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UPSC CSE Mains 2020 — Mathematics Questions with Answers

All 56 Mathematics previous-year questions from UPSC CSE Mains 2020, each with the correct answer and a full explanation. Practise them as a free, timed mock test with instant scoring.

1. Q1.Linear Algebra

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

   Explanation: Prove the set of n×n real magic squares forms a vector space over R and give two distinct 2×2 examples.

2. Q2.Linear Algebra

   Let M₂(R) be the vector space of all 2 × 2 real matrices. Let B = [[1, −1], [−4, 4]]. Suppose T: M₂(R) → M₂(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.

   Explanation: Determine rank and nullity of the linear map T(A)=BA on 2×2 matrices and find a nonzero A in its kernel.

3. Q3.Calculus

   Evaluate lim_{x→π/4} (tan x)^{tan 2x}.

   Explanation: Evaluate the indeterminate-form limit of (tan x)^(tan 2x) as x approaches π/4.

4. Q4.Calculus

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

   Explanation: Determine all asymptotes (vertical and oblique) of the given rational curve.

5. Q5.Analytic Geometry

   Find the equations of the tangent plane to the ellipsoid 2x² + 6y² + 3z² = 27 which passes through the line x − y − z = 0 = x − y + 2z − 9.

   Explanation: Find tangent planes to the ellipsoid that contain the given line of intersection of two planes.

6. Q6.Calculus

   Evaluate ∫₀¹ tan⁻¹(1 − 1/x) dx.

   Explanation: Evaluate the definite integral of arctan(1 − 1/x) over [0,1].

7. Q7.Linear Algebra

   Define an n × n matrix as A = I − 2u·uᵀ, where u is a unit column vector. (i) Examine if A is symmetric. (ii) Examine if A is orthogonal. (iii) Show that trace(A) = n − 2. (iv) Find A₃ₓ₃, when u = [1/3, 2/3, 2/3]ᵀ.

   Explanation: Analyse the Householder-type matrix A = I − 2uuᵀ for symmetry, orthogonality, trace, and compute a 3×3 instance.

8. Q8.Analytic Geometry

   Find the equation of the cylinder whose generators are parallel to the line x/1 = y/(−2) = z/3 and whose guiding curve is x² + y² = 4, z = 2.

   Explanation: Derive the equation of a cylinder with given generator direction and guiding curve.

9. Q9.Calculus

   Consider the function f(x) = ∫₀ˣ (t² − 5t + 4)(t² − 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].

   Explanation: Analyse critical points, local extrema, and zeros of an integral-defined function on [0,5].

10. Q10.Linear Algebra

    Let F be a subfield of complex numbers and T a function from F³ → F³ defined by T(x₁, x₂, x₃) = (x₁ + x₂ + 3x₃, 2x₁ − x₂, −3x₁ + x₂ − x₃). 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.

    Explanation: Find conditions on (a,b,c) lying in the null space of the given linear map and compute its nullity.

11. Q11.Analytic Geometry

    If the straight line x/1 = y/2 = 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.

    Explanation: Find the remaining two of three mutually perpendicular generators of the given cone.

12. Q12.Linear Algebra

    Let A = [[1, 0, 2], [2, −1, 3], [4, 1, 8]] and B = [[−11, 2, 2], [−4, 0, 1], [6, −1, −1]]. (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.

    Explanation: Compute matrix product, determinants, and solve the linear system using these matrices.

13. Q13.Analytic Geometry

    Find the locus of the point of intersection of the perpendicular generators of the hyperbolic paraboloid x²/a² − y²/b² = 2z.

    Explanation: Determine the locus of intersection points of perpendicular generators of the hyperbolic paraboloid.

14. Q14.Calculus

    Find an extreme value of the function u = x² + y² + z², subject to the condition 2x + 3y + 5z = 30, by using Lagrange's method of undetermined multiplier.

    Explanation: Use Lagrange multipliers to find the extreme value of a quadratic function under a linear constraint.

15. Q15.Ordinary Differential Equations

    Solve the following differential equation: x cos(y/x)(y dx + x dy) = y sin(y/x)(x dy − y dx).

    Explanation: Solve the given homogeneous first-order differential equation.

16. Q16.Ordinary Differential Equations

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

    Explanation: Find the orthogonal trajectories of the one-parameter family of circles through two fixed points.

17. Q17.Vector Analysis

    For what value of a, b, c is the vector field V̄ = (−4x − 3y + az)î + (bx + 3y + 5z)ĵ + (4x + cy + 3z)k̂ irrotational? Hence, express V̄ as the gradient of a scalar function φ. Determine φ.

    Explanation: Determine constants making the field irrotational and find its scalar potential.

18. Q18.Dynamics and Statics

    A uniform rod, in vertical position, can turn 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?

    Explanation: Find the equilibrium inclination of a hinged uniform rod under a horizontal force equal to half its weight.

19. Q19.Dynamics and Statics

    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 (1/2)(P²/m)·(a² − ab + b²)/(a² + ab + b²), where AB = a and BC = b.

    Explanation: Prove the impulsive kinetic energy generated when a particle-loaded rigid rod is struck by a blow.

20. Q20.Ordinary Differential Equations

    Using the method of variation of parameters, solve the differential equation y'' + (1 − cot x)y' − y cot x = sin² x, if y = e⁻ˣ is one solution of CF.

    Explanation: Solve a second-order linear ODE by variation of parameters given one CF solution.

21. Q21.Vector Analysis

    For the vector function Ā, where Ā = (3x² + 6y)î − 14yz ĵ + 20xz² k̂, calculate ∫_C Ā·dr̄ from (0, 0, 0) to (1, 1, 1) along the following paths: (i) x = t, y = t², z = t³; (ii) Straight lines joining (0, 0, 0) to (1, 0, 0), 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.

    Explanation: Compute the line integral of a vector field along three paths and discuss path-independence.

22. Q22.Dynamics and Statics

    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.

    Explanation: Use the tilting (moment) conditions about the supports to find the beam's weight.

23. Q23.Vector Analysis

    Verify the Stokes' theorem for the vector field F̄ = xy î + yz ĵ + xz k̂ on the surface S which is the part of the cylinder z = 1 − x² for 0 ≤ x ≤ 1, −2 ≤ y ≤ 2; S is oriented upwards.

    Explanation: Verify Stokes' theorem by computing both the line and surface integrals over a cylindrical patch.

24. Q24.Ordinary Differential Equations

    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?

    Explanation: Solve a variable-coefficient IVP via Laplace transform and discuss uniqueness.

25. Q25.Dynamics and Statics

    (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 λ, where V²λ = √(μ² + p²V⁴) − μ. Here μ is a constant.

    Explanation: Two-part rigid-body statics (thrust in light rod of a hung square frame) and central-orbit dynamics (least distance from a star).

26. Q26.Ordinary Differential Equations

    (i) Solve the following differential equation: (x + 1)² y'' − 4(x + 1)y' + 6y = 6(x + 1)² + sin log(x + 1). (ii) Find the general and singular solutions of the differential equation 9p²(2 − y)² = 4(3 − y), where p = dy/dx.

    Explanation: Solve a Cauchy-Euler ODE and find general and singular solutions of a Clairaut-type equation.

27. Q27.Vector Analysis

    Evaluate the surface integral ∬_S (∇ × F̄)·n̂ dS for F̄ = y î + (x − 2xz) ĵ − xy k̂ and S is the surface of the sphere x² + y² + z² = a² above the xy-plane.

    Explanation: Evaluate the flux of the curl of a vector field over the upper hemisphere.

28. Q28.Dynamics and Statics

    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 P / (M + 2mk²/r²), and find the horizontal force exerted on each axle by the truck. The axle friction and wind resistance are to be neglected.

    Explanation: Prove the acceleration of a railway truck accounting for rotational inertia of wheel-axle pairs and find the axle force.

29. Q29.Modern Algebra

    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.

    Explanation: Show only the trivial homomorphism exists from S₃ to Z₃.

30. Q30.Modern Algebra

    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.

    Explanation: Prove quotient rings of a PID have principal ideals and R/P is a PID for prime P.

31. Q31.Real Analysis

    Prove that the sequence (aₙ) satisfying the condition |aₙ₊₁ − aₙ| ≤ α|aₙ − aₙ₋₁|, 0 < α < 1 for all natural numbers n ≥ 2, is a Cauchy sequence.

    Explanation: Show a contractive sequence is Cauchy.

32. Q32.Complex Analysis

    Evaluate the integral ∫_C (z² + 3z)dz counterclockwise from (2, 0) to (0, 2) along the curve C, where C is the circle |z| = 2.

    Explanation: Evaluate a complex line integral along an arc of |z| = 2.

33. Q33.Linear Programming

    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 5 feet — 700 required; 9 feet — 400 required; 7 feet — 300 required. 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.

    Explanation: Formulate the trim-loss cutting-stock problem as a standard-form LP and give a basic feasible solution.

34. Q34.Modern Algebra

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

    Explanation: Prove a cyclic group of order n has exactly φ(n) generators.

35. Q35.Real Analysis

    Prove that the function f(x) = sin x² is not uniformly continuous on the interval [0, ∞).

    Explanation: Show sin x² fails uniform continuity on [0, ∞).

36. Q36.Complex Analysis

    Using contour integration, evaluate the integral ∫₀^{2π} 1/(3 + 2 sin θ) dθ.

    Explanation: Evaluate a real trigonometric integral via the residue theorem.

37. Q37.Modern Algebra

    Let R be a finite field of characteristic p (> 0). Show that the mapping f : R → R defined by f(a) = aᵖ, ∀ a ∈ R is an isomorphism.

    Explanation: Show the Frobenius map is an automorphism of a finite field.

38. Q38.Linear Programming

    Solve the linear programming problem using simplex method: Minimize z = − 6x₁ − 2x₂ − 5x₃ subject to 2x₁ − 3x₂ + x₃ ≤ 14, − 4x₁ + 4x₂ + 10x₃ ≤ 46, 2x₁ + 2x₂ − 4x₃ ≤ 37, x₁ ≥ 2, x₂ ≥ 1, x₃ ≥ 3.

    Explanation: Solve an LP minimization with lower-bounded variables by the simplex method.

39. Q39.Real Analysis

    If u = tan⁻¹((x³ + y³)/(x − y)), x ≠ y, then show that x²(∂²u/∂x²) + 2xy(∂²u/∂x∂y) + y²(∂²u/∂y²) = (1 − 4 sin²u) sin 2u.

    Explanation: Use Euler's theorem on homogeneous functions to verify the given second-order PDE identity.

40. Q40.Complex Analysis

    If v(r, θ) = (r − 1/r) sin θ, r ≠ 0, then find an analytic function f(z) = u(r, θ) + iv(r, θ).

    Explanation: Construct the analytic function from the given harmonic conjugate in polar form.

41. Q41.Real Analysis

    Show that ∫₀^{π/2} (sin²x)/(sin x + cos x) dx = (1/√2) logₑ(1 + √2).

    Explanation: Evaluate the definite trigonometric integral to the stated closed form.

42. Q42.Linear Programming

    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. Cost matrix (Sources S₁, S₂, S₃ to Destinations D₁, D₂, D₃, D₄): S₁: 10, 0, 20, 11 (Availability 15); S₂: 12, 8, 9, 20 (Availability 25); S₃: 0, 14, 16, 18 (Availability 10). Demand: D₁ = 5, D₂ = 20, D₃ = 15, D₄ = 10.

    Explanation: Find IBFS by VAM and obtain the optimal transportation cost.

43. Q43.Partial Differential Equations

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

    Explanation: Form the PDE by eliminating arbitrary functions and classify it.

44. Q44.Numerical Analysis

    Show that the equation f(x) = cos(π(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.

    Explanation: Locate roots and compute the negative root to 4 decimals by Newton-Raphson.

45. Q45.Computer Programming

    Let g(w, x, y, z) = (w + x + y)(x + ȳ + z)(w + ȳ) 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.

    Explanation: Obtain the CNF and express the Boolean function as a product of maxterms.

46. Q46.Partial Differential Equations

    Solve the partial differential equation: (D³ − 2D²D′ − DD′² + 2D′³)z = e^{2x+y} + sin(x − 2y); D ≡ ∂/∂x, D′ ≡ ∂/∂y.

    Explanation: Solve a third-order linear PDE with constant coefficients (CF plus PI).

47. Q47.Mechanics and Fluid Dynamics

    Prove that the moment of inertia of a triangular lamina ABC about any axis through A in its plane is (M/6)(β² + βγ + γ²) where M is the mass of the lamina and β, γ are respectively the length of perpendiculars from B and C on the axis.

    Explanation: Derive the moment of inertia of a triangular lamina about an axis through a vertex.

48. Q48.Partial Differential Equations

    Find the integral surface of the partial differential equation: (x − y)y²(∂z/∂x) + (y − x)x²(∂z/∂y) = (x² + y²)z that contains the curve xz = a³, y = 0 on it.

    Explanation: Find the integral surface of a Lagrange PDE through a given curve.

49. Q49.Numerical Analysis

    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.

    Explanation: Set up and iterate the Gauss-Seidel scheme and compare with the exact solution.

50. Q50.Mechanics and Fluid Dynamics

    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 x² + y² = R², 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 F⃗ = −kr⃗, k is a constant.

    Explanation: Write the Hamiltonian and derive the equations of motion for a particle on a cylinder under a central restoring force.

51. Q51.Partial Differential Equations

    Find the solution of the partial differential equation: z = ½(p² + q²) + (p − x)(q − y); p ≡ ∂z/∂x, q ≡ ∂z/∂y which passes through the x-axis.

    Explanation: Solve a non-linear first-order PDE by Charpit's method through the x-axis.

52. Q52.Numerical Analysis

    Find a quadrature formula ∫₀¹ f(x) dx/√(x(1 − x)) = α₁ f(0) + α₂ f(½) + α₃ f(1) which is exact for polynomials of highest possible degree. Then use the formula to evaluate ∫₀¹ dx/√(x − x³) (correct up to three decimal places).

    Explanation: Construct a weighted quadrature formula and apply it to a singular integral.

53. Q53.Mechanics and Fluid Dynamics

    A velocity potential in a two-dimensional fluid flow is given by φ(x, y) = xy + x² − y². Find the stream function for this flow.

    Explanation: Find the stream function from the given velocity potential using Cauchy-Riemann relations.

54. Q54.Partial Differential Equations

    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 = 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 horizontal tension / mass per unit length = c².

    Explanation: Solve the wave equation for a plucked string with triangular initial profile.

55. Q55.Numerical Analysis

    Write the three point Lagrangian interpolating polynomial relative to the points x₀, x₀ + ε and x₁. Then by taking the limit ε → 0, establish the relation f(x) = ((x₁ − x)(x + x₁ − 2x₀))/(x₁ − x₀)² f(x₀) + ((x − x₀)(x₁ − x))/(x₁ − x₀) f′(x₀) + ((x − x₀)²)/(x₁ − x₀)² f(x₁) + E(x) where E(x) = (1/6)(x − x₀)²(x − x₁)f‴(ξ) is the error function and min.(x₀, x₀ + ε, x₁) < ξ < max.(x₀, x₀ + ε, x₁).

    Explanation: Derive the confluent (Hermite-type) interpolation relation as a limit of three-point Lagrange interpolation.

56. Q56.Mechanics and Fluid Dynamics

    Two sources of strength m/2 are placed at the points (±a, 0). Show that at any point on the circle x² + y² = a², the velocity is parallel to the y-axis and is inversely proportional to y.

    Explanation: Show the velocity due to two equal sources is parallel to the y-axis and inversely proportional to y on the given circle.