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UPSC CSE Mains 2023 — Mathematics Questions with Answers

All 59 Mathematics previous-year questions from UPSC CSE Mains 2023, 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 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, 0, −1) ∈ span {V₁, V₂, V₃}? Justify your answer.

   Explanation: Tests whether a given vector lies in the span (linear combination) of three vectors in ℝ⁴.

2. Q2.Linear Algebra

   Find the rank and nullity of the linear transformation T : ℝ³ → ℝ³ given by T(x, y, z) = (x + z, x + y + 2z, 2x + y + 3z).

   Explanation: Determine rank and nullity of a given linear transformation on ℝ³.

3. Q3.Calculus

   Find the values of p and q for which lim(x→0) [x(1 + p·cos x) − q·sin x] / x³ exists and equals 1.

   Explanation: Determine parameters p, q so that a given limit exists and equals 1, using series expansion.

4. Q4.Calculus

   Examine the convergence of the integral ∫₀¹ (log x)/(1 + x) dx.

   Explanation: Test convergence of an improper integral with a singularity at x = 0.

5. Q5.Analytic Geometry

   A variable plane which is at a constant distance 3p from the origin O cuts the axes in the points A, B, C respectively. Show that the locus of the centroid of the tetrahedron OABC is 9(1/x² + 1/y² + 1/z²) = 16/p².

   Explanation: Find the locus of the centroid of a tetrahedron formed by a variable plane at fixed distance from origin.

6. Q6.Linear Algebra

   If the matrix of a linear transformation T : ℝ³ → ℝ³ relative to the basis {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is [[1, 1, 2], [−1, 2, 1], [0, 1, 3]], then find the matrix of T relative to the basis {(1, 1, 1), (0, 1, 1), (0, 0, 1)}.

   Explanation: Change of basis: find the matrix of a linear transformation with respect to a new basis.

7. Q7.Calculus

   Evaluate the triple integral which gives the volume of the solid enclosed between the two paraboloids Z = 5(x² + y²) and Z = 6 − 7x² − y².

   Explanation: Compute the volume between two paraboloids using a triple integral.

8. Q8.Analytic Geometry

   Show that the equation 2x² + 3y² − 8x + 6y − 12z + 11 = 0 represents an elliptic paraboloid. Also find its principal axis and principal planes.

   Explanation: Identify a quadric as an elliptic paraboloid and determine its principal axis and planes.

9. Q9.Analytic Geometry

   The plane x/a + y/b + z/c = 1 meets the coordinate axes in A, B, C respectively. Prove that the equation of the cone generated by the lines drawn from the origin O to meet the circle ABC is yz(b/c + c/b) + zx(c/a + a/c) + xy(b/a + a/b) = 0.

   Explanation: Derive the equation of the cone formed by joining the origin to a circle cut by a plane on the axes.

10. Q10.Linear Algebra

    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 = A^(n−2) + A² − I for n ≥ 3, where I is the identity matrix of order 3. Hence, find A⁴⁰.

    Explanation: Verify Cayley-Hamilton theorem and use the recurrence to compute a high power of a matrix.

11. Q11.Calculus

    Justify whether (0, 0) is an extreme point for the function f(x, y) = 2x⁴ − 3x²y + y².

    Explanation: Examine whether the origin is an extreme point of a two-variable function.

12. Q12.Analytic Geometry

    Find the equation of the sphere through the circle x² + y² + z² − 4x − 6y + 2z − 16 = 0; 3x + y + 3z − 4 = 0 in the following two cases. (i) the point (1, 0, −3) lies on the sphere. (ii) the given circle is a great circle of the sphere.

    Explanation: Find the sphere through a given circle under two conditions (point on sphere; circle as great circle).

13. Q13.Linear Algebra

    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 row-reduced echelon form.

    Explanation: Find the rank of a 4x4 matrix via row-reduced echelon form.

14. Q14.Calculus

    Trace the curve y²(x² − 1) = 2x − 1.

    Explanation: Trace a given algebraic curve, analysing symmetry, asymptotes and regions of existence.

15. Q15.Analytic Geometry

    Prove that the locus of a line which meets the lines y = mx, z = c; y = −mx, z = −c and the circle x² + y² = a², z = 0 is c²m²(cy − mzx)² + c²(yz − cmx)² = a²m²(z² − c²)².

    Explanation: Find the locus of a line meeting two given lines and a circle in 3D space.

16. Q16.Ordinary Differential Equations

    Obtain the solution of the initial-value problem dy/dx − 2xy = 2, y(0) = 1 in the form y = e^(x²)[1 + √π · erf(x)].

    Explanation: Solve a linear first-order ODE and express the solution using the error function.

17. Q17.Ordinary Differential Equations

    Given that L{f(t); p} = F(p). Show that ∫₀^∞ f(t)/t dt = ∫₀^∞ F(x) dx. Hence evaluate the integral ∫₀^∞ (e^(−t) − e^(−3t))/t dt.

    Explanation: Prove a Laplace-transform property for f(t)/t and use it to evaluate a Frullani-type integral.

18. Q18.Dynamics and Statics

    A cylinder of radius 'a' touches a vertical wall along a generating line. Axis of the cylinder is fixed horizontally. A uniform flat beam of length 'l' and weight 'W' rests with its extremities in contact with the wall and the cylinder, making an angle of 45° with the vertical. If frictional forces are neglected, then show that a/l = (√5 + 5)/(4√2). Also, find the reactions of the cylinder and wall.

    Explanation: Statics equilibrium of a beam resting against a smooth wall and cylinder; find ratio and reactions.

19. Q19.Dynamics and Statics

    A particle is moving under Simple Harmonic Motion of period T about a centre O. It passes through the point P with velocity v along the direction OP and OP = p. Find the time that elapses before the particle returns to the point P. What will be the value of p when the elapsed time is T/2?

    Explanation: Simple harmonic motion: find return time to a point and the corresponding amplitude condition.

20. Q20.Vector Analysis

    If a⃗ = sin θ î + cos θ ĵ + θ k̂, b⃗ = cos θ î − sin θ ĵ − 3 k̂, c⃗ = 2 î + 3 ĵ − 3 k̂, then find the values of the derivative of the vector function a⃗ × (b⃗ × c⃗) w.r.t. θ at θ = π/2 and θ = π.

    Explanation: Differentiate a vector triple product with respect to a parameter and evaluate at given values.

21. Q21.Ordinary Differential Equations

    Solve the differential equation: d³y/dx³ − 3·d²y/dx² + 4·dy/dx − 2y = eˣ + cos x.

    Explanation: Solve a third-order linear ODE with constant coefficients and a mixed forcing term.

22. Q22.Dynamics and Statics

    When a particle is projected from a point O₁ on the sea level with a velocity v and angle of projection θ with the horizon in a vertical plane, its horizontal range is R₁. If it is further projected from a point O₂, which is vertically above O₁ at a height h in the same vertical plane, with the same velocity v and same angle θ with the horizon, its horizontal range is R₂. Prove that R₂ > R₁ and (R₂ − R₁) : R₁ is equal to (1/2){√(1 + 2gh/(v²sin²θ)) − 1} : 1.

    Explanation: Projectile motion: compare horizontal ranges from two heights and prove the given ratio.

23. Q23.Vector Analysis

    Evaluate the integral ∬_S (3y²z² î + 4z²x² ĵ + z²y² k̂) · n̂ dS, where S is the upper part of the surface 4x² + 4y² + 4z² = 1 above the plane z = 0 and bounded by the xy-plane. Hence, verify Gauss-Divergence theorem.

    Explanation: Evaluate a surface integral over a hemispherical region and verify the Gauss divergence theorem.

24. Q24.Ordinary Differential Equations

    Find the solution of the differential equation: dy/dx = −(2xy³ + 2)/(3x²y² + 8e^(4y)).

    Explanation: Solve a first-order ODE that reduces to an exact equation.

25. Q25.Ordinary Differential Equations

    Reduce the equation x²p² + y(2x + y)p + y² = 0 to Clairaut's form by the substitution y = u and xy = v. Hence solve the equation and show that y + 4x = 0 is a singular solution of the differential equation.

    Explanation: Reduce an ODE to Clairaut's form via substitution and find its general and singular solutions.

26. Q26.Dynamics and Statics

    A solid hemisphere is supported by a string fixed to a point on its rim and to a point on a smooth vertical wall with which the curved surface is in contact. If θ is the angle of inclination of the string with vertical and φ is the angle of inclination of the plane base of the hemisphere to the vertical, then find the value of (tan φ − tan θ).

    Explanation: Statics: equilibrium of a hemisphere supported by a string against a smooth wall.

27. Q27.Vector Analysis

    If the tangent to a curve makes a constant angle θ with a fixed line, then prove that the ratio of radius of torsion to radius of curvature is proportional to tan θ. Further prove that if this ratio is constant, then the tangent makes a constant angle with a fixed direction.

    Explanation: Differential geometry of space curves: relate radius of torsion to radius of curvature for a helix-like curve.

28. Q28.Ordinary Differential Equations

    Solve the following initial value problem by using Laplace transform technique: d²y/dt² − 4·dy/dt + 3y(t) = f(t), y(0) = 1, y'(0) = 0 and f(t) is a given function of t.

    Explanation: Solve a second-order linear IVP with a general forcing function using Laplace transforms and convolution.

29. Q29.Dynamics and Statics

    A particle is projected from an apse at a distance √c from the centre of force with a velocity √((2λ/3)c³) and is moving with central acceleration λ(r⁵ − c²r). Find the path of motion of this particle. Will that be the curve x⁴ + y⁴ = c²?

    Explanation: Central orbit problem: find the path of a particle under a given central acceleration starting from an apse.

30. Q30.Vector Analysis

    For a scalar point function φ and vector point function f⃗, prove the identity ∇·(φ f⃗) = ∇φ · f⃗ + φ(∇·f⃗). Also find the value of ∇·((f(r)/r) r⃗) and then verify stated identity.

    Explanation: Prove a divergence product identity and apply it to a radial vector field.

31. Q31.Modern Algebra

    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 G'.

    Explanation: Examine existence of an onto homomorphism from a group of order 10 to one of order 6.

32. Q32.Modern Algebra

    Express the ideal 4Z + 6Z as a principal ideal in the integral domain Z.

    Explanation: Write the sum of ideals 4Z + 6Z as a single principal ideal in Z.

33. Q33.Real Analysis

    Test the convergence of the series ∑_{n=1}^{∞} [1·3·5…(2n−1)] / [2·4·6…(2n)] · x^(2n+1)/(2n+1), x > 0.

    Explanation: Test convergence of the given positive-term power series for x > 0.

34. Q34.Complex Analysis

    State the sufficient conditions for a function f(z) = f(x + iy) = u(x, y) + iv(x, y) to be analytic in its domain. Hence, show that f(z) = log z is analytic in its domain and find df/dz.

    Explanation: State analyticity sufficient conditions, prove log z analytic, and find its derivative.

35. Q35.Linear Programming

    A person requires 24, 24 and 20 units of chemicals A, B and C respectively for his garden. Product P contains 2, 4 and 1 units of chemicals A, B and C respectively per jar and product Q contains 2, 1 and 5 units of chemicals A, B and C respectively per jar. If a jar of P costs ₹30 and a jar of Q costs ₹50, then how many jars of each should be purchased in order to minimize the cost and meet the requirements?

    Explanation: Formulate and solve an LP to minimize cost meeting chemical requirement constraints.

36. Q36.Modern Algebra

    Prove that a non-commutative group of order 2p, where p is an odd prime, must have a subgroup of order p.

    Explanation: Prove existence of an order-p subgroup in a non-abelian group of order 2p.

37. Q37.Real Analysis

    Using the method of Lagrange's multipliers, find the minimum and maximum distances of the point P(2, 6, 3) from the sphere x² + y² + z² = 4.

    Explanation: Find extreme distances from a point to a sphere using Lagrange multipliers.

38. Q38.Complex Analysis

    Evaluate ∫_0^{2π} cos2θ / (5 + 4cosθ) dθ using contour integration.

    Explanation: Evaluate a real trigonometric integral via contour integration on the unit circle.

39. Q39.Modern Algebra

    Prove that x² + 1 is an irreducible polynomial in Z_3[x]. Further show that the quotient ring Z_3[x] / ⟨x² + 1⟩ is a field of 9 elements.

    Explanation: Prove irreducibility of x²+1 over Z_3 and that the quotient ring is a 9-element field.

40. Q40.Complex Analysis

    Prove that u(x, y) = e^x (x cos y − y sin y) is harmonic. Find its conjugate harmonic function v(x, y) and express the corresponding analytic function f(z) in terms of z.

    Explanation: Show u harmonic, construct conjugate harmonic v, and write analytic f(z) in z.

41. Q41.Linear Programming

    Solve the following linear programming problem by Big M method: Minimize Z = 2x₁ + 3x₂ subject to x₁ + x₂ ≥ 9, x₁ + 2x₂ ≥ 15, 2x₁ − 3x₂ ≤ 9, x₁, x₂ ≥ 0. Is the optimal solution unique? Justify your answer.

    Explanation: Solve LP by Big-M simplex and determine uniqueness of the optimal solution.

42. Q42.Real Analysis

    Prove that the oscillation of a real-valued bounded function f defined on [a, b] is the supremum of the set { |f(x₁) − f(x₂)| : x₁, x₂ ∈ [a, b] }.

    Explanation: Prove oscillation of a bounded function equals the supremum of pairwise value differences.

43. Q43.Complex Analysis

    Classify the singular point z = 0 of the function f(z) = e^z / (z − sin z) and obtain the principal part of its Laurent series expansion.

    Explanation: Classify singularity at z=0 and find the principal part of the Laurent expansion.

44. Q44.Linear Programming

    A department head has 5 subordinates and 5 jobs to be performed. The time (in hours) that each subordinate will take to perform each job is given in the matrix below: Subordinate I: A=4, B=9, C=4, D=12, E=4; Subordinate II: A=15, B=11, C=20, D=5, E=8; Subordinate III: A=17, B=7, C=15, D=12, E=18; Subordinate IV: A=9, B=13, C=11, D=9, E=14; Subordinate V: A=6, B=11, C=12, D=9, E=14. How should the jobs be assigned, one to each subordinate, so as to minimize the total time? Also, obtain the total minimum time to perform all the jobs if the subordinate IV cannot be assigned job C.

    Explanation: Solve an assignment problem (Hungarian method) including a restricted assignment case.

45. Q45.Partial Differential Equations

    By eliminating the arbitrary functions f and g from z = f(x² − y) + g(x² + y), form partial differential equation.

    Explanation: Form a PDE by eliminating two arbitrary functions from the given relation.

46. Q46.Numerical Analysis

    Given dy/dx = (y² − x)/(y² + x) with initial condition y = 1 at x = 0. Find the value of y for x = 0·4 by Euler's method, correct to 4 decimal places, taking step length h = 0·1.

    Explanation: Solve an IVP numerically using Euler's method to 4 decimal places.

47. Q47.Computer Programming

    Evaluate, using the binary arithmetic, the following numbers in their given system: (i) (634·235)_8 − (132·223)_8 (ii) (7AB·432)_16 − (5CA·D61)_16.

    Explanation: Perform octal and hexadecimal subtraction using binary arithmetic.

48. Q48.Mechanics and Fluid Dynamics

    A planet of mass m is revolving around the sun of mass M. The kinetic energy T and the potential energy V of the planet are given by T = (1/2) m(ṙ² + r²θ̇²) and V = G Mm (1/(2a) − 1/r), where (r, θ) are the polar coordinates of the planet at time t, G is the gravitational constant and 2a is the major axis of the ellipse (the path of the planet). Find the Hamiltonian and the Hamilton equations of the planet's motion.

    Explanation: Construct the Hamiltonian and Hamilton's equations for orbital planetary motion.

49. Q49.Mechanics and Fluid Dynamics

    In a fluid motion, there is a source of strength 2m placed at z = 2 and two sinks of strength m are placed at z = 2 + i and z = 2 − i. Find the streamlines.

    Explanation: Find streamlines for a source and two sinks using complex potential.

50. Q50.Partial Differential Equations

    Find the surface passing through the two lines z = x = 0 and z − 1 = x − y = 0, and satisfying the partial differential equation ∂²z/∂x² − 4 ∂²z/(∂x ∂y) + 4 ∂²z/∂y² = 0.

    Explanation: Find a surface satisfying a second-order PDE and passing through two given lines.

51. Q51.Numerical Analysis

    Solve the system of linear equations 7x₁ − x₂ + 2x₃ = 11, 2x₁ + 8x₂ − x₃ = 9, x₁ − 2x₂ + 9x₃ = 7 correct up to 4 significant figures by the Gauss-Seidel iterative method. Take initially guessed solution as x₁ = x₂ = x₃ = 0.

    Explanation: Solve a 3x3 linear system iteratively by Gauss-Seidel to 4 significant figures.

52. Q52.Mechanics and Fluid Dynamics

    A mechanical system with 2 degrees of freedom has the Lagrangian L = (1/2) m(ẋ² + ẏ²) − (1/2) m(w₁²x² + w₂²y²) + kxy where m, w₁, w₂, k are constants. Find the parameter θ so that under the transformation x = q₁cosθ − q₂sinθ, y = q₁sinθ + q₂cosθ the Lagrangian in terms of q₁, q₂ will not contain the product term q₁q₂. Find the Lagrange's equations w.r.t. q₁ and q₂ independent of parameter θ.

    Explanation: Find rotation angle decoupling the Lagrangian and derive the Lagrange equations.

53. Q53.Computer Programming

    Find the conjunctive normal form (CNF) of the following Boolean function: f(x, y, z, t) = x·y·z + x̄·y·(t + z̄).

    Explanation: Convert the given Boolean function to conjunctive normal form.

54. Q54.Computer Programming

    Express the Boolean function f(x, y, z) = x + (x̄·ȳ + x̄·z) + z in disjunctive normal form (DNF) and construct the truth table for the function.

    Explanation: Convert the Boolean function to disjunctive normal form and build its truth table.

55. Q55.Mechanics and Fluid Dynamics

    A perfectly rough ball is at rest within a hollow cylindrical roller. The roller is drawn along a level path with uniform velocity V. Let a and b be the radii of the ball and the roller respectively. If V² > (27/7) g(b − a), then show that the ball will roll completely round the inside of the roller.

    Explanation: Prove the velocity condition under which the ball completes a full loop inside the roller.

56. Q56.Partial Differential Equations

    Solve the partial differential equation a² ∂²u/∂x² = ∂²u/∂t², 0 < x < L, t > 0 subject to the conditions u(0, t) = 0, u(L, t) = 0, t > 0; u(x, 0) = x, (∂u/∂t)_{t=0} = 1, 0 < x < L.

    Explanation: Solve the one-dimensional wave equation with given boundary and initial conditions.

57. Q57.Partial Differential Equations

    Reduce the partial differential equation ∂²z/∂y² − ∂²z/(∂x ∂y) + ∂z/∂x − ∂z/∂y (1 + 1/x) + z/x = 0 to canonical form.

    Explanation: Reduce the given second-order PDE to its canonical form.

58. Q58.Numerical Analysis

    Compute a root of the equation log₁₀(2x + 1) − x² + 3 = 0, in the interval [0, 3], by Regula-Falsi method, correct to 6 decimal places.

    Explanation: Find a root via the Regula-Falsi method to 6 decimal places.

59. Q59.Mechanics and Fluid Dynamics

    Determine under what conditions the velocity field u = c(x² − y²), v = −2cxy, w = 0 is a solution to the Navier-Stokes momentum equations. Assuming that the conditions are met, determine the resulting pressure distribution, when z is up and the external body forces are B_x = 0 = B_y, B_z = −g.

    Explanation: Check the velocity field against Navier-Stokes equations and derive the pressure distribution.