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UPSC CSE Mains 2024 — Mathematics Questions with Answers

All 58 Mathematics previous-year questions from UPSC CSE Mains 2024, each with the correct answer and a full explanation. Practise them as a free, timed mock test with instant scoring.

1. Q1.Linear Algebra

   Let H be a subspace of R^4 spanned by the vectors v1 = (1, -2, 5, -3), v2 = (2, 3, 1, -4), v3 = (3, 8, -3, -5). Then find a basis and dimension of H, and extend the basis of H to a basis of R^4.

   Explanation: Row-reduce the spanning vectors to find a basis and dimension of the subspace, then extend to a basis of R^4 by adding standard vectors.

2. Q2.Linear Algebra

   Let T : R^3 -> R^3 be a linear operator and B = {v1, v2, v3} be a basis of R^3 over R. Suppose that Tv1 = (1, 1, 0), Tv2 = (1, 0, -1), Tv3 = (2, 1, -1). Find a basis for the range space and null space of T.

   Explanation: Use the images of the basis vectors to determine the rank and a basis of the range, then apply rank-nullity to find a basis of the null space.

3. Q3.Calculus

   Discuss the continuity of the function f(x) = 1/(1 - e^(-1/x)) for x != 0, and f(x) = 0 for x = 0, for all values of x.

   Explanation: Examine left-hand and right-hand limits at x = 0 (where e^(-1/x) behaves differently) and continuity elsewhere.

4. Q4.Calculus

   Expand ln(x) in powers of (x - 1) by Taylor's theorem and hence find the value of ln(1.1) correct up to four decimal places.

   Explanation: Apply Taylor's series expansion of ln(x) about x = 1 and evaluate at x = 1.1 to four-decimal accuracy.

5. Q5.Analytic Geometry

   Find the equation of the right circular cylinder which passes through the circle x^2 + y^2 + z^2 = 9, x - y + z = 3.

   Explanation: Determine the axis (normal to the plane through the sphere's centre) and use the radius of the given circle to form the right circular cylinder equation.

6. Q6.Linear Algebra

   Consider a linear operator T on R^3 over R defined by T(x, y, z) = (2x, 4x - y, 2x + 3y - z). Is T invertible? If yes, justify your answer and find T^(-1).

   Explanation: Form the matrix of T, check non-zero determinant for invertibility, then invert to obtain T^(-1).

7. Q7.Calculus

   If u = (x + y)/(1 - xy) and v = tan^(-1) x + tan^(-1) y, then find d(u, v)/d(x, y). Are u and v functionally related? If yes, find the relationship.

   Explanation: Compute the Jacobian d(u,v)/d(x,y); a zero Jacobian implies functional dependence, found via the tangent addition identity u = tan(v).

8. Q8.Analytic Geometry

   Find the image of the line x = 3 - 6t, y = 2t, z = 3 + 2t in the plane 3x + 4y - 5z + 26 = 0.

   Explanation: Find where the line meets the plane and reflect a point of the line across the plane to obtain the image line.

9. Q9.Linear Algebra

   Let V = M_{2x2}(R) denote a vector space over the field of real numbers. Find the matrix of the linear mapping phi : V -> V given by phi(v) = [[1, 2], [3, -1]] v with respect to the standard basis of M_{2x2}(R), and hence find the rank of phi. Is phi invertible? Justify your answer.

   Explanation: Represent left-multiplication by the 2x2 matrix as a 4x4 matrix on the standard basis of M_{2x2}, then read off rank and invertibility.

10. Q10.Calculus

    Find the volume of the greatest cylinder which can be inscribed in a cone of height h and semi-vertical angle alpha.

    Explanation: Set up the cylinder volume as a function of its height inside the cone and maximize using calculus (maxima-minima).

11. Q11.Analytic Geometry

    Find the vertex of the cone 4x^2 - y^2 + 2z^2 + 2xy - 3yz + 12x - 11y + 6z + 4 = 0.

    Explanation: The vertex is found by setting the partial derivatives of the conic with respect to x, y, z (and the homogeneity condition) to zero and solving.

12. Q12.Linear Algebra

    Let A = [[3, 2, 4], [2, 0, 2], [4, 2, 3]] be a 3x3 matrix. Find the eigenvalues and the corresponding eigenvectors of A. Hence find the eigenvalues and the corresponding eigenvectors of A^(-15), where A^(-15) = (A^(-1))^15.

    Explanation: Solve the characteristic equation for eigenvalues/eigenvectors of A; eigenvectors are preserved while eigenvalues map to lambda^(-15) for A^(-15).

13. Q13.Calculus

    Using double integration, find the area lying inside the cardioid r = a(1 + cos theta) and outside the circle r = a.

    Explanation: Set up a polar double integral between the circle r = a and the cardioid over the appropriate theta range and evaluate the area.

14. Q14.Analytic Geometry

    Find the equation of the sphere which touches the plane 3x + 2y - z + 2 = 0 at the point (1, -2, 1) and cuts orthogonally the sphere x^2 + y^2 + z^2 - 4x + 6y + 4 = 0.

    Explanation: Use the tangency condition at the given point on the plane plus the orthogonal-intersection condition to determine the sphere's equation.

15. Q15.Ordinary Differential Equations

    Find the orthogonal trajectories of the family of curves r = c(sec theta + tan theta), where c is a parameter.

    Explanation: Eliminate the parameter to get the differential equation of the family, replace dr/dtheta by the orthogonality relation, and solve for the orthogonal trajectories.

16. Q16.Ordinary Differential Equations

    Solve the integral equation y(t) = cos t + integral from 0 to t of y(x) cos(t - x) dx using Laplace transform.

    Explanation: Apply the Laplace transform using the convolution theorem on the integral term, solve for Y(s), and invert to obtain y(t).

17. Q17.Vector Analysis

    At any time t (in seconds), the coterminous edges of a variable parallelepiped are represented by the vectors alpha = t i + (t + 1) j + (2t + 1) k, beta = 2t i + (3t - 1) j + t k, gamma = i + 3t j + k. What is the rate of change of the vectorial area of the parallelogram, whose coterminous edges are alpha and gamma? Also find the rate of change of the volume of the parallelepiped at t = 1 second.

    Explanation: Vectorial area = (1/2)(alpha x gamma) and volume = scalar triple product [alpha beta gamma]; differentiate each with respect to t and evaluate the volume rate at t = 1.

18. Q18.Dynamics and Statics

    A solid hemisphere rests in equilibrium on a solid sphere of equal radius. Determine the stability of the equilibrium in the two situations - (i) when the curved surface and (ii) when the flat surface of the hemisphere rests on the sphere.

    Explanation: Apply the stability criterion comparing the height of the centre of gravity with the radii of curvature of the contacting surfaces for both orientations.

19. Q19.Vector Analysis

    Let C be a plane curve r(t) = f(t) i + g(t) j, where f and g have second-order derivatives. Show that the curvature at a point is given by kappa = |f'(t) g''(t) - g'(t) f''(t)| / ([f'(t)]^2 + [g'(t)]^2)^(3/2). What is the value of torsion tau at any point of this curve?

    Explanation: Derive curvature from the cross product of r' and r'' for a plane curve; torsion is zero because the curve is planar.

20. Q20.Vector Analysis

    Show that the principal normals at two consecutive points of a curve do not intersect unless torsion tau is zero.

    Explanation: Use the Serret-Frenet formulae to show consecutive principal normals are skew unless the torsion vanishes.

21. Q21.Dynamics and Statics

    A regular tetrahedron, formed of six light rods, each of length l, rests on a smooth horizontal plane. A ring of weight W and radius r is supported by the slant sides. Using the principle of virtual work, find the stress in any of the horizontal sides.

    Explanation: Express the geometry of the tetrahedron, give a virtual displacement, and equate virtual work of W and the rod stresses to solve for the stress in a horizontal rod.

22. Q22.Dynamics and Statics

    A particle executes simple harmonic motion such that in two of its positions, velocities are u and v, and the two corresponding accelerations are f1 and f2. For what value(s) of k, the distance between the two positions is k(v^2 - u^2)? Show also that the amplitude of the motion is (1/(f2^2 - f1^2)) [(u^2 - v^2)(u^2 f2^2 - v^2 f1^2)]^(1/2).

    Explanation: Use the SHM relations v^2 = omega^2(a^2 - x^2) and f = omega^2 x to relate the two positions, find k, and derive the amplitude.

23. Q23.Ordinary Differential Equations

    Find the second solution of the differential equation xy'' + (x - 1)y' - y = 0 using u(x) = -e^(-x) as one of the solutions.

    Explanation: Apply reduction of order with the known solution u(x) = -e^(-x) to obtain the linearly independent second solution.

24. Q24.Ordinary Differential Equations

    Find the general solution of the differential equation x^2 y'' - 2xy' + 2y = x^3 sin x by the method of variation of parameters.

    Explanation: Solve the Cauchy-Euler homogeneous part, then apply variation of parameters with the non-homogeneous term x^3 sin x.

25. Q25.Ordinary Differential Equations

    State uniqueness theorem for the existence of unique solution of the initial value problem dy/dx = f(x, y), y(x0) = y0 in the rectangular region R : |x - x0| <= a, |y - y0| <= b. Test the existence and uniqueness of the solution of the initial value problem dy/dx = 2*sqrt(y), y(1) = 0, in a suitable rectangle R. If more than one solution exist, then find all the solutions.

    Explanation: State the Lipschitz/Picard uniqueness theorem; show f = 2*sqrt(y) is not Lipschitz at y = 0, giving non-uniqueness, then list the multiple solutions.

26. Q26.Dynamics and Statics

    A heavy particle hanging vertically from a fixed point by a light inextensible string of length l starts to move with initial velocity u in a circle so as to make a complete revolution in a vertical plane. Show that the sum of tensions at the ends of any diameter is constant.

    Explanation: Use energy conservation and the radial equation of motion for vertical circular motion; sum the tensions at diametrically opposite points to show it is constant.

27. Q27.Vector Analysis

    State Stokes' theorem and verify it for the vector field F = xy i + yz j + zx k over the surface S, which is the upwardly oriented part of the cylinder z = 1 - x^2, for 0 <= x <= 1, -2 <= y <= 2.

    Explanation: Compute both the surface integral of curl F over the parabolic cylinder patch and the line integral of F around its boundary to verify Stokes' theorem.

28. Q28.Ordinary Differential Equations

    Using Laplace transform, solve the initial value problem y'' + 2y' + 5y = delta(t - 2), y(0) = 0, y'(0) = 0, where delta(t - 2) denotes the Dirac delta function.

    Explanation: Take Laplace transforms (delta(t-2) gives e^(-2s)), solve for Y(s), and invert using the second shifting theorem to get the response after t = 2.

29. Q29.Vector Analysis

    Using Gauss divergence theorem, evaluate the integral of (y^2 i + xz^3 j + (z - 1)^2 k) . n dS over the surface S of the region bounded by the cylinder x^2 + y^2 = 16 and the planes z = 1 and z = 5.

    Explanation: Convert the surface integral to a volume integral of div F over the cylinder using the divergence theorem and evaluate in cylindrical coordinates.

30. Q30.Dynamics and Statics

    A particle moves with a central acceleration mu(3/r^3 + d^2/r^5) being projected from a distance d at an angle 45 degrees with a velocity equal to that in a circle at the same distance. Prove that the time it takes to reach the centre of force is (d^2/sqrt(2*mu)) (2 - pi/2).

    Explanation: Use the central-orbit differential equation and energy/angular-momentum relations with the given projection conditions to integrate the time to reach the centre of force.

31. Q31.Modern Algebra

    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 subgroup of order m.

    Explanation: Modern Algebra: use Sylow theory / counting of subgroups of prime order m in a group of order mn.

32. Q32.Complex Analysis

    If w = f(z) is an analytic function of z, then show that (∂²/∂x² + ∂²/∂y²) log |f'(z)| = 0.

    Explanation: Complex Analysis: log|f'(z)| is harmonic since f'(z) is analytic and non-zero; apply Laplacian.

33. Q33.Real Analysis

    Test the convergence of the integral ∫₀² (log x)/√(2 − x) dx.

    Explanation: Real Analysis: improper integral with singularities at x=0 (log) and x=2 (1/√(2-x)); test convergence at both limits.

34. Q34.Complex Analysis

    If φ and ψ are functions of x and y satisfying Laplace equation, then show that f(z) = p + iq, i = √(−1), is an analytic function, where p = ∂φ/∂y − ∂ψ/∂x and q = ∂φ/∂x + ∂ψ/∂y.

    Explanation: Complex Analysis: verify Cauchy-Riemann equations for p and q using the fact that φ, ψ are harmonic.

35. Q35.Linear Programming

    Use two phase method to solve the following linear programming problem: Maximize z = x₁ + 2x₂ subject to x₁ − x₂ ≥ 3, 2x₁ + x₂ ≤ 10, x₁, x₂ ≥ 0.

    Explanation: Linear Programming: apply the two-phase simplex method (Phase I to find a feasible basis, Phase II to optimize).

36. Q36.Real Analysis

    Using Cauchy's general principle of convergence, examine the convergence of the sequence ⟨fₙ⟩, where fₙ = 1 + 1/1! + 1/2! + … + 1/n!.

    Explanation: Real Analysis: show ⟨fₙ⟩ is Cauchy (the tail Σ1/k! → 0), hence convergent (limit e).

37. Q37.Modern Algebra

    Show that every homomorphic image of an abelian group is abelian, but the converse is not necessarily true.

    Explanation: Modern Algebra: homomorphic image of abelian is abelian; counterexample for converse (non-abelian group with abelian image, e.g. S₃ → S₃/A₃).

38. Q38.Complex Analysis

    Find the function which is analytic inside and on the circle C : z = e^{iθ}, 0 ≤ θ ≤ 2π and has the value ((a² − 1)cos θ + i(a² + 1)sin θ)/(a⁴ − 2a²cos 2θ + 1) on the circumference of C, where a² > 1.

    Explanation: Complex Analysis: express boundary value in terms of z = e^{iθ} and identify the analytic function (partial fractions / Poisson-type reconstruction).

39. Q39.Complex Analysis

    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 these poles.

    Explanation: Complex Analysis: poles at z = integers (from sin πz), z = 0 (double, combined with z factor) and z = −1/2; compute orders and residues.

40. Q40.Real Analysis

    Consider the series Σₙ₌₁^∞ Uₙ(x), 0 ≤ x ≤ 1, the sum of whose first n terms is given by Sₙ(x) = (1/2n²) log(1 + n⁴x²), x ∈ [0, 1]. Show that the given series can be differentiated term-by-term, though Σₙ₌₁^∞ Uₙ′(x) does not converge uniformly on [0, 1].

    Explanation: Real Analysis: term-by-term differentiation valid as Sₙ→0 and Sₙ′→0 pointwise to the correct limit, yet ΣUₙ′ fails uniform convergence on [0,1].

41. Q41.Linear Programming

    Using duality principle, solve the following linear programming problem: Minimize z = 4x₁ + 3x₂ + x₃ subject to x₁ + 2x₂ + 4x₃ ≥ 12, 3x₁ + 2x₂ + x₃ ≥ 8, x₁, x₂, x₃ ≥ 0.

    Explanation: Linear Programming: form the dual (a maximization with 2 variables), solve it, and read the primal optimum from complementary slackness.

42. Q42.Modern Algebra

    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 S is prime but not a maximal ideal of Z[x].

    Explanation: Modern Algebra (Rings): Z[x]/(x) ≅ Z is an integral domain (so (x) prime) but not a field (so (x) not maximal).

43. Q43.Real Analysis

    Find the upper and lower Riemann integrals for the function f defined on [0, 1] as follows: f(x) = (1 − x²)^{1/2} if x is rational, and f(x) = (1 − x) if x is irrational. Hence, show that f is not Riemann integrable on [0, 1].

    Explanation: Real Analysis: compute upper integral (using (1−x²)^{1/2}) and lower integral (using (1−x)); they differ, so f is not Riemann integrable.

44. Q44.Operations Research

    The personnel manager of a company wants to assign officers A, B and C to the regional offices at Delhi, Mumbai, Kolkata and Chennai. The cost of relocation (in thousand Rupees) of the three officers at the four regional offices are: A → Delhi 16, Mumbai 22, Kolkata 24, Chennai 20; B → Delhi 10, Mumbai 32, Kolkata 26, Chennai 16; C → Delhi 10, Mumbai 20, Kolkata 46, Chennai 30. Find the assignment which minimizes the total cost of relocation and also determine the minimum cost.

    Explanation: Operations Research: unbalanced assignment problem (3 officers, 4 offices) — add a dummy officer and apply the Hungarian method.

45. Q45.Partial Differential Equations

    Show that if f and g are arbitrary functions of their respective arguments, then u = f(x − kt + iαy) + g(x − kt − iαy) is a solution of ∂²u/∂x² + ∂²u/∂y² = (1/C²) ∂²u/∂t², where α² = 1 − k²/C².

    Explanation: PDE: compute the second partials of u and substitute to verify it satisfies the given wave-type equation.

46. Q46.Numerical Analysis

    Solve the following system of linear equations by Gauss-Jordan method: 2x + 3y − z = 5, 4x + 4y − 3z = 3, 2x − 3y + 2z = 2.

    Explanation: Numerical Analysis / Linear Algebra: reduce the augmented matrix to reduced row echelon form to read off x, y, z.

47. Q47.Computer Programming

    (i) Determine the decimal equivalent in sign magnitude form of (8D)₁₆ and (FF)₁₆. (ii) Determine the decimal equivalent of (9B2.1A)₁₆.

    Explanation: Computer Science / Number Systems: convert hexadecimal to binary then to decimal; interpret sign-magnitude for the 8-bit values.

48. Q48.Mechanics

    A rough uniform board of mass m and length 2a rests on a smooth horizontal plane and a man of mass M walks on it from one end to the other. Find the distance covered by the board during this time.

    Explanation: Mechanics (Dynamics): conservation of momentum / centre of mass stays fixed on a smooth plane; board moves 2aM/(M+m).

49. Q49.Fluid Dynamics

    The velocity potential φ of a flow is given by φ = (1/2)(x² + y² − 2z²). Determine the streamlines.

    Explanation: Fluid Dynamics: velocity = ∇φ; integrate dx/u = dy/v = dz/w to obtain the streamlines.

50. Q50.Partial Differential Equations

    Show that the solution of the two-dimensional Laplace's equation ∂²φ/∂x² + ∂²φ/∂y² = 0, x ∈ (−∞, ∞), y ≥ 0, subject to the boundary condition φ(x, 0) = f(x), x ∈ (−∞, ∞), along with φ(x, y) → 0 for |x| → ∞ and y → ∞, can be written in the form φ(x, y) = (y/π) ∫_{−∞}^{∞} f(ξ) dξ / (y² + (x − ξ)²).

    Explanation: PDE: solve the half-plane Dirichlet problem via Fourier transform; arrive at the Poisson integral formula for the upper half-plane.

51. Q51.Computer Programming

    Draw the logical circuit for the Boolean expression Y = AB̄C̄ + BC̄ + ĀB. Also, obtain the output Y (truth table) for the three input bit sequences A = 10001111, B = 00111100, C = 11000100.

    Explanation: Computer Science: draw AND/OR/NOT gates for Y, then evaluate Y bitwise across the 8-bit input sequences to fill the truth table.

52. Q52.Mechanics

    Find the moment of inertia of a quadrant of an elliptic disk x²/a² + y²/b² = 1, of mass M, about the line passing through its centre and perpendicular to its plane. Given that the density at any point is proportional to xy.

    Explanation: Mechanics: set density ρ = kxy, compute mass M and the moment of inertia ∫∫ρ(x²+y²)dA over the quadrant; express I in terms of M, a, b.

53. Q53.Partial Differential Equations

    Find the integral surface of the following quasi-linear equation (y − φ) ∂φ/∂x + (φ − x) ∂φ/∂y = x − y, which passes through the curve φ = 0, xy = 1 and through the circle x + y + φ = 0, x² + y² + φ² = a².

    Explanation: PDE: use Lagrange's auxiliary equations dx/(y−φ) = dy/(φ−x) = dφ/(x−y); find two independent integrals and fit the given curves.

54. Q54.Numerical Analysis

    Integrate f(x) = 5x³ − 3x² + 2x + 1 from x = −2 to x = 4 using (i) Simpson's 3/8 rule with width h = 1, and (ii) Trapezoidal rule with width h = 1.

    Explanation: Numerical Analysis: tabulate f at x = −2,…,4 (h=1, 6 intervals) and apply Simpson's 3/8 and the trapezoidal rule; compare with exact value.

55. Q55.Fluid Dynamics

    Let the velocity field u(x, y) = B(x² − y²)/(x² + y²)², v(x, y) = 2Bxy/(x² + y²)², w(x, y) = 0, where B is a constant, satisfy the equations of motion for inviscid incompressible flow. Determine the pressure associated with this velocity field.

    Explanation: Fluid Dynamics: substitute the velocity field into Euler's equations for inviscid incompressible flow and integrate to find the pressure p.

56. Q56.Partial Differential Equations

    Solve the partial differential equation ∂/∂y(∂φ/∂x + φ) + 2x²y(∂φ/∂x + φ) = 0 by transforming it to the canonical form.

    Explanation: PDE: substitute ψ = ∂φ/∂x + φ to reduce to a first-order ODE in y; solve and back-substitute to get φ.

57. Q57.Numerical Analysis

    Using Newton's forward difference formula for interpolation, estimate the value of f(2.5) from the following data: x : 1, 2, 3, 4, 5, 6 ; f(x) : 0, 1, 8, 27, 64, 125.

    Explanation: Numerical Analysis: build the forward difference table and apply Newton's forward interpolation at x = 2.5.

58. Q58.Fluid Dynamics

    Suppose an infinite liquid contains two parallel, equal and opposite rectilinear vortices at a distance 2a. Show that the streamlines relative to the vortex are given by the equation log[(x² + (y − a)²)/(x² + (y + a)²)] + y/a = C, where C is a constant, the origin is the middle point of the join, and the line joining the vortices is the axis of y.

    Explanation: Fluid Dynamics: write the stream function for a vortex pair and account for the relative (moving) frame to derive the given streamline equation.