Previous Year Question Papers

Asymptotes, pedal equations, curvature & radius of curvature, tracing of curves (Cartesian & polar), Beta & Gamma functions, quadrature, rectification, partial differentiation, Euler's theorem, Lagrange's multiplier, Jacobians, double & triple integrals, Dirichlet's theorem, volume & surface of solid of revolution

Groups: definitions, order of elements, subgroups; permutation groups, cyclic groups, even & odd permutations; cosets, Lagrange's theorem, normal subgroups, simple groups; quotient groups, homomorphism & isomorphism, Cayley's theorem, fundamental theorem of homomorphism; automorphisms, three isomorphism theorems; rings, integral domain, division ring, field, characteristic of a ring

First order ODEs: variable separation, homogeneous, linear, exact; Clairaut's form, singular solutions; linear DEs with constant coefficients; homogeneous linear ODEs, simultaneous DEs; second order ODEs by inspection, change of variables, removal of first derivative, operational factors; PDEs: Lagrange's method, Charpit's method, standard forms

Continuity & derivability: Cauchy definition, Darboux theorem; real number system: ordered field, supremum & infimum, completeness, Archimedean property, Bolzano-Weierstrass theorem, Heine-Borel theorem, compact & connected sets; sequences: convergence, Cauchy sequences, uniform convergence; series: convergence tests, Leibnitz test, Weierstrass M-test, Abel's & Dirichlet's tests, Fourier series; Riemann integration: Darboux sums, fundamental theorem of calculus

Rolle's theorem, mean value theorems, Taylor's theorem; equivalent sets, denumerable & non-denumerable sets, countable & uncountable sets; non-denumerability of [0,1] and reals; improper integrals: comparison test, μ-test, Abel's test; uniform convergence: term-by-term differentiation & integration

Matrices: symmetric, skew-symmetric, Hermitian; row & column rank, eigenvalues & eigenvectors, Cayley-Hamilton theorem; vector spaces: subspaces, linear span, dependence & independence; basis & dimension, extension theorem; quotient space, linear transformations, rank & nullity, Sylvester's law; algebra of linear transformations, dual space & dual basis

Sets: cardinality, induction, inclusion-exclusion, pigeon-hole, permutations & combinations; relations: binary relations, lattices, Hasse diagrams; graphs: Eulerian paths, planar graphs; trees: rooted, binary, spanning, minimal spanning; finite state machines; recurrence relations: generating functions, homogeneous & particular solutions; Boolean algebra: duality, propositional calculus, switching circuits

Divisibility: GCD, LCM, Euclidean algorithm, linear Diophantine equation ax+by=c, fundamental theorem of arithmetic; congruences: Fermat's little theorem, Euler's theorem, Wilson's theorem, Chinese remainder theorem; number-theoretic functions: τ, σ, Möbius inversion; primitive roots & indices; quadratic residues, Gauss lemma, quadratic reciprocity law; Fibonacci, Fermat & perfect numbers; sums of squares

Differentiation of vectors: unit tangent, velocity & acceleration; gradient, directional derivative, tangent plane & normal; divergence & curl, solenoidal & irrotational vectors; line integrals, work done by force, surface integrals; Gauss's divergence theorem, Stokes' theorem, Green's theorem

Equilibrium: Lami's theorem, triangle of forces, moments; friction, limiting equilibrium on inclined plane; catenary; virtual work; projectile motion, impact, conservation of momentum; SHM, Hooke's law, elastic strings; constrained motion on circle & cycloid, motion in resisting medium; fluid pressure: specific gravity, plane surfaces, centre of pressure

Conics: eccentricity, foci, tracing; ellipse: tangent, normal, pole & polar, conjugate diameters, auxiliary & director circle; hyperbola: asymptotes, conjugate & rectangular hyperbola; polar equations of conics; 3D geometry: planes, skew lines, shortest distance; sphere, cone, cylinder; conicoids: tangent plane, diametral plane, principal planes & directions

LPP formulation, graphical method; simplex algorithm, artificial variables, two-phase & Big-M methods; duality: primal-dual relationships, economic interpretation; transportation problem: northwest-corner, least cost, Vogel's method; assignment problem: Hungarian method; game theory: two-person zero-sum games, mixed strategies

Nikhilam & Urdhvatiryak multiplication, Pravartya division; factorisation of quadratic & cubic equations, HCF by Vedic method; Shunyam Samuchhye & Pravartya methods; quadratic, cubic & biquadratic equations; Vedic methods in differentiation, integration by Anshikbhinna; Sahayakbhinna, divisibility of large numbers, square roots & cube roots, Vedic geometry