integral ∮_c dz/(2z^2 − 5z + 2).

    Explanation: Evaluate a contour integral over a circle traversed twice using the residue theorem.

38. Q38.Linear Programming

    A department of a company has five employees with five jobs to be performed. The time (in hours) that each man takes to perform each job is given in the effectiveness matrix. Assign all the jobs to these five employees to minimize the total processing time. (Jobs A–E vs Employees I–V; matrix rows A:[10,5,13,15,16], B:[3,9,18,13,6], C:[10,7,2,2,2], D:[7,11,9,7,12], E:[7,9,10,4,12].)

    Explanation: Solve a 5x5 assignment problem to minimize total processing time using the Hungarian method.

39. Q39.Real Analysis

    Find the maximum and minimum values of f(x) = x^3 − 9x^2 + 26x − 24 for 0 ≤ x ≤ 1.

    Explanation: Find the absolute extrema of a cubic polynomial on a closed interval.

40. Q40.Modern Algebra

    Let F be a field and f(x) ∈ F[x] a polynomial of degree > 0 over F. Show that there is a field F' and an imbedding q : F → F' such that the polynomial f^q ∈ F'[x] has a root in F', where f^q is obtained by replacing each coefficient a of f by q(a).

    Explanation: Construct a field extension in which a given polynomial acquires a root (Kronecker's theorem).

41. Q41.Complex Analysis

    Find the Laurent series expansion of f(z) = (z^2 − z + 1)/(z(z^2 − 3z + 2)) in the powers of (z + 1) in the region |z + 1| > 3.

    Explanation: Expand a rational function as a Laurent series in powers of (z+1) on an annular region.

42. Q42.Complex Analysis

    Let f be an entire function whose Taylor series expansion with centre z = 0 has infinitely many terms. Show that z = 0 is an essential singularity of f(1/z).

    Explanation: Show that composing a transcendental entire function with 1/z produces an essential singularity at 0.

43. Q43.Real Analysis

    Find the stationary values of x^2 + y^2 + z^2 subject to the conditions ax^2 + by^2 + cz^2 = 1 and lx + my + nz = 0. Interpret the result geometrically.

    Explanation: Find constrained stationary values using Lagrange multipliers and interpret geometrically.

44. Q44.Linear Programming

    Convert the following LPP into dual LPP: Minimize Z = x_1 − 3x_2 − 2x_3 subject to 3x_1 − x_2 + 2x_3 ≤ 7, 2x_1 − 4x_2 ≥ 12, −4x_1 + 3x_2 + 8x_3 = 10, where x_1, x_2 ≥ 0 and x_3 is unrestricted in sign.

    Explanation: Form the dual of a primal LPP with mixed constraints and an unrestricted variable.

45. Q45.Modern Algebra

    Show that there are infinitely many subgroups of the additive group ℚ of rational numbers.

    Explanation: Prove the additive group of rationals has infinitely many subgroups.

46. Q46.Complex Analysis

    Using contour integration, evaluate the integral ∫_{−∞}^{∞} sin x dx / (x(x^2 + a^2)), a > 0.

    Explanation: Evaluate a real improper integral with a pole on the real axis via contour integration.

47. Q47.Linear Programming

    Solve the following linear programming problem using Big M method: Maximize Z = 4x_1 + 5x_2 + 2x_3 subject to 2x_1 + x_2 + x_3 ≥ 10, x_1 + 3x_2 + x_3 ≤ 12, x_1 + x_2 + x_3 = 6, x_1, x_2, x_3 ≥ 0.

    Explanation: Solve an LPP with mixed constraints using the Big-M (penalty) simplex method.

48. Q48.Partial Differential Equations

    Obtain the partial differential equation by eliminating arbitrary function f from the equation f(x + y + z, x^2 + y^2 + z^2) = 0.

    Explanation: Form a first-order PDE by eliminating an arbitrary function of two arguments.

49. Q49.Numerical Analysis

    Find a positive root of the equation 3x = 1 + cos x by a numerical technique using initial values 0, π/2; and further improve the result using Newton-Raphson method correct to 8 significant figures.

    Explanation: Find a root of a transcendental equation and refine it via Newton-Raphson to 8 significant figures.

50. Q50.Computer Programming

    (i) Convert (3798.3875)_10 into octal and hexadecimal equivalents. (ii) Obtain the principal conjunctive normal form of (¬P → R) ∧ (Q ↔ P).

    Explanation: Perform base conversion of a decimal number and find the principal conjunctive normal form of a logical expression.

51. Q51.Mechanics and Fluid Dynamics

    A particle is constrained to move along a circle lying in the vertical xy-plane. With the help of the D'Alembert's principle, show that its equation of motion is ẍy − ÿx − gx = 0, where g is the acceleration due to gravity.

    Explanation: Derive the equation of motion of a constrained particle on a vertical circle using D'Alembert's principle.

52. Q52.Mechanics and Fluid Dynamics

    What arrangements of sources and sinks can have the velocity potential w = log_e(z − a^2/z)? Draw the corresponding sketch of the streamlines and prove that two of them subdivide into the circle r = a and the axis of y.

    Explanation: Identify source-sink arrangement from a complex potential and analyze the resulting streamlines.

53. Q53.Partial Differential Equations

    Solve the wave equation a^2 ∂²u/∂x² = ∂²u/∂t², 0 < x < L, t > 0 subject to the conditions u(0, t) = 0, u(L, t) = 0, u(x, 0) = (1/4)x(L − x), (∂u/∂t)|_{t=0} = 0.

    Explanation: Solve the one-dimensional wave equation with fixed ends and given initial displacement by separation of variables.

54. Q54.Computer Programming

    Obtain the Boolean function F(x, y, z) based on the table given below. Then simplify F(x, y, z) and draw the corresponding GATE network. (Truth table: 111→1, 110→1, 101→1, 100→0, 011→1, 010→0, 001→0, 000→0.)

    Explanation: Derive and minimize a Boolean function from a truth table and draw its logic gate network.

55. Q55.Mechanics and Fluid Dynamics

    Obtain the Lagrangian equation for the motion of a system of two particles of unequal masses connected by an inextensible string passing over a small smooth pulley.

    Explanation: Derive the Lagrangian equation of motion for an Atwood-machine system of two unequal masses.

56. Q56.Partial Differential Equations

    Find the general solution of the partial differential equation (D^2 − D'^2 − 3D + 3D')z = xy + e^(x+2y), where D ≡ ∂/∂x and D' ≡ ∂/∂y.

    Explanation: Find the complete solution of a linear PDE with constant coefficients (CF plus particular integral).

57. Q57.Numerical Analysis

    Solve the system of equations 3x_1 + 9x_2 − 2x_3 = 11, 4x_1 + 2x_2 + 13x_3 = 24, 4x_1 − 2x_2 + x_3 = −8 correct up to 4 significant figures by using Gauss-Seidel method after verifying whether the method is applicable in your transformed form of the system.

    Explanation: Solve a linear system iteratively by Gauss-Seidel after ensuring diagonal dominance.

58. Q58.Mechanics and Fluid Dynamics

    Show that q = λ(−y î + x ĵ)/(x^2 + y^2), (λ = constant) is a possible incompressible fluid motion. Determine the streamlines. Is the kind of the motion potential? If yes, then find the velocity potential.

    Explanation: Verify a velocity field as incompressible flow, find streamlines, and determine the velocity potential.

59. Q59.Partial Differential Equations

    Find a complete integral of the partial differential equation p = (z + qy)^2 by using Charpit's method.

    Explanation: Find a complete integral of a nonlinear first-order PDE via Charpit's method.

60. Q60.Numerical Analysis

    Derive Newton's backward difference interpolation formula and also do error analysis.

    Explanation: Derive the Newton backward-difference interpolation formula and analyze its error term.

61. Q61.Mechanics and Fluid Dynamics

    Show that for the complex potential tan⁻¹ z, the streamlines and equipotential curves are circles. Find the velocity at any point and check the singularities at z = ±i.

    Explanation: Analyze streamlines, equipotentials, velocity, and singularities for the complex potential arctan z.