scra 2013 mathematics syllabus scra syllabus

UPSC’S SCRA 2013 MATHEMATICS SYLLABUS

Syllabus of SCRA paper III named as MATHEMATICS consist of following topics

1. Algebra :

Concept of a set, Union and Intersection of sets, Complement of a set, Null set, Universal set and Power set, Venn diagrams and simple applications. Cartesian product of two sets, relation and mapping — examples, Binary operation on a set— examples.

Representation of real numbers on a line. Complex numbers: Modulus, Argument, Algebraic operations on complex numbers. Cube roots of unity.

Binary system of numbers, Conversion of a decimal number to a binary number and vice-versa. Arithmetic, Geometric and Harmonic progressions.

Summation of series involving A.P., G.P., and H.P.. Quadratic equations with real co-efficients. Quadratic expressions: extreme values. Permutation and Combination, Binomial theorem and its applications.

Matrices and Determinants: Types of matrices, equality, matrix addition and scalar multiplication- properties. Matrix multiplication — non-commutative and distributive property over addition. Transpose of a matrix, Determinant of a matrix. Minors and Cofactors. Properties of determinants. Singular and non-singular matrices. Adjoint and Inverse of a square-matrix, Solution of a system of linear equations in two and three variables-elimination method, Cramer’s rule and Matrix inversion method (Matrices with m rows and n columns where m, n < to 3 are to be considered).

Idea of a Group, Order of a Group, Abelian Group. Identitiy and inverse elements-Illustration by simple examples.

2. Trigonometry :

Addition and subtraction formulae, multiple and sub-multiple angles. Product and factoring formulae. Inverse trigonometric functions — Domains, Ranges and Graphs. DeMoivre's theorem, expansion of Sin n0 and Cos n0 in a series of multiples of Sines and Cosines.S olution of simple trigonometric equations. Applications: Heights and Distance.

3. Analytic Geometry (two dimensions):

Rectangular Cartesian. Coordinate system, distance between two points, equation of a straight line in various forms, angle between two lines, distance of a point from a line. Transformation of axes. Pair of straight lines, general equation of second degree in x and y — condition to represent a pair of straight lines, point of intersection, angle between two lines. Equation of a circle in standard and in general form, equations of tangent and normal at a point, orthogonality of two cricles. Standard equations of parabola, ellipse and hyperbola — parametric equations, equations of tangent and normal at a point in both cartesian and parametric forms.

4. Differential Calculus:

Concept of a real valued function — domain, range and graph. Composite functions,one to one, onto and inverse functions, algebra of real functions, examples of polynomial,rational, trigonometric, exponential and logarithmic functions. Notion of limit,Standard limits - examples. Continuity of functions - examples, algebraic operations oncontinuous functions. Derivative of a function at a point, geometrical and physical interpretationof a derivative - applications. Derivative of sum, product and quotient of functions,derivative of a function with respect to another function, derivative of a compositefunction, chain rule. Second order derivatives. Rolle's theorem (statement only),increasing and decreasing functions. Application of derivatives in problems of maxima,minima, greatest and least values of a function.

5. Integral Calculus and Differential equations :

Integral Calculus : Integration as inverse of differential, integration by substitution and by parts, standard integrals involving algebraic expression, trigonometric, exponential and hyperbolic functions. Evaluation of definite integrals-determination of areas of plane regions bounded by curves - applications. Differential equations : Definition of order and degree of a differential equation, formation of a differential equation by examples. General and particular solution of a differential equation, solution of first order and first degree differential equation of various types - examples. Solution of second order homogeneous differential equation with constant co-efficients.

6. Vectors and its applications :

Magnitude and direction of a vector, equal vectors, unit vector, zero vector, vectors in two and three dimensions, position vector. Multiplication of a vector by a scalar, sum and difference of two vectors, Parallelogram law and triangle law of addition. Multiplication of vectors — scalar product or dot product of two vectors, perpendicularity, commutative and distributive properties. Vector product or cross product of two vectors.

Scalar and vector triple products. Equations of a line, plane and sphere in vector form - simple problems. Area of a triangle, parallelogram and problems of plane geometry and trigonometry using vector methods. Work done by a force and moment of a force.

7. Statistics and probability :

Statistics : Frequency distribution, cumulative frequency distribution - examples. Graphical representation - Histogram, frequency polygon - examples. Measure of central tendency - mean, median and mode. Variance and standard deviation – determination and comparison. Correlation and regression.

Probability : Random experiment, outcomes and associated sample space, events, mutually exclusive and exhaustive events, impossible and certain events. Union and Intersection of events. Complementary, elementary and composite events. Definition of probability : classical and statistical - examples. Elementary theorems on probability - simple problems. Conditional probability, Bayes' theorem - simple problems. Random variable as function on a sample space. Binomial distribution, examples of random experiments giving rise to Binomial distribution.