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UPSC Mathematics — Previous Year Questions

Every question from Mathematics papers in UPSC Civil Services Mains (2021–2026), organised by year. 96 questions in total, each with directive-word analysis, model answer points, and a full evaluation rubric.

96Questions
6Years covered

2026 Mains

16 questions · 800 marks

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Paper I 8 Qs · 400 marksPaper II 8 Qs · 400 marks
All 16 questions from 2026

Paper I

  1. Q1 (a) Find all the values of a for which the system of homogeneous equations ax + y + z = 0 x + y - z = 0 x + y - az = 0… 50 marks
  2. Q2 (a) Let T : P₂[x] → P₂[x] be a linear transformation defined by T(a₀ + a₁x + a₂x²) = –(a₀ + 2a₁ + a₂) + (2a₀ + 3a₁) x –… 50 marks
  3. Q3 (a) Given P₃[x] = a₀ + a₁x + a₂x² + a₃x³ | a₀, a₁, a₂, a₃ ∈ ℝ as a vector space over the field ℝ. (i) Let H = p(x) ∈… 50 marks
  4. Q4 (a) Let F be a subfield of complex numbers and let T be the linear transformation from F³ into F³, defined by T(x, y… 50 marks
  5. Q5 (a) If y₁ and y₂ are two solutions of dy/dx + P(x)y = Q(x), and y₂ = y₁u, then show that u = 1 + a e⁻∫(Q/y₁) dx, where… 50 marks
  6. Q6 (a) Solve x² d²y/dx² − 3x dy/dx + y = (sin(log x) + 1)/x log x. (15 marks) (b) (i) A battleship is steaming ahead with… 50 marks
  7. Q7 (a) A light elastic string of natural length l has one extremity fixed at a point A and the other attached to a stone… 50 marks
  8. Q8 (a) Solve (d^2y)/(dx²)+a^2y=sec ax by the method of variation of parameters. (15 marks) (b) AB is a uniform rod of… 50 marks

Paper II

  1. Q1 (a) Let G be a group of order pq, where p and q are primes. Show that either G is an abelian group or no non-identity… 50 marks
  2. Q2 (a) Let G be a finite group generated by two elements u and v of order 2 each. Show that G is isomorphic to the… 50 marks
  3. Q3 (a) Prove that cosh ( z + 1/z ) = a₀ + Σₙ₌₁^∞ aₙ ( zⁿ + 1/zⁿ ) , where aₙ = (1 / 2π) ∫₀²π cos nθ cosh (2 cos θ) dθ, for… 50 marks
  4. Q4 (a) Let n be a positive integer. Show that the following statements are equivalent : (i) ℤ/nℤ is an integral domain… 50 marks
  5. Q5 (a) Find the integral surface of the equations dx/(xz − y) = dy/(yz − x) = dz/(1 − z²) . (10 marks) (b) By using the… 50 marks
  6. Q6 (a) Reduce the partial differential equation y ∂²z/∂x² + (x + y) ∂²z/∂x∂y + x ∂²z/∂y² = 0 to canonical form and hence… 50 marks
  7. Q7 (a) Solve the partial differential equation (2D² - 5DD' + 2D'²) z = 24 (y - x) + sin (y - x) where D ≡ ∂/∂x , D' ≡ ∂/∂y… 50 marks
  8. Q8 (a) Two-dimensional harmonic equation in plane polar coordinates (r, θ) takes the form : ∂²V/∂r² + (1/r) ∂V/∂r + (1/r²)… 50 marks

2025 Mains

16 questions · 800 marks

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Paper I 8 Qs · 400 marksPaper II 8 Qs · 400 marks
All 16 questions from 2025

Paper I

  1. Q1 (a) Can the set {(0, 0, 0, 3), (1, 1, 0, 0), (0, 1, –1, 0)} be extended to form a basis of the vector space ℝ⁴? Justify… 50 marks
  2. Q2 (a) Let T : ℝ³ → ℝ² be a linear transformation such that T(1, 1, -1) = (1, 0), T(4, 1, 1) = (0, 1) and T(1, -1, 2) =… 50 marks
  3. Q3 (a) Reduce the following matrix to echelon form: A = 2 & -2 & 2 & 1 -3 & 6 & 0 & -1 1 & -7 & 10 & 2 (15 marks) (b) Find… 50 marks
  4. Q4 (a) Show that there is no tangent plane to the sphere x² + y² + z² - 4x + 2y - 4z + 4 = 0 that can be passed through… 50 marks
  5. Q5 (a) Solve (1-y²+(y⁴)/(x²))(dy/dx)²-2y/xdy/dx+(y²)/(x²)=0. 10 marks (b) Form the differential equation of all ellipses… 50 marks
  6. Q6 (a) If F(s) and G(s) are Laplace transforms of f(t) and g(t) respectively, then prove that L∫₀^t f(x) g(t-x) dx = F(s)… 50 marks
  7. Q7 (a) A solid sphere rests inside a fixed rough and hemispherical bowl of twice its radius. If a large amount of weight… 50 marks
  8. Q8 (a) Solve the differential equation (x + 2)(d^2y)/(dx²) - (2x + 5)dy/dx + 2y = (1 + x) e^x by the method of variation… 50 marks

Paper II

  1. Q1 (a) Let H and K be two subgroups of a group G such that o(H) > √o(G) and o(K) > √o(G). Show that H ∩ K ≠ {e}, where e… 50 marks
  2. Q2 (a) Define Cauchy sequence and prove that every convergent sequence of real numbers is a Cauchy sequence. What is the… 50 marks
  3. Q3 (a) Evaluate the integral ∮_C e^z/(z²(z+1)³) dz, C : |z| = 2. (15 marks) (b) Show that the volume of the greatest… 50 marks
  4. Q4 (a) Examine whether the mapping φ: Z[x] → Z defined by φ(f(x)) = f(0), for f(x) ∈ Z[x], is a homomorphism. Deduce that… 50 marks
  5. Q5 (a) Find the solution of the equation (D² + DD' - 2D'²)z = ysin x, where D ≡ (∂)/(∂ x) and D' ≡ (∂)/(∂ y). (10 marks)… 50 marks
  6. Q6 (a) Solve (∂² u)/(∂ x²) + (∂² u)/(∂ y²) = 0 for a rectangular plate subject to the boundary conditions u(0,y) = 0… 50 marks
  7. Q7 (a) Find the complete integral of z(p²-q²) = x-y; p≡∂z/∂x, q≡∂z/∂y. (15 marks) (b) Find the unique polynomial of degree… 50 marks
  8. Q8 (a) Find the characteristics of the partial differential equation p² + q² = 2; p ≡ ∂z/∂x, q ≡ ∂z/∂y and determine the… 50 marks

2024 Mains

16 questions · 800 marks

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Paper I 8 Qs · 400 marksPaper II 8 Qs · 400 marks
All 16 questions from 2024

Paper I

  1. Q1 (a) Let H be a subspace of R⁴ spanned by the vectors v₁ = (1, –2, 5, –3), v₂ = (2, 3, 1, –4), v₃ = (3, 8, –3, –5). Then… 50 marks
  2. Q2 (a) Consider a linear operator T on R³ over R defined by T(x, y, z) = (2x, 4x – y, 2x + 3y – z). Is T invertible? If… 50 marks
  3. Q3 (a) Let V = M₂ₓ₂(ℝ) denote a vector space over the field of real numbers. Find the matrix of the linear mapping φ: V →… 50 marks
  4. Q4 (a) Let A = 3 & 2 & 4 2 & 0 & 2 4 & 2 & 3 be a 3×3 matrix. Find the eigenvalues and the corresponding eigenvectors of… 50 marks
  5. Q5 (a) Find the orthogonal trajectories of the family of curves r = c(sec θ + tan θ), where c is a parameter. (10 marks)… 50 marks
  6. Q6 (a) A regular tetrahedron, formed of six light rods, each of length l, rests on a smooth horizontal plane. A ring of… 50 marks
  7. Q7 (a) State uniqueness theorem for the existence of unique solution of the initial value problem dy/dx = f(x, y), y(x₀) =… 50 marks
  8. Q8 (a) Using Laplace transform, solve the initial value problem y'' + 2y' + 5y = δ(t-2), y(0) = 0, y'(0) = 0 where δ(t-2)… 50 marks

Paper II

  1. Q1 (a) Let G be a finite group of order mn, where m and n are prime numbers with m > n. Show that G has at most one… 50 marks
  2. Q2 (a) Using Cauchy's general principle of convergence, examine the convergence of the sequence < fₙ >, where fₙ = 1 +… 50 marks
  3. Q3 (a) Locate the poles and their order for the function f(z) = 1/[z(sin πz)(z + 1/2)]. Also, find the residue of f(z) at… 50 marks
  4. Q4 (a) Consider the polynomial ring Z[x] over the ring Z of integers. Let S be an ideal of Z[x] generated by x. Show that… 50 marks
  5. Q5 (a) Show that if f and g are arbitrary functions of their respective arguments, then u = f(x - kt + iαy) + g(x - kt … 50 marks
  6. Q6 (a) Show that the solution of the two-dimensional Laplace's equation ∂²φ(x,y)/∂x² + ∂²φ(x,y)/∂y² = 0, x ∈ (-∞, ∞), y ≥… 50 marks
  7. Q7 (a) Find the integral surface of the following quasi-linear equation (y - φ) (∂ φ)/(∂ x) + (φ - x) (∂ φ)/(∂ y) = x - y… 50 marks
  8. Q8 (a) Solve the partial differential equation (∂)/(∂ y)((∂ φ)/(∂ x) + φ) + 2x^2y((∂ φ)/(∂ x) + φ) = 0 by transforming it… 50 marks

2023 Mains

16 questions · 800 marks

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Paper I 8 Qs · 400 marksPaper II 8 Qs · 400 marks
All 16 questions from 2023

Paper I

  1. Q1 (a) Let V₁ = (2, -1, 3, 2), V₂ = (-1, 1, 1, -3) and V₃ = (1, 1, 9, -5) be three vectors of the space ℝ⁴. Does (3, -1… 50 marks
  2. Q2 (a) If the matrix of a linear transformation T : IR³→IR³ relative to the basis (1, 0, 0), (0, 1, 0), (0, 0, 1) is 1 & 1… 50 marks
  3. Q3 Let A = 1 & 0 & 0 1 & 0 & 1 0 & 1 & 0 (i) Verify the Cayley-Hamilton theorem for the matrix A. (ii) Show that Aⁿ = Aⁿ⁻²… 50 marks
  4. Q4 (a) Find the rank of the matrix A = 1 & 2 & -1 & 0 -1 & 3 & 0 & -4 2 & 1 & 3 & -2 1 & 1 & 1 & -1 by reducing it to… 50 marks
  5. Q5 (a) Obtain the solution of the initial-value problem dy/dx - 2xy = 2, y(0) = 1 in the form y = eˣ²[1 + √π erf(x)]. (10… 50 marks
  6. Q6 (a) Solve the differential equation: d³y/dx³ - 3d²y/dx² + 4dy/dx - 2y = eˣ + cos x. (15 marks) (b) When a particle is… 50 marks
  7. Q7 (a)(i) Find the solution of the differential equation : dy/dx=-(2xy³+2)/(3x²y²+8e^(4y)) 10 (a)(ii) Reduce the equation… 50 marks
  8. Q8 (a) Solve the following initial value problem by using Laplace transform technique : d²y/dt² - 4 dy/dt + 3y(t) = f(t)… 50 marks

Paper II

  1. Q1 (a) Let G be a group of order 10 and G′ be a group of order 6. Examine whether there exists a homomorphism of G onto… 50 marks
  2. Q2 (a) Prove that a non-commutative group of order 2p, where p is an odd prime, must have a subgroup of order p. (15… 50 marks
  3. Q3 (a) Prove that x² + 1 is an irreducible polynomial in Z₃[x]. Further show that the quotient ring (Z₃[x])/(⟨ x²+1 ⟩) is… 50 marks
  4. Q4 (a) Prove that the oscillation of a real-valued bounded function f defined on [a, b] is the supremum of the set… 50 marks
  5. Q5 (a) By eliminating the arbitrary functions f and g from z = f(x² - y) + g(x² + y), form partial differential equation… 50 marks
  6. Q6 (a) Find the surface passing through the two lines z = x = 0 and z-1 = x-y = 0, and satisfying the partial differential… 50 marks
  7. Q7 (a) (i) Find the conjunctive normal form (CNF) of the following Boolean function: f(x, y, z, t) = x · y · z + x̄ · y ·… 50 marks
  8. Q8 (a) Reduce the partial differential equation ∂²z/∂y² - ∂²z/∂x∂y + ∂z/∂x - ∂z/∂y(1+1/x) + z/x = 0 to canonical form. (15… 50 marks

2022 Mains

16 questions · 800 marks

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Paper I 8 Qs · 400 marksPaper II 8 Qs · 400 marks
All 16 questions from 2022

Paper I

  1. Q1 (a) Prove that any set of n linearly independent vectors in a vector space V of dimension n constitutes a basis for V… 50 marks
  2. Q2 (a) Find all solutions to the following system of equations by row-reduced method : x₁ + 2x₂ − x₃ = 2, 2x₁ + 3x₂ + 5x₃… 50 marks
  3. Q3 (a) Let the set P = {(x, y, z)ᵀ : x − y − z = 0 and 2x − y + z = 0} be the collection of vectors of a vector space… 50 marks
  4. Q4 (a) Find a linear map T : R² → R² which rotates each vector of R² by an angle θ. Also, prove that for θ = (π)/2, T has… 50 marks
  5. Q5 (a) Show that the general solution of the differential equation dy/dx + Py = Q can be written in the form y = Q/P −… 50 marks
  6. Q6 (a) A cable of weight w per unit length and length 2l hangs from two points P and Q in the same horizontal line. Show… 50 marks
  7. Q7 (a) Verify Stokes' theorem for F⃗ = xî + z²ĵ + y²k̂ over the plane surface : x + y + z = 1 lying in the first octant… 50 marks
  8. Q8 (a) (i) Find the general and singular solutions of the differential equation : (x² - a²)p² - 2xyp + y² + a² = 0, where… 50 marks

Paper II

  1. Q1 (a) Show that the multiplicative group G = {1, -1, i, -i}, where i = √(-1), is isomorphic to the group G′ = ({0, 1, 2… 50 marks
  2. Q2 (a) Let f(x) = x² on [0, k], k > 0. Show that f is Riemann integrable on the closed interval [0, k] and ∫₀ᵏ f dx =… 50 marks
  3. Q3 (a) Evaluate ∫_C (z+4)/(z² + 2z + 5) dz, where C is |z + 1 - i| = 2. (15 marks) (b) Find the maximum and minimum values… 50 marks
  4. Q4 (a) Let R be a field of real numbers and S, the field of all those polynomials f(x) ∈ R[x] such that f(0) = 0 = f(1)… 50 marks
  5. Q5 (a) 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… 50 marks
  6. Q6 (a) Solve the heat equation ∂u/∂t = ∂²u/∂x², 0 < x < l, t > 0 subject to the conditions u(0, t) = u(l, t) = 0, u(x, 0)… 50 marks
  7. Q7 (a) Find the general solution of the partial differential equation (D² + DD′ - 6D′²)z = x² sin(x+y) where D ≡ ∂/∂x and… 50 marks
  8. Q8 (a) Reduce the following partial differential equation to a canonical form and hence solve it: yu_xx + (x+y)u_xy +… 50 marks

2021 Mains

16 questions · 800 marks

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Paper I 8 Qs · 400 marksPaper II 8 Qs · 400 marks
All 16 questions from 2021

Paper I

  1. Q1 (a) If A= 1 & -1 & 1 2 & -1 & 0 1 & 0 & 0 , then show that A² = A⁻¹ (without finding A⁻¹). (10 marks) (b) Find the… 50 marks
  2. Q2 (a) Show that the planes, which cut the cone ax² + by² + cz² = 0 in perpendicular generators, touch the cone (x²)/(b+c)… 50 marks
  3. Q3 (a)(i) If u = x² + y², v = x² - y², where x = rcosθ, y = rsinθ, then find (∂(u,v))/(∂(r,θ)). (7 marks) (a)(ii) If… 50 marks
  4. Q4 (a)(i) 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… 50 marks
  5. Q5 Solve the differential equation: d²y/dx² + 2y = x²e^(3x) + e^x cos 2x (10 marks) Solve the initial value problem… 50 marks
  6. Q6 A heavy string, which is not of uniform density, is hung up from two points. Let T₁, T₂, T₃ be the tensions at the… 50 marks
  7. Q7 (a) Verify Gauss divergence theorem for F⃗ = 2x^2yî - y^2ĵ + 4xz^2k̂ taken over the region in the first octant… 50 marks
  8. Q8 (a)(i) Find the orthogonal trajectories of the family of confocal conics (x²)/(a²+λ) + (y²)/(b²+λ) = 1; a > b > 0 are… 50 marks

Paper II

  1. Q1 (a) Let m₁, m₂, …, mₖ be positive integers and d > 0 the greatest common divisor of m₁, m₂, …, mₖ. Show that there… 50 marks
  2. Q2 (a) Find the maximum and minimum values of f(x) = x³ - 9x² + 26x - 24 for 0 ≤ x ≤ 1. (15 marks) (b) Let F be a field… 50 marks
  3. Q3 (a) Let f be an entire function whose Taylor series expansion with centre z = 0 has infinitely many terms. Show that z… 50 marks
  4. Q4 (a) Show that there are infinitely many subgroups of the additive group Q of rational numbers. (15 marks) (b) Using… 50 marks
  5. Q5 (a) Obtain the partial differential equation by eliminating arbitrary function f from the equation f(x+y+z, x²+y²+z²) =… 50 marks
  6. Q6 (a) Solve the wave equation a²∂²u/∂x² = ∂²u/∂t², 0<x<L, t>0 subject to the conditions u(0,t)=0, u(L,t)=0… 50 marks
  7. Q7 (a) Find the general solution of the partial differential equation (D² - D'² - 3D + 3D')z = xy + e^(x+2y) where D ≡… 50 marks
  8. Q8 (a) Find a complete integral of the partial differential equation p = (z + qy)² by using Charpit's method. 15 marks (b)… 50 marks

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