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UPSC CSE Mains 2022 — Mathematics Questions with Answers

All 57 Mathematics previous-year questions from UPSC CSE Mains 2022, each with the correct answer and a full explanation. Practise them as a free, timed mock test with instant scoring.

1. Q1.Linear Algebra

   Prove that any set of n linearly independent vectors in a vector space V of dimension n constitutes a basis for V.

   Explanation: Prove n linearly independent vectors form a basis of an n-dimensional vector space.

2. Q2.Linear Algebra

   Let T : ℝ² → ℝ³ be a linear transformation such that T(1, 0)ᵀ = (1, 2, 3)ᵀ and T(1, 1)ᵀ = (−3, 2, 8)ᵀ. Find T(2, 4)ᵀ.

   Explanation: Determine the image of a given vector under a linear transformation specified on a basis.

3. Q3.Calculus

   Evaluate lim_{x→∞} (eˣ + x)^(1/x).

   Explanation: Evaluate a limit of an exponential-type expression as x tends to infinity.

4. Q4.Calculus

   Examine the convergence of ∫₀² dx / (2x − x²).

   Explanation: Test the convergence of an improper integral with singularities at the endpoints.

5. Q5.Analytic Geometry

   A variable plane passes through a fixed point (a, b, c) and meets the axes at points A, B and C respectively. Find the locus of the centre of the sphere passing through the points O, A, B and C, O being the origin.

   Explanation: Find the locus of the centre of a sphere through the origin and the axis intercepts of a variable plane.

6. Q6.Linear Algebra

   Find all solutions to the following system of equations by row-reduced method: x₁ + 2x₂ − x₃ = 2; 2x₁ + 3x₂ + 5x₃ = 5; −x₁ − 3x₂ + 8x₃ = −1.

   Explanation: Solve a system of linear equations using row-reduced echelon form.

7. Q7.Calculus

   A wire of length l is cut into two parts which are bent in the form of a square and a circle respectively. Using Lagrange's method of undetermined multipliers, find the least value of the sum of the areas so formed.

   Explanation: Minimize the sum of areas of a square and a circle using Lagrange multipliers.

8. Q8.Analytic Geometry

   If P, Q, R; P′, Q′, R′ are feet of the six normals drawn from a point to the ellipsoid x²/a² + y²/b² + z²/c² = 1, and the plane PQR is represented by lx + my + nz = p, show that the plane P′Q′R′ is given by x/(a²l) + y/(b²m) + z/(c²n) + 1/p = 0.

   Explanation: Show the plane through the remaining three normal feet of an ellipsoid satisfies a given equation.

9. Q9.Linear Algebra

   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 ℝ³(ℝ). Then (i) prove that P is a subspace of ℝ³; (ii) find a basis and dimension of P.

   Explanation: Prove a solution set is a subspace and find its basis and dimension.

10. Q10.Calculus

    Use double integration to calculate the area common to the circle x² + y² = 4 and the parabola y² = 3x.

    Explanation: Compute the common area of a circle and a parabola using double integration.

11. Q11.Analytic Geometry

    Find the equation of the sphere of smallest possible radius which touches the straight lines (x−3)/3 = (y−8)/(−1) = (z−3)/1 and (x+3)/(−3) = (y+7)/2 = (z−6)/4.

    Explanation: Find the smallest sphere tangent to two given skew straight lines.

12. Q12.Linear Algebra

    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 no eigenvalue in ℝ.

    Explanation: Find the rotation linear map and show it has no real eigenvalue for a right-angle rotation.

13. Q13.Calculus

    Trace the curve y²x² = x² − a², where a is a real constant.

    Explanation: Trace and describe the shape of a given algebraic curve.

14. Q14.Analytic Geometry

    If the plane ux + vy + wz = 0 cuts the cone ax² + by² + cz² = 0 in perpendicular generators, then prove that (b + c)u² + (c + a)v² + (a + b)w² = 0.

    Explanation: Prove a condition on plane coefficients for a plane to cut a cone in perpendicular generators.

15. Q15.Ordinary Differential Equations

    Show that the general solution of the differential equation dy/dx + Py = Q can be written in the form y = Q/P − e^(−∫P dx){C + ∫e^(∫P dx) d(Q/P)}, where P, Q are non-zero functions of x and C, an arbitrary constant.

    Explanation: Derive an alternative closed form of the general solution of a linear first-order ODE.

16. Q16.Ordinary Differential Equations

    Show that the orthogonal trajectories of the system of parabolas x² = 4a(y + a) belong to the same system.

    Explanation: Prove a family of parabolas is self-orthogonal.

17. Q17.Dynamics and Statics

    A body of weight w rests on a rough inclined plane of inclination θ, the coefficient of friction, μ, being greater than tan θ. Find the work done in slowly dragging the body a distance 'b' up the plane and then dragging it back to the starting point, the applied force being in each case parallel to the plane.

    Explanation: Find the work done dragging a body up and back on a rough inclined plane.

18. Q18.Dynamics and Statics

    A projectile is fired from a point O with velocity √(2gh) and hits a tangent at the point P(x, y) in the plane, the axes OX and OY being horizontal and vertically downward lines through the point O, respectively. Show that if the two possible directions of projection be at right angles, then x² = 2hy and then one of the possible directions of projection bisects the angle POX.

    Explanation: Establish projectile relations when the two projection directions are perpendicular.

19. Q19.Vector Analysis

    Show that A = (6xy + z³)î + (3x² − z)ĵ + (3xz² − y)k̂ is irrotational. Also find φ such that A = ∇φ.

    Explanation: Prove a vector field is irrotational and find its scalar potential.

20. Q20.Dynamics and Statics

    A cable of weight w per unit length and length 2l hangs from two points P and Q in the same horizontal line. Show that the span of the cable is 2l(1 − 2h²/(3l²)), where h is the sag in the middle of the tightly stretched position.

    Explanation: Derive the span of a hanging cable (catenary) in terms of its sag.

21. Q21.Ordinary Differential Equations

    Solve the following differential equation by using the method of variation of parameters: (x² − 1)d²y/dx² − 2x dy/dx + 2y = (x² − 1)², given that y = x is one solution of the reduced equation.

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

22. Q22.Vector Analysis

    Verify Green's theorem in the plane for ∮_C (3x² − 8y²)dx + (4y − 6xy)dy, where C is the boundary curve of the region defined by x = 0, y = 0, x + y = 1.

    Explanation: Verify Green's theorem for a line integral over a triangular region.

23. Q23.Vector Analysis

    Verify Stokes' theorem for F = xî + z²ĵ + y²k̂ over the plane surface: x + y + z = 1 lying in the first octant.

    Explanation: Verify Stokes' theorem for a vector field over a plane surface in the first octant.

24. Q24.Ordinary Differential Equations

    Solve the following initial value problem by using Laplace's transformation d²y/dt² − 3 dy/dt + 2y = h(t), where h(t) = 2 for 0 < t < 4 and h(t) = 0 for t > 4, y(0) = 0, y'(0) = 0.

    Explanation: Solve an initial value problem with a piecewise forcing function using Laplace transforms.

25. Q25.Dynamics and Statics

    Suppose a cylinder of any cross-section is balanced on another fixed cylinder, the contact of curved surfaces being rough and the common tangent line horizontal. Let ρ and ρ′ be the radii of curvature of the two cylinders at the point of contact and h be the height of centre of gravity of the upper cylinder above the point of contact. Show that the upper cylinder is balanced in stable equilibrium if h < ρρ′/(ρ + ρ′).

    Explanation: Derive the stability condition for one cylinder balanced on another.

26. Q26.Ordinary Differential Equations

    Find the general and singular solutions of the differential equation: (x² − a²)p² − 2xyp + y² + a² = 0, where p = dy/dx. Also give the geometric relation between the general and singular solutions.

    Explanation: Find the general and singular solutions of a Clairaut-type ODE and their geometric relation.

27. Q27.Ordinary Differential Equations

    Solve the following differential equation: (3x + 2)²d²y/dx² + 5(3x + 2)dy/dx − 3y = x² + x + 1.

    Explanation: Solve a Cauchy-Euler (Legendre) type linear ODE with variable coefficients.

28. Q28.Dynamics and Statics

    A chain of n equal uniform rods is smoothly jointed together and suspended from its one end A₁. A horizontal force P is applied to the other end A_{n+1} of the chain. Find the inclinations of the rods to the downward vertical line in the equilibrium configuration.

    Explanation: Find the equilibrium inclinations of a jointed chain of rods under a horizontal force.

29. Q29.Vector Analysis

    Using Gauss' divergence theorem, evaluate ∬_S F·n̂ dS, where F = xî − yĵ + (z² − 1)k̂ and S is the cylinder formed by the surfaces z = 0, z = 1, x² + y² = 4.

    Explanation: Evaluate a surface integral over a cylinder using the divergence theorem.

30. Q30.Modern Algebra

    Show that the multiplicative group G = {1, -1, i, -i}, where i = √(-1), is isomorphic to the group G' = ({0, 1, 2, 3}, +₄).

    Explanation: Establish a group isomorphism between the cyclic multiplicative group of fourth roots of unity and (Z4, +).

31. Q31.Complex Analysis

    If f(z) = u + iv is an analytic function of z, and u - v = (cos x + sin x - e^(-y)) / (2 cos x - e^y - e^(-y)), then find f(z) subject to the condition f(π/2) = 0.

    Explanation: Use Milne-Thomson method on the analytic function given u - v to reconstruct f(z).

32. Q32.Real Analysis

    Test the convergence of ∫₀^∞ (cos x / (1 + x²)) dx.

    Explanation: Examine convergence of the improper integral over [0, ∞).

33. Q33.Complex Analysis

    Expand f(z) = 1 / ((z - 1)²(z - 3)) in a Laurent series valid for the regions (i) 0 < |z - 1| < 2 and (ii) 0 < |z - 3| < 2.

    Explanation: Develop Laurent series expansions valid in two annular/punctured-disk regions about z = 1 and z = 3.

34. Q34.Linear Programming

    Use two-phase method to solve the following linear programming problem: Minimize Z = x₁ + x₂ subject to 2x₁ + x₂ ≥ 4, x₁ + 7x₂ ≥ 7, x₁, x₂ ≥ 0.

    Explanation: Solve the LPP via the two-phase simplex method handling artificial variables.

35. Q35.Real Analysis

    Let f(x) = x² on [0, k], k > 0. Show that f is Riemann integrable on the closed interval [0, k] and ∫₀^k f dx = k³/3.

    Explanation: Prove Riemann integrability from definition and evaluate the integral as k³/3.

36. Q36.Modern Algebra

    Prove that every homomorphic image of a group G is isomorphic to some quotient group of G.

    Explanation: Prove the First Isomorphism Theorem for groups.

37. Q37.Complex Analysis

    Apply the calculus of residues to evaluate ∫_{-∞}^{∞} (cos x dx) / ((x² + a²)(x² + b²)), a > b > 0.

    Explanation: Evaluate the real integral using contour integration and the residue theorem.

38. Q38.Complex Analysis

    Evaluate ∫_C (z + 4) / (z² + 2z + 5) dz, where C is |z + 1 - i| = 2.

    Explanation: Use Cauchy's residue theorem on the contour to evaluate the integral.

39. Q39.Real Analysis

    Find the maximum and minimum values of x²/a⁴ + y²/b⁴ + z²/c⁴, when lx + my + nz = 0 and x²/a² + y²/b² + z²/c² = 1. Interpret the result geometrically.

    Explanation: Use Lagrange multipliers for constrained extrema and give the geometric interpretation.

40. Q40.Linear Programming

    Solve the following linear programming problem by the simplex method. Write its dual. Also, write the optimal solution of the dual from the optimal table of the given problem: Maximize Z = x₁ + x₂ + x₃ subject to 2x₁ + x₂ + x₃ ≤ 2, 4x₁ + 2x₂ + x₃ ≤ 2, x₁, x₂, x₃ ≥ 0.

    Explanation: Solve the primal LPP by simplex, formulate the dual, and read the dual optimum from the primal optimal tableau.

41. Q41.Modern Algebra

    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). Prove that S is an ideal of R[x]. Is the residue class ring R[x]/S an integral domain? Give justification for your answer.

    Explanation: Show S is an ideal of the polynomial ring and analyze whether the quotient is an integral domain.

42. Q42.Real Analysis

    Test for convergence or divergence of the series x + (2²x²)/2! + (3³x³)/3! + (4⁴x⁴)/4! + (5⁵x⁵)/5! + ··· (x > 0).

    Explanation: Apply the ratio/root test to determine convergence of the power-type series.

43. Q43.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: Source S1 to destinations A,B,C,D costs 21,16,25,13 with availability 11; Source S2 costs 17,18,14,23 with availability 13; Source S3 costs 32,27,18,41 with availability 19; requirements A=6, B=10, C=12, D=15 (total 43).

    Explanation: Obtain IBFS by VAM, then optimize using MODI/u-v to find the minimum transportation cost.

44. Q44.Partial Differential Equations

    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 differential equation of the cone.

    Explanation: Eliminate the arbitrary function to derive the first-order PDE of the cone.

45. Q45.Numerical Analysis

    Solve, by Gauss elimination method, the system of equations 2x + 2y + 4z = 18, x + 3y + 2z = 13, 3x + y + 3z = 14.

    Explanation: Apply Gaussian elimination with back-substitution to solve the linear system.

46. Q46.Computer Programming

    (i) Convert the number (1093.21875)₁₀ into octal and the number (1693.0628)₁₀ into hexadecimal systems. (ii) Express the Boolean function F(x, y, z) = xy + x'z in a product of maxterms form.

    Explanation: Perform radix conversions and express the Boolean function as a product of maxterms (canonical POS).

47. Q47.Mechanics and Fluid Dynamics

    A particle at a distance r from the centre of force moves under the influence of the central force F = -k/r², where k is a constant. Obtain the Lagrangian and derive the equations of motion.

    Explanation: Form the Lagrangian for central-force motion and derive the Euler-Lagrange equations of motion.

48. Q48.Mechanics and Fluid Dynamics

    The velocity components of an incompressible fluid in spherical polar coordinates (r, θ, ψ) are (2Mr^(-3) cos θ, Mr^(-2) sin θ, 0), where M is a constant. Show that the velocity is of the potential kind. Find the velocity potential and the equations of the streamlines.

    Explanation: Show the flow is irrotational, find the velocity potential and streamline equations in spherical coordinates.

49. Q49.Partial Differential Equations

    Solve the heat equation ∂u/∂t = ∂²u/∂x², 0 < x < l, t > 0 subject to the conditions u(0, t) = u(l, t) = 0 and u(x, 0) = x(l - x), 0 ≤ x ≤ l.

    Explanation: Solve the one-dimensional heat equation by separation of variables and Fourier sine series.

50. Q50.Computer Programming

    Find a combinatorial circuit corresponding to the Boolean function f(x, y, z) = [x · (y' + z)] + y and write the input/output table for the circuit.

    Explanation: Design the logic gate circuit for the Boolean function and tabulate its truth table.

51. Q51.Mechanics and Fluid Dynamics

    Find the moment of inertia of a right circular solid cone about one of its slant sides (generator) in terms of its mass M, height h and the radius of base as a.

    Explanation: Compute the moment of inertia of a solid cone about its generator line.

52. Q52.Partial Differential Equations

    Find the general solution of the partial differential equation (D² + DD' - 6D'²) z = x² sin(x + y), where D ≡ ∂/∂x and D' ≡ ∂/∂y.

    Explanation: Find the complementary function and particular integral of the linear homogeneous PDE with constant coefficients.

53. Q53.Numerical Analysis

    The velocity of a train which starts from rest is given by the following table, the time being reckoned in minutes from the start and the velocity in km/hour: t (minutes) = 2,4,6,8,10,12,14,16,18,20; v (km/hour) = 16, 28.8, 40, 46.4, 51.2, 32, 17.6, 8, 3.2, 0. Using Simpson's 1/3rd rule, estimate approximately in km the total distance run in 20 minutes.

    Explanation: Apply Simpson's one-third rule for numerical integration of the velocity-time data to estimate distance.

54. Q54.Mechanics and Fluid Dynamics

    Two point vortices each of strength k are situated at (±a, 0) and a point vortex of strength -k/2 is situated at the origin. Show that the fluid motion is stationary and also find the equations of streamlines. If the streamlines, which pass through the stagnation points, meet the x-axis at (±b, 0), then show that 3√3(b² - a²)² = 16a³b.

    Explanation: Analyze a point-vortex system, prove stationarity, derive streamlines, and establish the stagnation-point relation.

55. Q55.Partial Differential Equations

    Reduce the following partial differential equation to a canonical form and hence solve it: y u_xx + (x + y) u_xy + x u_yy = 0.

    Explanation: Classify the second-order PDE, reduce to canonical form via characteristics, and solve.

56. Q56.Numerical Analysis

    Using Runge-Kutta method of fourth order, solve the differential equation dy/dx = x + y² with y(0) = 1, at x = 0.2. Use four decimal places for calculation and step length 0.1.

    Explanation: Apply the fourth-order Runge-Kutta method over two steps to compute y(0.2).

57. Q57.Mechanics and Fluid Dynamics

    Verify that w = ik log{(z - ia)/(z + ia)} is the complex potential of a steady flow of fluid about a circular cylinder, where the plane y = 0 is a rigid boundary. Find also the force exerted by the fluid on unit length of the cylinder.

    Explanation: Verify the complex potential satisfies the boundary condition and compute the hydrodynamic force via Blasius' theorem.