UPPSC GIC Lecturer Notification 2025

1. 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. 2. Algebra:
(i) Matrices: Types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric Matrices Addition, multiplication & scalar multiplication of matrix Singular and non-singular matrices Ir-revartable matrices
(ii) Determinants: Determinants of a square matrix (up to 3×3 matrix) Properties of determinants, Adjoin and inverse of a square matrix Consistency and number of solutions of system of linear equations by examples. Solving system of linear equations in two of three variables (having unique solutions)
(iii) Theory of equations of degree greater than or equal to two: Arthmatical, Geometrical and Hormonical progressions. Permutations and combinations, Binomial theorem Sum of exponential and logarithmic senes
(iv) Probability: Multiplication theorem on probability, Conditional probability, Independent events Total probability Bayes’s theorem distribution 3. Calculus:
(i) Limit of a function: Continuity & differentiability, derivative of composite functions, and differentiation of different types of functions. Chain rule, Roles theorem and lagrange mean value theorem, Maciaunns & Taylor’s series L. Hospitals rule, partial differentiation, successive differentiation, Leibnitz theorem, equation of tangent & normal to a given curve, Maxima, minima, increasing and decreasing function
(ii) Integration: Various methods of integration of definite integration as a limit of sum, Basic properties of definite integrals & evalution of definite integrals. Application in finding the area under simple curves of spheres, cones & cylinders.
(iii) Differential equations: Order and degree of a differential equation. Formation of differential equations whose general solution is given. Solution of differential equations of Ist order & Ist degree. Linear differential equations with constant coefficients Homogeneous differential equations. 4. Co-ordinate geometry of two dimensions:
Equation of the pair of straight in homogeneous and non homogeneous form. Conditions when a non-homogeneous equation of 2nd degree represents circle, parabola, ellipse and hyperbola equation of tangents & normals to the above conics Common tangents to the two conics, Pair of tangents Chord of contacts, polarlines to the above conics 5. Vectors and three dimensional geometry:
(i) Vectors: Vector & scalars. Unit vectors, Direction considerations of a vector Multiplication of a Vector by scalar, dot product cross product of vector and their in physics (work done and moments, angular velocity, projection of a vector on a line Angle between two vector)
(ii) Three dimension Geometry: Direction cosine ratios of line joining two points, Cartesian and vector equation of a line. Coplanar and skew Ines, shortest distance between two lines, cartesian and vector equation of a plane Angle between (a) two lines (b) two planes (c) Aline and a plane Distance of a point from a plane intersection of two Iine, Intersection of a line of plane & intersection of two plane. Equation of a plane passing through the intersection of two planes
(iii) Equation of a sphere, cones, cylinders. 6. Group:
Examples especially the group of nith roots of unity, group of residue class modulo n and modulo p where p is a prime subgroup. Homomorphism and somorphisms properties of Homomorphism. Subgroups generated by a subset. Order of element in a group, Cyclic group, Symmetric group Sn Lagrange theorem, Fermat’s theorem with application point of view. Normal subgroups Fundamental theorem of Homomorphism, Endomorphism, automorphism, First isomorphism theorem and second isomorphism theorem
Ring and field with simple examples as (2n,⋅,)δ(20,⋅)
Linear Algebra: vector space with examples, subspace, linear dependence and Independence, Basis and dimension of a vector space, Quotient space, Sum and direct sum of spaces. Linear transformation, Karmel and image of a-linear transformation, Rank and nullity of linear transformations, Rank nullity theorem. Composite of linear transformations and its rank & nulity Singular and non singular linear transformation. Transpose of a linear transformation, Matrix of a linear transformation
Vector differentiation: Gradient, divergence, curl, first order vector identities Directional derivatives (with application point of view)
Vector integration: Line integral surface integral, volume integral, Green’s theorem, Gauss-divergence theorem, stokes’s theorem, (From application point of view)
Riemann integration: Integration of discontinuous functions, Lower and upper integrals of a bounded functions, Integration of a step function and signum function
Statics: Equilibrium of a body under the action of three forces, coplanar forces. Equilibrium of a body Under the Action of a system of coplanar forces. Centre of gravity. common catenary Friction
Dynamics: Motion of a projectile in a vertical plane under gravity, Work power and energy. Direct impact of smooth bodies, Radial and transverse Velocity and acceleration Tangential and normal acceleration
Trigonometry: Trigonometric equations, Properties of triangles Inverse circular functions. Height and distance, Complex numbers. D-movers theorem & its application, nth roots of unity
Linear programming, logarithmic differentiation, derivatives of implicit function. approximation, application of derivatives in the form of rate of change of quantities. straight ine, various form of the equation of ine (parallel of axes, point shape form perpendicular form, slope intercept form)