We also provide the UPSC Mains Mathematics at Rajendra Nagar near Karol Bagh in central Delhi under the guidance of our Director and eminent Mathematician Tyagi Sir, who has been teaching Mathematics for more than 25 years known as the best mentor for UPSC in Mathematics. Tyagi Sir, has presented more than hundred seminars on UPSC all across the country. Maximum students from Rising Star Academy who cleared UPSC have scored very high marks in Mathematics, as can be seen in our results given below. Now-a-days after implementation of CSAT with new pattern of mains exam, students from Science background are on the top of UPSC results so in that case Mathematics plays vital role in clearing UPSC with high marks and good rank.

1.Our Toppers


Educational Qualification

The candidate must hold a degree of any of Universities incorporated by an Act of the Central or State Legislature in India or other educational institutions established by an Act of Parliament or declared to be deemed as a University Under Section-3 of the University Grants Commission Act, 1956, or possess an equivalent qualification.

NOTE I :Candidates who have appeared at an examination the passing of which would render them educationally qualified for the Commission’s examination but have not been informed of the results as also the candidates who intend to appear at such a qualifying examination will also be eligible for admission to the Preliminary Examination. All candidates who are declared qualified by the Commission for taking the Civil Services (Main) Examination will be required to produce proof of passing the requisite examination with their application for the Main Examination failing which such candidates will not be admitted to the Main Examination. The applications for the Main Examination will be called sometime in the month of August/September.

NOTE II : In exceptional cases the Union Public Service Commission may treat a candidate who has not any of the foregoing qualifications as a qualified candidate provided that he/she has passed examination conducted by the other Institutions, the standard of which in the opinion of the Commission justifies his/her admission to the examination.

NOTE III :Candidates possessing professional and technical qualifications which are recognised by Government as equivalent to professional and technical degree would also be eligible for admission to the examination.

NOTE IV : Candidates who have passed the final professional M.B.B.S. or any other Medical Examination but have not completed their internship by the time of submission of their applications for the Civil Services (Main) Examination, will be provisionally admitted to the Examination provided they submit along with their application a copy of certificate from the concerned authority of the University/ Institution that they had passed the requisite final professional medical examination. In such cases, the candidates will be required to produce at the time of their interview original Degree or a certificate from the concerned competent authority of the University/Institution that they had completed all requirements (including completion of internship) for the award of the Degree.

For the Indian Administrative Service and the Indian Police Service, a candidate must be a citizen of India.
For other services, a candidate must be either:—
a citizen of India, or
a subject of Nepal, or
a subject of Bhutan, or
a Tibetan refugee who came over to India before 1st January, 1962 with the intention of permanently settling in India,

a person of Indian origin who has migrated from Pakistan, Burma, Sri Lanka, East African countries of Kenya, Uganda, the United Republic of Tanzania, Zambia, Malawi, Zaire, Ethiopia and Vietnam with the intention of permanently settling in India.
Provided that a candidate belonging to categories (b), (c), (d) and (e) shall be a person in whose favour a certificate of eligibility has been issued by the Government of India.

Provided further that candidates belonging to categories (b), (c) and (d) above will not be eligible for appointment to the Indian Foreign Service.

A candidate in whose case a certificate of eligibility is necessary, may be admitted to the examination but the offer of appointment may be given only after the necessary eligibility certificate has been issued to him/her by the Government of India.


(a) A candidate must have attained the age of 21 years and must not have attained the age of 30 years on 1st August, i.e. he/she must have been born not earlier than 2nd August, 1983 and not later than 1st August, 1992.

(b) The upper age limit prescribed above will be relaxable :

(i) upto a maximum of five years if a candidate belongs to a Scheduled Caste or a Scheduled Tribe.

(ii) upto a maximum of three years in the case of candidates belonging to Other Backward Classes who are eligible to avail of reservation applicable to such candidates.

(iii) upto a maximum of five years if a candidate had ordinarily been domiciled in the State of Jammu & Kashmir during the period from the 1st January, 1980 to the 31st day of December, 1989.

(iv) upto a maximum of three years in the case of Defence Services personnel disabled in operations during hostilities with any foreign country or in a disturbed area and released as a consequence thereof.

(v) upto a maximum of five years in the case of ex-servicemen including Commission Officers and ECOs/ SSCOs who have rendered at least five years Military Service as on 1st August and have been released (i) on completion of assignment (including those whose assignment is due to be completed within one year from 1st August) otherwise than by way of dismissal or discharge on account of misconduct or inefficiency, or (ii) on account of physical disability attributable to Military Service, or (iii) on invalidment.

(vi) Upto a maximum of five years in the case of ECOs/SSCOs who have completed an initial period of assignment of five years Military Service as on 1st August and whose assignment has been extended beyond five years and in whose case the Ministry of Defence issues a certificate that they can apply for civil employment and that they will be released on three months notice on selection from the date of receipt of offer of appointment.

(vii) upto a maximum of 10 years in the case of blind, deaf-mute and orthopaedically handicapped persons.

NOTE I : Candidates belonging to the Scheduled Castes and the Scheduled Tribes and the Other Backward Classes who are also covered under any other clauses of para 3(ii)(b) above, viz. those coming under the category of Ex-servicemen, persons domiciled in the State of J & K, blind, deaf-mute and orthopaedically handicapped etc. will be eligible for grant of cumulative age-relaxation under both the categories.

NOTE II : The term ex-servicemen will apply to the persons who are defined as ex-servicemen in the Ex-servicemen (Reemployment in Civil Services and Posts) Rules, 1979, as amended from time to time.

NOTE III : The age concession under para 3(ii)(b)(v) and (vi) will not be admissible to Ex-Servicemen and Commissioned Officers including ECOs/SSCOs who are released on own request.

NOTE IV : Notwithstanding the provision of age-relaxation under para 3 (ii) (b) (vii) above, a physically disabled candidate will be considered to be eligible for appointment only if he/she (after such physical examination as the Government or appointing authority, as the case may be, may prescribe) is found to satisfy the requirements of physical and medical standards for the concerned Services/posts to be allocated to the physically disabled candidates by the Government.


The date of birth accepted by the Commission is that entered in the Matriculation or Secondary School Leaving Certificate or in a certificate recognised by an Indian University as equivalent to Matriculation or in an extract from a Register of Matriculates maintained by a University, which extract must be certified by the proper authority of the University or in the Higher Secondary or an equivalent examination certificate.

These certificates are required to be submitted only at the time of applying for the Civil Services (Main) Examination.

No other document relating to age like horoscopes, affidavits, birth extracts from Municipal Corporation, service records and the like will be accepted.

The expression Matriculation/Secondary Examination Certificate in this part of the instruction includes the alternative certificates mentioned-above.

NOTE 1 : Candidates should note that only the Date of Birth as recorded in the Matriculation/Secondary Examination Certificate or an equivalent certificate as on the date of submission of applications will be accepted by the Commission and no subsequent request for its change will be considered or granted.

NOTE 2 : Candidates should also note that once a Date of Birth has been claimed by them and entered in the records of the Commission for the purpose of admission to an examination, no change will be allowed subsequently (or at any other examination of the Commission) on any grounds whatsoever.

NOTE 3 : The candidate should exercise due care while entering their date of birth in column 3 of the Application Form for the Preliminary Examination. If on verification at any subsequent stage, any variation is found in their date of birth from the one entered in their matriculation or equivalent Examination certificate, disciplinary action will be taken against them by the Commission under the Rules.


Every candidate appearing at the examination who is otherwise eligible, shall be permitted four attempts at the examination.

Provided that this restriction on the number of attempts will not apply in the case of Scheduled Castes and Scheduled Tribes candidates who are otherwise eligible.

Provided further that the number of attempts permissible to candidates belonging to Other Backward Classes, who are otherwise eligible shall be seven. The relaxation will be available to the candidates who are eligible to avail of reservation applicable to such candidates.

Provided further that a physically handicapped will get as many attempts as are available to other nonphysically handicapped candidates of his or her community, subject to the condition that a physically handicapped candidate belonging to the General Category shall be eligible for seven attempts. The relaxation will be available to the physically handicapped candidates who are eligible to avail of reservation applicable to such candidates.


(i) An attempt at a Preliminary Examination shall be deemed to be an attempt at the Examination.

(ii) If a candidate actually appears in any one paper in the Preliminary Examination, he/she shall be deemed to have made an attempt at the Examination.

(iii) Notwithstanding the disqualification/cancellation of candidature, the fact of appearance of the candidate at the examination will count as an attempt.


A candidate who is appointed to the Indian Administrative Service or the Indian Foreign Service on the results of an earlier examination and continues to be a member of that service will not be eligible to compete at this examination.

In case such a candidate is appointed to the IAS/IFS after the Preliminary Examination of Civil Services Examination, is over and he/she continues to be a member of that service, he/she shall not be eligible to appear in the Civil Services (Main) Examination, notwithstanding his/her having qualified in the Preliminary Examination.

Also provided that if such a candidate is appointed to IAS/IFS after the commencement of the Civil Services (Main) Examination, but before the result thereof and continues to be a member of that service, he/she shall not be considered for appointment to any service/post on the basis of the result of this examination viz. Civil Services Examination.


Candidates must be physically fit according to physical standards for admission to Civil Services Examination. as per guidelines given in Appendix-III of Rules for Examination published in the Gazette of India Extraordinary .



These are very comprehensive programmes which run over the span of 4-6 months depending on the course. Under these programmes the classes are conducted 4-6 hrs per day for 5 to 6 days in a week. The class schedules are designed in a manner that every student in the class gets equal opportunity to learn and apply.

As the understanding level of all the students in a class may not be uniform, we try to distinguish weaker so called average students in the class and work over them. For such students we have developed Basic Building Measures (BBM) which has two components

1. BBM- Tests: After 2-3 lectures on any topic the academy offers a test to all the students. This test (Called BBM Test) contains very basic and fundamental questions about that topic. The performances over this test are analyzed comprehensively and on the basis of their performance, those students are identified who could not cross minimum thresh hold and requires extra care.

2. BBM Classes: The students Identified by BBM tests are clubbed into various groups and dedicated faculty are assigned to each group. These groups are given extra classes (BBM Classes). In these classes, this faculty helps the students clear their doubts and concept so that they too can equally participate in the regular classes along with the so called average plus students.

The RCPs of Rising Star Academy are complete in its nature. In the span of almost half a year we try to bridge the gap between your degree and learning so that you can appear in any competitive exam of your eligibility and interest.

Batches UPSC Mains
Registration Open 01 June 2017
Open Seminar/Scholarship Test 23-July-2017
Batch Starts 23-July-2017

Course Fee
UPSC 40,000/-

IAS Mains Mathematics Syllabus


Linear Algebra: Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimension; Linear transformations, rank and nullity, matrix of a linear transformation.
Algebra of Matrices; Row and column reduction, Echelon form, congruence’s and similarity; Rank of a matrix; Inverse of a matrix; Solution of system of linear equations; Eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, Symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues.

Calculus: Real numbers, functions of a real variable, limits, continuity, differentiability, mean-value theorem, Taylor's theorem with remainders, indeterminate forms, maxima and minima, asymptotes; Curve tracing; Functions of two or three variables: limits, continuity, partial derivatives, maxima and minima, Lagrange's method of multipliers, Jacobian.
Riemann's definition of definite integrals; Indefinite integrals; Infinite and improper integrals; Double and triple integrals (evaluation techniques only); Areas, surface and volumes.

Analytic Geometry: Cartesian and polar coordinates in three dimensions, second degree equations in three variables, reduction to canonical forms, straight lines, shortest distance between two skew lines; Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

Ordinary Differential Equations: Formulation of differential equations; Equations of first order and first degree, integrating factor; Orthogonal trajectory; Equations of first order but not of first degree, Clairaut's equation, singular solution. Second and higher order linear equations with constant coefficients, complementary function, particular integral and general solution.
Second order linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using method of variation of parameters.

Laplace and Inverse Laplace transforms and their properties; Laplace transforms of elementary functions. Application to initial value problems for 2nd order linear equations with constant coefficients.

Dynamics & Statics: Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; constrained motion; Work and energy, conservation of energy; Kepler's laws, orbits under central forces.

Equilibrium of a system of particles; Work and potential energy, friction; common catenary; Principle of virtual work; Stability of equilibrium, equilibrium of forces in three dimensions.

Vector Analysis: Scalar and vector fields, differentiation of vector field of a scalar variable; Gradient, divergence and curl in cartesian and cylindrical coordinates; Higher order derivatives; Vector identities and vector equations. Application to geometry: Curves in space, Curvature and torsion; Serret-Frenet’s formulae. Gauss and Stokes’ theorems, Green’s identities.


(1) Algebra: Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayley’s theorem.
Rings, subrings and ideals, homomorphisms of rings; Integral domains, principal ideal domains, Euclidean domains and unique factorization domains; Fields, quotient fields.

Real Analysis: Real number system as an ordered field with least upper bound property; Sequences, limit of a sequence, Cauchy sequence, completeness of real line; Series and its convergence, absolute and conditional convergence of series of real and complex terms, rearrangement of series.

Continuity and uniform continuity of functions, properties of continuous functions on compact sets.

Riemann integral, improper integrals; Fundamental theorems of integral calculus.
Uniform convergence, continuity, differentiability and integrability for sequences and series of functions; Partial derivatives of functions of several (two or three) variables, maxima and minima.

Complex Analysis: Analytic functions, Cauchy-Riemann equations, Cauchy's theorem, Cauchy's integral formula, power series representation of an analytic function, Taylor’s series; Singularities; Laurent's series; Cauchy's residue theorem; Contour integration.

Linear Programming: Linear programming problems, basic solution, basic feasible solution and optimal solution; Graphical method and simplex method of solutions; Duality.
Transportation and assignment problems.

Partial differential equations: Family of surfaces in three dimensions and formulation of partial differential equations; Solution of quasilinear partial differential equations of the first order, Cauchy's method of characteristics; Linear partial differential equations of the second order with constant coefficients, canonical form; Equation of a vibrating string, heat equation, Laplace equation and their solutions.

Numerical Analysis and Computer programming: Numerical methods: Solution of algebraic and transcendental equations of one variable by bisection, Regula-Falsi and Newton-Raphson methods; solution of system of linear equations by Gaussian elimination and Gauss-Jordan (direct), Gauss-Seidel(iterative) methods. Newton's (forward and backward) interpolation, Lagrange's interpolation.

Numerical integration: Trapezoidal rule, Simpson's rules, Gaussian quadrature formula.
Numerical solution of ordinary differential equations: Euler and Runga Kutta-methods.
Computer Programming: Binary system; Arithmetic and logical operations on numbers; Octal and Hexadecimal systems; Conversion to and from decimal systems; Algebra of binary numbers.
Elements of computer systems and concept of memory; Basic logic gates and truth tables, Boolean algebra, normal forms.
Representation of unsigned integers, signed integers and reals, double precision reals and long integers.
Algorithms and flow charts for solving numerical analysis problems.

Mechanics and Fluid Dynamics: Generalized coordinates; D' Alembert's principle and Lagrange's equations; Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions.
Equation of continuity; Euler's equation of motion for inviscid flow; Stream-lines, path of a particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid.

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