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UPSC CSE Mains 2018 — Mathematics Questions with Answers

All 62 Mathematics previous-year questions from UPSC CSE Mains 2018, 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 A be a 3 × 2 matrix and B a 2 × 3 matrix. Show that C = A · B is a singular matrix.

   Explanation: Product of a 3×2 and 2×3 matrix has rank at most 2, hence the 3×3 matrix C is singular.

2. Q2.Linear Algebra

   Express basis vectors e₁ = (1, 0) and e₂ = (0, 1) as linear combinations of α₁ = (2, −1) and α₂ = (1, 3).

   Explanation: Solve for scalars expressing standard basis vectors in terms of the given vectors via change of basis.

3. Q3.Calculus

   Determine if lim_{z→1} (1 − z) tan(πz/2) exists or not. If the limit exists, then find its value.

   Explanation: Evaluate the 0·∞ indeterminate limit via substitution/L'Hôpital; value is 2/π.

4. Q4.Calculus

   Find the limit lim_{n→∞} (1/n²) Σ_{r=0}^{n−1} √(n² − r²).

   Explanation: Express as a Riemann sum for ∫₀¹ √(1 − x²) dx, giving π/4.

5. Q5.Analytic Geometry

   Find the projection of the straight line (x − 1)/2 = (y − 1)/3 = (z + 1)/(−1) on the plane x + y + 2z = 6.

   Explanation: Find the plane through the line perpendicular to the given plane; its intersection is the projection.

6. Q6.Linear Algebra

   Show that if A and B are similar n × n matrices, then they have the same eigenvalues.

   Explanation: Similar matrices share the same characteristic polynomial since det(B − λI) = det(P⁻¹(A − λI)P).

7. Q7.Calculus

   Find the shortest distance from the point (1, 0) to the parabola y² = 4x.

   Explanation: Minimize the squared distance from (1,0) to a parametric point on the parabola.

8. Q8.Calculus

   The ellipse x²/a² + y²/b² = 1 revolves about the x-axis. Find the volume of the solid of revolution.

   Explanation: Volume of the prolate spheroid by integration is (4/3)πab².

9. Q9.Analytic Geometry

   Find the shortest distance between the lines a₁x + b₁y + c₁z + d₁ = 0, a₂x + b₂y + c₂z + d₂ = 0 and the z-axis.

   Explanation: Apply the shortest-distance-between-two-skew-lines formula using direction ratios and a connecting vector.

10. Q10.Linear Algebra

    For the system of linear equations x + 3y − 2z = −1, 5y + 3z = −8, x − 2y − 5z = 7, determine which of the following statements are true and which are false: (i) The system has no solution. (ii) The system has a unique solution. (iii) The system has infinitely many solutions.

    Explanation: Determine consistency by comparing rank of coefficient matrix and augmented matrix.

11. Q11.Calculus

    Let f(x, y) = xy² if y > 0, = −xy² if y ≤ 0. Determine which of ∂f/∂x (0, 1) and ∂f/∂y (0, 1) exists and which does not exist.

    Explanation: Examine partial derivatives at (0,1) from the definition for the piecewise function.

12. Q12.Analytic Geometry

    Find the equations to the generating lines of the paraboloid (x + y + z)(2x + y − z) = 6z which pass through the point (1, 1, 1).

    Explanation: Find the two generators of the hyperbolic paraboloid passing through the given point.

13. Q13.Analytic Geometry

    Find the equation of the sphere in xyz-plane passing through the points (0, 0, 0), (0, 1, −1), (−1, 2, 0) and (1, 2, 3).

    Explanation: Substitute the four points into the general sphere equation and solve for its coefficients.

14. Q14.Calculus

    Find the maximum and the minimum values of x⁴ − 5x² + 4 on the interval [2, 3].

    Explanation: Evaluate at critical points and endpoints; on [2,3] the function is increasing.

15. Q15.Calculus

    Evaluate the integral ∫₀ᵃ ∫_{x/a}^{x} (x dy dx)/(x² + y²).

    Explanation: Integrate over y first using arctan, then integrate the resulting expression in x.

16. Q16.Analytic Geometry

    Find the equation of the cone with (0, 0, 1) as the vertex and 2x² − y² = 4, z = 0 as the guiding curve.

    Explanation: Use the line through the vertex and a point on the guiding curve to eliminate the parameter and form the cone.

17. Q17.Analytic Geometry

    Find the equation of the plane parallel to 3x − y + 3z = 8 and passing through the point (1, 1, 1).

    Explanation: Parallel plane has the same normal; substitute the point to find the constant term.

18. Q18.Ordinary Differential Equations

    Solve: y″ − y = x²e^{2x}.

    Explanation: Find complementary function and particular integral using the operator/undetermined-coefficients method.

19. Q19.Vector Analysis

    Find the angle between the tangent at a general point of the curve whose equations are x = 3t, y = 3t², z = 3t³ and the line y = z − x = 0.

    Explanation: Compute the tangent direction by differentiation and find the angle with the line's direction via dot product.

20. Q20.Ordinary Differential Equations

    Solve: y‴ − 6y″ + 12y′ − 8y = 12e^{2x} + 27e^{−x}.

    Explanation: Third-order constant-coefficient ODE with repeated root; handle the resonant e^{2x} term specially.

21. Q21.Ordinary Differential Equations

    (i) Find the Laplace transform of f(t) = 1/√t. (ii) Find the inverse Laplace transform of (5s² + 3s − 16)/((s − 1)(s − 2)(s + 3)).

    Explanation: Use the Gamma function for L{t^{-1/2}} and partial fractions for the inverse transform.

22. Q22.Dynamics and Statics

    A particle projected from a given point on the ground just clears a wall of height h at a distance d from the point of projection. If the particle moves in a vertical plane and if the horizontal range is R, find the elevation of the projection.

    Explanation: Use the projectile trajectory equation with range R to express the angle of projection in terms of h, d, R.

23. Q23.Ordinary Differential Equations

    Solve: (dy/dx)² y + 2(dy/dx)x − y = 0.

    Explanation: First-order ODE of higher degree in p = dy/dx; solve for p and integrate.

24. Q24.Dynamics and Statics

    A particle moving with simple harmonic motion in a straight line has velocities v₁ and v₂ at distances x₁ and x₂ respectively from the centre of its path. Find the period of its motion.

    Explanation: Use v² = ω²(a² − x²) for two states to eliminate amplitude and find ω, hence the period.

25. Q25.Ordinary Differential Equations

    Solve: y″ + 16y = 32 sec 2x.

    Explanation: Solve the non-homogeneous ODE using variation of parameters for the sec 2x forcing term.

26. Q26.Vector Analysis

    If S is the surface of the sphere x² + y² + z² = a², then evaluate ∬_S [(x + z) dy dz + (y + z) dz dx + (x + y) dx dy] using Gauss' divergence theorem.

    Explanation: Apply the divergence theorem; divergence equals 2, so integral equals 2 times the sphere's volume.

27. Q27.Ordinary Differential Equations

    Solve: (1 + x)² y″ + (1 + x) y′ + y = 4 cos(log(1 + x)).

    Explanation: Cauchy-Euler type equation; substitute 1 + x = e^t to obtain a constant-coefficient ODE.

28. Q28.Vector Analysis

    Find the curvature and torsion of the curve r⃗ = a(u − sin u)i⃗ + a(1 − cos u)j⃗ + bu k⃗.

    Explanation: Use curvature κ = |r′×r″|/|r′|³ and torsion τ = (r′×r″)·r‴/|r′×r″|².

29. Q29.Ordinary Differential Equations

    Solve the initial value problem y″ − 5y′ + 4y = e^{2t}, y(0) = 19/12, y′(0) = 8/3.

    Explanation: Solve the constant-coefficient ODE and apply initial conditions to fix the arbitrary constants.

30. Q30.Ordinary Differential Equations

    Find α and β such that xᵅyᵝ is an integrating factor of (4y² + 3xy) dx − (3xy + 2x²) dy = 0 and solve the equation.

    Explanation: Impose the exactness condition on the multiplied equation to determine α, β, then integrate.

31. Q31.Vector Analysis

    Let v⃗ = v₁i⃗ + v₂j⃗ + v₃k⃗. Show that curl(curl v⃗) = grad(div v⃗) − ∇²v⃗.

    Explanation: Prove the vector identity by expanding both sides in Cartesian components.

32. Q32.Vector Analysis

    Evaluate the line integral ∮_C (−y³ dx + x³ dy + z³ dz) using Stokes' theorem. Here C is the intersection of the cylinder x² + y² = 1 and the plane x + y + z = 1. The orientation on C corresponds to counterclockwise motion in the xy-plane.

    Explanation: Convert the line integral to a surface integral of the curl over the disk using Stokes' theorem.

33. Q33.Vector Analysis

    Let F⃗ = xy²i⃗ + (y + x)j⃗. Integrate (∇ × F⃗) · k⃗ over the region in the first quadrant bounded by the curves y = x² and y = x using Green's theorem.

    Explanation: Apply Green's theorem to convert the double integral of the curl's k-component to a line integral around the region.

34. Q34.Ordinary Differential Equations

    Find f(y) such that (2xe^y + 3y²) dy + (3x² + f(y)) dx = 0 is exact and hence solve.

    Explanation: Apply the exactness condition to determine f(y), then integrate to obtain the solution.

35. Q35.Modern Algebra

    Let R be an integral domain with unit element. Show that any unit in R[x] is a unit in R.

    Explanation: Units of the polynomial ring over an integral domain are exactly the units of the base ring.

36. Q36.Real Analysis

    Prove the inequality: π²/9 < ∫_{π/6}^{π/2} (x / sin x) dx < 2π²/9.

    Explanation: Bound the integrand x/sin x between linear functions and integrate to obtain the stated bounds.

37. Q37.Complex Analysis

    Prove that the function u(x, y) = (x − 1)³ − 3xy² + 3y² is harmonic and find its harmonic conjugate and the corresponding analytic function f(z) in terms of z.

    Explanation: Verify Laplace's equation, integrate Cauchy–Riemann equations to get the conjugate, then express f(z).

38. Q38.Real Analysis

    Find the range of p(>0) for which the series 1/(1+a)^p − 1/(2+a)^p + 1/(3+a)^p − …, a > 0, is (i) absolutely convergent and (ii) conditionally convergent.

    Explanation: Use p-series comparison for absolute convergence and the Leibniz test for conditional convergence.

39. Q39.Linear Programming

    An agricultural firm has 180 tons of nitrogen fertilizer, 250 tons of phosphate and 220 tons of potash. It will be able to sell a mixture of these substances in their respective ratio 3 : 3 : 4 at a profit of Rs. 1500 per ton and a mixture in the ratio 2 : 4 : 2 at a profit of Rs. 1200 per ton. Pose a linear programming problem to show how many tons of these two mixtures should be prepared to obtain the maximum profit.

    Explanation: Formulate the resource-constrained mixture problem as an LPP maximizing total profit.

40. Q40.Modern Algebra

    Show that the quotient group of (ℝ, +) modulo ℤ is isomorphic to the multiplicative group of complex numbers on the unit circle in the complex plane. Here ℝ is the set of real numbers and ℤ is the set of integers.

    Explanation: Map x + ℤ to e^{2πix} and verify it is a well-defined isomorphism onto the unit circle group.

41. Q41.Linear Programming

    Solve the following linear programming problem by Big M-method and show that the problem has finite optimal solutions. Also find the value of the objective function: Minimize z = 3x₁ + 5x₂ subject to x₁ + 2x₂ ≥ 8, 3x₁ + 2x₂ ≥ 12, 5x₁ + 6x₂ ≤ 60, x₁, x₂ ≥ 0.

    Explanation: Add artificial variables with the Big-M penalty and run the simplex iterations to the optimum.

42. Q42.Real Analysis

    Show that if a function f defined on an open interval (a, b) of ℝ is convex, then f is continuous. Show, by example, if the condition of open interval is dropped, then the convex function need not be continuous.

    Explanation: Convexity forces local Lipschitz behaviour on an open interval; a boundary jump gives a discontinuous convex example on a closed interval.

43. Q43.Modern Algebra

    Find all the proper subgroups of the multiplicative group of the field (ℤ₁₃, +₁₃, ×₁₃), where +₁₃ and ×₁₃ represent addition modulo 13 and multiplication modulo 13 respectively.

    Explanation: The multiplicative group is cyclic of order 12; list subgroups for each divisor of 12.

44. Q44.Complex Analysis

    Show by applying the residue theorem that ∫₀^∞ dx/(x² + a²)² = π/(4a³), a > 0.

    Explanation: Use a semicircular contour and the residue at the double pole z = ia to evaluate the real integral.

45. Q45.Linear Programming

    How many basic solutions are there in the following linearly independent set of equations? Find all of them. 2x₁ − x₂ + 3x₃ + x₄ = 6, 4x₁ − 2x₂ − x₃ + 2x₄ = 10.

    Explanation: Choose all C(4,2) variable pairs as basic; count and solve those giving valid basic solutions.

46. Q46.Real Analysis

    Suppose ℝ be the set of all real numbers and f : ℝ → ℝ is a function such that the following equations hold for all x, y ∈ ℝ: (i) f(x + y) = f(x) + f(y), (ii) f(xy) = f(x) f(y). Show that ∀ x ∈ ℝ either f(x) = 0, or, f(x) = x.

    Explanation: An additive and multiplicative map preserves order via squares; hence it is monotone and forced to be 0 or the identity.

47. Q47.Complex Analysis

    Find the Laurent's series which represent the function 1/((1 + z²)(z + 2)) when (i) |z| < 1, (ii) 1 < |z| < 2, (iii) |z| > 2.

    Explanation: Apply partial fractions then expand each term in the appropriate annular region.

48. Q48.Linear Programming

    In a factory there are five operators O₁, O₂, O₃, O₄, O₅ and five machines M₁, M₂, M₃, M₄, M₅. The operating costs are given when the Oᵢ operator operates the Mⱼ machine (i, j = 1, 2, …, 5). But there is a restriction that O₃ cannot be allowed to operate the third machine M₃ and O₂ cannot be allowed to operate the fifth machine M₅. The cost matrix is: O₁: (24, 29, 18, 32, 19); O₂: (17, 26, 34, 22, 21); O₃: (27, 16, 28, 17, 25); O₄: (22, 18, 28, 30, 24); O₅: (28, 16, 31, 24, 27) for machines M₁..M₅. Find the optimal assignment and the optimal assignment cost also.

    Explanation: Block the two forbidden cells with high cost and solve the assignment by the Hungarian method.

49. Q49.Partial Differential Equations

    Find the partial differential equation of the family of all tangent planes to the ellipsoid x² + 4y² + 4z² = 4, which are not perpendicular to the xy plane.

    Explanation: Write the tangent-plane family with p, q and eliminate the parameters to form the PDE.

50. Q50.Numerical Analysis

    Using Newton's forward difference formula find the lowest degree polynomial uₓ when it is given that u₁ = 1, u₂ = 9, u₃ = 25, u₄ = 55 and u₅ = 105.

    Explanation: Build the forward difference table and substitute into Newton's forward interpolation polynomial.

51. Q51.Mechanics and Fluid Dynamics

    For an incompressible fluid flow, two components of velocity (u, v, w) are given by u = x² + 2y² + 3z², v = x²y − y²z + zx. Determine the third component w so that they satisfy the equation of continuity. Also, find the z-component of acceleration.

    Explanation: Integrate the continuity equation for ∂w/∂z to get w, then compute the material derivative for a_z.

52. Q52.Numerical Analysis

    Starting from rest in the beginning, the speed (in Km/h) of a train at different times (in minutes) is given by the table: Time (Minutes) 2, 4, 6, 8, 10, 12, 14, 16, 18, 20; Speed (Km/h) 10, 18, 25, 29, 32, 20, 11, 5, 2, 8.5. Using Simpson's 1/3rd rule, find the approximate distance travelled (in Km) in 20 minutes from the beginning.

    Explanation: Apply Simpson's one-third rule to the speed-time data with consistent units to approximate the integral (distance).

53. Q53.Numerical Analysis

    Write down the basic algorithm for solving the equation xeˣ − 1 = 0 by bisection method, correct to 4 decimal places.

    Explanation: State the bracketing, midpoint, sign-test, and tolerance-stopping steps of the bisection algorithm.

54. Q54.Partial Differential Equations

    Find the general solution of the partial differential equation (y³x − 2x⁴)p + (2y⁴ − x³y)q = 9z(x³ − y³), where p = ∂z/∂x, q = ∂z/∂y, and find its integral surface that passes through the curve: x = t, y = t², z = 1.

    Explanation: Solve Lagrange's auxiliary equations for two independent integrals and apply the curve condition.

55. Q55.Computer Programming

    Find the equivalent of numbers given in a specified number system to the system mentioned against them: (i) (111011.101)₂ to decimal system, (ii) (1000111110000.00101100)₂ to hexadecimal system, (iii) (C4F2)₁₆ to decimal system, (iv) (418)₁₀ to binary system.

    Explanation: Apply positional-value and grouping conversions between binary, decimal and hexadecimal.

56. Q56.Mechanics and Fluid Dynamics

    Suppose the Lagrangian of a mechanical system is given by L = ½ m(a ẋ² + 2b ẋẏ + c ẏ²) − ½ k(a x² + 2b xy + c y²), where a, b, c, m(>0), k(>0) are constants and b² ≠ ac. Write down the Lagrangian equations of motion and identify the system.

    Explanation: Apply the Euler–Lagrange equations to obtain coupled linear equations identifying a two-DOF harmonic oscillator.

57. Q57.Partial Differential Equations

    Solve the partial differential equation (2D² − 5DD' + 2D'²)z = 5 sin(2x + y) + 24(y − x) + e^{3x+4y}, where D ≡ ∂/∂x, D' ≡ ∂/∂y.

    Explanation: Find the complementary function and particular integrals for each forcing term of this linear PDE with constant coefficients.

58. Q58.Numerical Analysis

    Find the values of the constants a, b, c such that the quadrature formula ∫₀^h f(x) dx = h[a f(0) + b f(h/3) + c f(h)] is exact for polynomials of as high degree as possible, and hence find the order of the truncation error.

    Explanation: Impose exactness on 1, x, x² to solve for a, b, c, then test higher powers to fix the truncation order.

59. Q59.Mechanics and Fluid Dynamics

    The Hamiltonian of a mechanical system is given by H = p₁q₁ − a q₁² + b q₂² − p₂q₂, where a, b are the constants. Solve the Hamiltonian equations and show that (p₂ − b q₂)/q₁ = constant.

    Explanation: Write Hamilton's canonical equations, integrate them, and verify the stated combination is conserved.

60. Q60.Computer Programming

    Simplify the boolean expression (a + b)·(b̄ + c) + b·(ā + c̄) by using the laws of boolean algebra. From its truth table write it in minterm normal form.

    Explanation: Apply Boolean identities to simplify, then enumerate minterms where the expression is 1.

61. Q61.Mechanics and Fluid Dynamics

    For a two-dimensional potential flow, the velocity potential is given by φ = x²y − xy² + ⅓(x³ − y³). Determine the velocity components along the directions x and y. Also, determine the stream function ψ and check whether φ represents a possible case of flow or not.

    Explanation: Differentiate φ for velocities, check Laplace's equation for validity, and integrate Cauchy–Riemann relations for ψ.

62. Q62.Partial Differential Equations

    A thin annulus occupies the region 0 < a ≤ r ≤ b, 0 ≤ θ ≤ 2π. The faces are insulated. Along the inner edge the temperature is maintained at 0°, while along the outer edge the temperature is held at T = K cos(θ/2), where K is a constant. Determine the temperature distribution in the annulus.

    Explanation: Solve Laplace's equation in polar coordinates by Fourier series matching the boundary data on r = a and r = b.