Lakshya Batch – UPPSC GIC Lecturer Mathematics Course

Relation and Functions:
Types of relations: reflexive, symmetric, transitive, and equivalence relations.
Equivalence class, one-one and onto functions, composite of functions, inverse of function, binary operation.
Algebra:
(i) Matrices: Types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices; addition, multiplication, scalar multiplication; singular and non-singular matrices; invertible matrices.
(ii) Determinants: Determinants of a square matrix (up to 3×3), properties of determinants, adjoint and inverse of a square matrix, consistency and number of solutions of systems of linear equations by examples, solving systems in two or three variables (unique solutions).
(iii) Theory of Equations (degree ≥ 2): Arithmetical, geometrical, and harmonical progressions; permutations and combinations; binomial theorem; sum of exponential and logarithmic series.
(iv) Probability: Multiplication theorem, conditional probability, independent events, total probability, Bayes’s theorem, distributions.
Calculus:
(i) Limit of a function: Continuity & differentiability, derivative of composite functions, differentiation of different types of functions, chain rule, Rolle’s theorem and Lagrange mean value theorem, Maclaurin & Taylor’s series, L’Hospital’s rule, partial differentiation, successive differentiation, Leibnitz theorem, equation of tangent & normal to a curve, maxima, minima, increasing and decreasing functions.
(ii) Integration: Various methods of integration, definite integration as a limit of sum, basic properties of definite integrals, evaluation of definite integrals, application in finding area under simple curves, spheres, cones & cylinders.
(iii) Differential equations: Order and degree, formation of differential equations from general solutions, solution of first order & first degree, linear differential equations with constant coefficients, homogeneous differential equations.
Co-ordinate Geometry (2D):
Equation of pair of straight lines (homogeneous and non-homogeneous forms).
Conditions for second-degree equation to represent circle, parabola, ellipse, hyperbola.
Equations of tangents & normals, common tangents to two conics, pair of tangents, chord of contact, polar lines.
Vectors & 3D Geometry:
(i) Vectors: Scalars & vectors, unit vectors, direction cosines/ratios, multiplication by scalar, dot & cross product, applications (work done, moments, angular velocity, projection), angle between two vectors.
(ii) 3D Geometry: Direction cosines/ratios of line joining two points, Cartesian & vector equation of a line, coplanar & skew lines, shortest distance between lines, Cartesian & vector equation of a plane, angle between two lines/planes/line & plane, distance of a point from a plane, intersection of line-plane & plane-plane, plane through intersection of two planes.
**(iii) Equations of sphere, cone, cylinder.
Group Theory:
Examples: group of nth roots of unity, residue classes modulo n, modulo p (p prime).
Subgroups, homomorphisms, isomorphisms, subgroup generated by a subset, order of an element, cyclic groups, symmetric group Sn, Lagrange theorem, Fermat theorem applications, normal subgroups, fundamental theorem of homomorphisms, endomorphism, automorphism, first and second isomorphism theorems.
Rings & Fields:
Simple examples such as (2n,·), (Z₀,·).
Linear Algebra:
Vector spaces (examples), subspaces, linear dependence/independence, basis & dimension, quotient space, sum & direct sum of spaces, linear transformations, kernel & image, rank & nullity, rank-nullity theorem, composite transformations, singular/non-singular transformations, transpose, matrix representation.
Vector Calculus:
Vector differentiation: Gradient, divergence, curl, first-order vector identities, directional derivatives (applications).
Vector integration: Line, surface, volume integrals; Green’s, Gauss-divergence, and Stokes theorems (applications).
Riemann Integration:
Integration of discontinuous functions, lower & upper integrals of bounded functions, integration of step and signum functions.
Statics:
Equilibrium under three forces, coplanar systems, equilibrium under systems of coplanar forces, center of gravity, common catenary, friction.
Dynamics:
Projectile motion, work, power & energy, direct impact of smooth bodies, radial & transverse velocity/acceleration, tangential & normal acceleration.
Trigonometry:
Trigonometric equations, properties of triangles, inverse circular functions, height & distance, complex numbers, De Moivre’s theorem & applications, nth roots of unity.
Miscellaneous:
Linear programming, logarithmic differentiation, derivatives of implicit functions, approximation, applications of derivatives in rate of change, equations of straight line (various forms: parallel to axes, point-slope, perpendicular, slope-intercept).