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17 previous year questions for Mathematics from 3 years. Practice with year-wise breakdown.
17
Questions
3
Years
2
Papers
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 is the identity element. Here o(H), o(K) and o(G) denote the order of H, K and G respectively. Let G = {e, x, x², y, yx, yx²} be a non-Abelian group with o(x) = 3 and o(y) = 2. Show that xy = yx² (where e is the identity element of G and o(x), o(y) denote the order of the elements x, y respectively). Examine whether the series Σ_{n=1}^{∞} (–1)^{n-1}/n is absolutely or conditionally convergent. Expand f(z) = 1/[(z+1)(z+3)] in a Laurent series valid for 1 < |z| < 3. How many basic solutions are there for the following system of equations? 2x₁ – x₂ + 3x₃ + x₄ = 6 4x₁ – 2x₂ – x₃ + 2x₄ = 10 Find all of them. Furthermore, find the number of basic solutions, which are feasible/non-feasible/non-degenerate.
(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}. [10M]
(b) Let G = {e, x, x², y, yx, yx²} be a non-Abelian group with o(x) = 3 and o(y) = 2. Show that xy = yx². [10M]
(c) Examine whether the series Σ_{n=1}^{∞} (–1)^{n-1} / n is absolutely or conditionally convergent. [10M]
(d) Expand f(z) = 1/[(z+1)(z+3)] in a Laurent series valid for 1 < |z| < 3. [10M]
(e) For the system 2x₁ – x₂ + 3x₃ + x₄ = 6, 4x₁ – 2x₂ – x₃ + 2x₄ = 10, find all basic solutions and determine how many are feasible, non-feasible and non-degenerate. [10M]
Define Cauchy sequence and prove that every convergent sequence of real numbers is a Cauchy sequence. What is the importance of Cauchy condition? Show that 3 is an irreducible element in the integral domain Z[i]. Use the method of contour integration to prove that ∫_{–∞}^{∞} (x² – x + 2)/(x⁴ + 10x² + 9) dx = 5π/12.
(a) Define Cauchy sequence and prove that every convergent sequence of real numbers is a Cauchy sequence. Explain the importance of Cauchy condition. [15M]
(b) Show that 3 is an irreducible element in the integral domain Z[i]. [15M]
(c) Using contour integration, prove that ∫_{–∞}^{∞} (x² – x + 2)/(x⁴ + 10x² + 9) dx = 5π/12. [20M]
Evaluate the integral ∮_C e^z / [z² (z+1)³] dz, C : |z| = 2. Show that the volume of the greatest rectangular parallelepiped that can be inscribed in the ellipsoid x²/a² + y²/b² + z²/c² = 1 is 8abc / (3√3). Apply the principle of duality to solve the following linear programming problem : Maximize Z = 3x₁ + 4x₂ subject to x₁ – x₂ ≤ 1, x₁ + x₂ ≥ 4, x₁ – 3x₂ ≤ 3, x₁, x₂ ≥ 0.
(a) Evaluate ∮_C e^z / [z² (z+1)³] dz for C : |z| = 2. [15M]
(b) Show that the volume of the greatest rectangular parallelepiped inscribed in the ellipsoid x²/a² + y²/b² + z²/c² = 1 is 8abc / (3√3). [20M]
(c) Using the principle of duality, solve the linear programming problem: Maximize Z = 3x₁ + 4x₂ subject to x₁ – x₂ ≤ 1, x₁ + x₂ ≥ 4, x₁ – 3x₂ ≤ 3, x₁, x₂ ≥ 0. [15M]
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 the ideal (x) is a prime ideal in Z[x], but not a maximal ideal in Z[x]. Prove that every continuous function is Riemann integrable. The following table shows all the necessary information on the available supply to each warehouse, the requirement of each market and the unit transportation cost from each warehouse to each market : Market I II III IV Supply Warehouse A 5 2 4 3 22 B 4 8 1 6 15 C 4 6 7 5 8 Requirement 7 12 17 9 The shipping clerk has worked out the following schedule from experience : 12 units from A to II, 1 unit from A to III, 9 units from A to IV, 15 units from B to III, 7 units from C to I and 1 unit from C to III. Find the optimal schedule and minimum total shipping cost.
(a) Examine whether φ : Z[x] → Z defined by φ(f(x)) = f(0) is a homomorphism. Deduce that the ideal (x) is prime but not maximal in Z[x]. [15M]
(b) Prove that every continuous function is Riemann integrable. [15M]
(c) Using the data given in the transportation table, find the optimal shipping schedule and the minimum total transportation cost. [20M]
Find the solution of the equation (D² + DD′ – 2D′²) z = y sin x, where D = ∂/∂x and D′ = ∂/∂y. Solve the following system of linear equations by Gauss-Seidel method: 10x + 2y + z = 9 2x + 20y – 2z = –44 –2x + 3y +10z = 22 (i) Convert the number (3479)₁₀ into binary system and the number (7AE·9F)₁₆ into decimal system. (ii) Determine the truth table for the Boolean function F(x, y, z) = (x + y + z′)(x′ + y′). Also derive the full disjunctive normal form of F(x, y, z) from the truth table.
(a) Solve (D² + DD′ – 2D′²) z = y sin x with D = ∂/∂x, D′ = ∂/∂y. [10M]
(b) Using Gauss-Seidel method, solve the system: 10x + 2y + z = 9; 2x + 20y – 2z = –44; –2x + 3y + 10z = 22. [10M]
(c) (i) Convert (3479)₁₀ into binary and (7AE·9F)₁₆ into decimal. (ii) Construct the truth table of F(x, y, z) = (x + y + z′)(x′ + y′) and derive its full disjunctive normal form. [10M]
Answer all the following sub-parts.
(a) Let H be a subspace of ℝ⁴ spanned by the vectors v₁ = (1, −2, 5, −3), v₂ = (2, 3, 1, −4) and v₃ = (3, 8, −3, −5). Find a basis and the dimension of H, and extend the basis of H to a basis of ℝ⁴. [10M]
(b) Let T : ℝ³ → ℝ³ be a linear operator and B = {v₁, v₂, v₃} be a basis of ℝ³ over ℝ. Suppose that Tv₁ = (1, 1, 0), Tv₂ = (1, 0, −1) and Tv₃ = (2, 1, −1). Find a basis for the range space and the null space of T. [10M]
(c) Discuss the continuity, for all values of x, of the function f(x) = {(1 − e^{−1/x})/x, x ≠ 0; 0, x = 0}. [10M]
(d) 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. [10M]
(e) Find the equation of the right circular cylinder which passes through the circle x² + y² + z² = 9, x − y + z = 3. [10M]
Answer all the following sub-parts.
(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 yes, justify your answer and find T⁻¹. [15M]
(b) If u = (x + y)/(1 − x y) and v = tan⁻¹x + tan⁻¹y, find ∂(u, v)/∂(x, y). Are u and v functionally related? If so, find the relationship. [15M]
(c) Find the image of the line x = 3 − 6t, y = 2t, z = 3 + 2t in the plane 3x + 4y − 5z + 26 = 0. [20M]
Answer all the following sub-parts.
(a) Let V = M₂×₂(ℝ) denote the vector space of 2 × 2 real matrices. Find the matrix of the linear mapping φ : V → V given by φ(v) = [1 2; 3 −1] v with respect to the standard basis of M₂×₂(ℝ), and hence find the rank of φ. Is φ invertible? Justify your answer. [15M]
(b) Find the volume of the greatest right circular cylinder that can be inscribed in a cone of height h and semi-vertical angle α. [20M]
(c) Find the vertex of the cone 4x² − y² + 2z² + 2xy − 3yz + 12x − 11y + 6z + 4 = 0. [15M]
Answer all the following sub-parts.
(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 A. Hence find the eigenvalues and eigenvectors of A⁻¹⁵, where A⁻¹⁵ = (A⁻¹)¹⁵. [20M]
(b) Using double integration, find the area lying inside the cardioid r = a(1 + cos θ) and outside the circle r = a. [15M]
(c) 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 x² + y² + z² − 4x + 6y + 4 = 0. [15M]
Answer all the following sub-parts.
(a) Find the orthogonal trajectories of the family of curves r = c(sec θ + tan θ), where c is a parameter. [10M]
(b) Solve the integral equation y(t) = cos t + ∫₀ᵗ y(x) cos (t − x) dx using Laplace transform. [10M]
(c) At any time t (in seconds) the coterminous edges of a variable parallelepiped are represented by the vectors α = t î + (t + 1) ĵ + (2t + 1) k̂, β = 2t î + (3t − 1) ĵ + t k̂, γ = î + 3t ĵ + k̂. What is the rate of change of the vectorial area of the parallelogram whose coterminous edges are α and γ? Also, find the rate of change of the volume of the parallelepiped at t = 1 second. [10M]
(d) 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. [10M]
(e(i)) Let C be a plane curve r(t) = f(t) î + g(t) ĵ, where f and g have second-order derivatives. Show that the curvature at a point is k = |f′(t) g″(t) − g′(t) f″(t)| / ( (f′(t)² + g′(t)²)^{3/2} ). What is the value of torsion τ at any point of this curve? [5M]
(e(ii)) Show that the principal normals at two consecutive points of a curve do not intersect unless the torsion τ is zero. [5M]
Answer all the following sub-parts.
(a) 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. [15M]
(b) A particle executes simple harmonic motion such that in two of its positions the velocities are u and v and the corresponding accelerations are f₁ and f₂. For what value(s) of k is the distance between the two positions k(v² − u²)? Show also that the amplitude of the motion is (1 / (f₂² − f₁²)) [(u² − v²)(u²f₂² − v²f₁²)]^{1/2}. [15M]
(c(i)) Using u(x) = e^{−x} as one solution, find the second solution of the differential equation x y″ + (x − 1) y′ − y = 0. [10M]
(c(ii)) Find the general solution of the differential equation x² y″ − 2x y′ + 2y = x³ sin x by the method of variation of parameters. [10M]
Answer all the following sub-parts.
(a) State the uniqueness theorem for the existence of a 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√y, y(1) = 0 in a suitable rectangle R. If more than one solution exists, find all the solutions. [15M]
(b) A heavy particle hanging vertically from a fixed point by a light inextensible string of length l starts with initial velocity u and moves in a vertical circle so as to make a complete revolution. Show that the sum of the tensions at the ends of any diameter is constant. [15M]
(c) State Stokes’ theorem and verify it for the vector field 𝐅 = x y î + y z ĵ + z x k̂ over the surface S, which is the upwardly oriented part of the cylinder z = 1 − x², for 0 ≤ x ≤ 1, −2 ≤ y ≤ 2. [20M]
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 / (v² sin²θ)) − 1] : 1.
Find the solution of the differential equation: dy/dx = −(2xy³ + 2)/(3x²y² + 8e^{4y}).
(a) (i) Find the conjunctive normal form (CNF) of the following Boolean function: f(x, y, z, t) = x·y·z + x̄·y·(t + z) (ii) Express the Boolean function f(x, y, z) = x + (x̄·ȳ + x̄·z̄) + z in disjunctive normal form (DNF) and construct the truth table for the function. (b) 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. (c) 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.
(a) (i) Find the conjunctive normal form (CNF) of the Boolean function f(x, y, z, t) = x·y·z + x̄·y·(t + z). (ii) Express the Boolean function f(x, y, z) = x + (x̄·ȳ + x̄·z̄) + z in disjunctive normal form (DNF) and construct the truth table for the function. [15M]
(b) 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. [15M]
(c) 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. [20M]
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.
(a) Reduce the partial differential equation ∂²z/∂y² − ∂²z/∂x∂y + ∂z/∂x − ∂z/∂y (1 + 1/x) + z/x = 0 to canonical form. (b) Compute a root of the equation log₁₀(2x + 1) − x² + 3 = 0, in the interval [0, 3], by the Regula-Falsi method, correct to 6 decimal places. (c) 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 Bₓ = 0 = B_y, B_z = −g.
(a) Reduce the partial differential equation ∂²z/∂y² − ∂²z/∂x∂y + ∂z/∂x − ∂z/∂y (1 + 1/x) + z/x = 0 to canonical form. [15M]
(b) Compute a root of the equation log₁₀(2x + 1) − x² + 3 = 0, in the interval [0, 3], by the Regula-Falsi method, correct to 6 decimal places. [15M]
(c) 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 Bₓ = 0 = B_y, B_z = −g. [20M]
We have 17 UPSC Mains Mathematics optional subject questions spanning 3 years (2023–2025).
Mathematics has 2 papers in UPSC Mains: Mathematics-II, Mathematics-I. Each paper carries 250 marks.