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UPSC SCRA M athem at ics Syllabus

Algebra

Concept of a set, Union and Intersection of sets, Complement of a set, Null set, Universal set and Power

set, Venn d iagrams and simp le applicatio ns. Cartesian p rod uct o f two sets, relatio n and m app ing -

exam ples, Binary op eration o n a set - examp les. Representatio n o f real num bers on a l ine.

Complex numbers: M od ulus, Argum ent, Algebraic operations on com plex num bers. Cub e roots of unity.

Binary system o f num bers, Con version o f a decimal num ber to a binary num ber and vice-versa.

Ar i thm etic, Geom etr ic and Harmonic progressions. Summation of series involving A.P., G.P.,

and H.P..Quadratic equations with real co-efficients. Quadratic expression s: extreme values. Permut ation

and Combination, Binomial theorem and its applications.

M atrices and Determ inants: Types of matrices, equality, matrix addition and scalar multipl ication -prop ert ies. M atr ix mu lt ip l icat ion non-com mut ative and d istr ibutive prop erty over add i t ion. Transpose

of a m atrix, Determ inant o f a matrix. M inors and Cof acto rs. Prop erties of d eterm inants. Singu lar and no n-

singular matrices. Adjoint and Inverse of a square-matrix, Solution of a system of l inear equations in two

and thr ee variables-elimination m etho d, Cram ers rule and M atrix inversion m etho d (M atrices with m ro ws

and n columns where m , n < to 3 are to b e considered). Idea of a Group, Order of a Group, Abel ian Group.

Ident itiy and inverse element s Il lustration by simp le examp les.

Trigonometry

Addit ion and subtraction formulae, mult ip le and sub-mult ip le angles. Product and factor ing formulae.Inverse tr igo nom etric functio ns - Do m ains, Ranges and Graphs. DeM oivre's theorem , expansion o f Sin n0

and Cos n0 in a series of m ultip les of Sines and Cosines. Solu tion of sim ple t r igon om etric equatio ns.

App lication s: Heigh ts and Distance.

Analytic Geom etry (t wo d imensions)

Rectangu lar Cartesian. Coo rdinate system, distance between tw o p oint s, equatio n o f a straigh t l ine in

various forms, angle between two l ines, distance of a point from a l ine. Transformation of axes. Pair of

straight l ines, general equation of second degree in x and y - condition to represent a pair of straight

l ines, point of intersection, angle between two l ines. Equation of a circle in standard and in general form,

equations of tangent and norm al at a p oint, or thog onal ity o f tw o cr icles. Standard equations of parabo la,

el l ipse and hyperbola - parametr ic equations, equations of tangent and normal at a point in both

cartesian and param etric form s.

Diff erential Calculus

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Concept of a real valued function - dom ain, range and g raph. Com posite functions, one to one, onto and

inverse functions, algebra of real functions, examples of polynomial, rational, tr igonometric, exponential

and lo gar i thmic functions. Notion of l imit, Standard l imits - exam ples. Cont inui ty of functions - examples,

algebraic operations on con tinuo us function s. Derivative of a fun ction at a po int, geom etrical and ph ysical

interpretation of a derivative - applications. Derivative of sum, product and quotient of functions,

der ivative of a function with respect to another function, der ivative of a composite

fun ction , chain ru le. Second ord er derivatives. Rolle's theorem (statem ent o nly), increasing and d ecreasing

fun ctions. Application o f d erivatives in prob lems of m axim a, m inim a, greatest and least values of a

funct ion .

Integ ral Calculus and D ifferential equatio ns

Integral Calculus: Integration as inverse of differential, integration by substitution and by parts, standard

integrals involving algebraic expression, tr igonometric, exponential and hyperbolic functions. Evaluation

of definite integrals-determination of areas of plane regions bounded by curves - applications.

Differential equations: Defin i t ion of order and degree of a di f ferentia l equation, formation of adifferential equation by examples. General and particular solution of a differential equation, solution of

first ord er and first d egree differential equation of various typ es - examp les. Solut ion o f second ord er

hom og eneous d i f ferential equation with constant co-eff icients.

Vectors and its applications

M agnitu de and d irection of a vector, equal vector s, unit vector, zero vector, vectors in tw o and thr ee

dim ensions, po sition vector. Mu ltipl ication o f a vector b y a scalar, sum and difference of t wo vecto rs,

Paral lelogram law and tr iangle law of addi t ion. M ult ip l icat ion of vectors scalar produ ct or dot prod uct

of two vectors, perpendicularity, commutative and distributive properties. Vector product or cross productof two vectors. Scalar and vector tr iple products. Equations of a l ine, plane and sphere in vector form -

simp le problem s. Area of a tr iangle, paral lelogram and prob lems of p lane geom etry and tr igon om etry

using vector m ethods. Work d one by a force and m om ent of a force.

Statistics and probability

Statistics: Frequency distribution, cumulative frequency distribution - examples. Graphical representation

- Histog ram, frequency po lygon - examp les. M easure of central tendency - m ean, m edian and m od e.

Variance and stand ard deviation - det erminatio n and com parison. Correlation and regression .

Probability: Rando m experim ent, ou tcom es and associated sam ple space, events, m utu ally exclusive and

exhaustive events, impo ssible and certain events. Union and Intersection of events. Com plem entary,

element ary and com po site events. Definitio n o f pro bab il i ty : classical and statistical - exam ples.

Elementary theorems on probabil i ty - simple problems. Conditional probabil i ty, Bayes' theorem - simple

pro blem s. Rando m variable as funct ion on a sam ple space. Binom ial distr ibutio n, examp les of

random exper iments giving r ise to Binom ial d istr ibut ion.