: Speical Class Railway Examination
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
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 Co-factors.
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, Cramers rule and Matrix inversion method
(Matrices with m rows and n columns where m, n less than equal to 3 are to be
Idea of a Group, Order of a Group, Abelian group. Identitiy and inverse
elements-Illustration by simple examples.
Addition and subtraction formulae,
multiple and sub-multiple angles. Products and factoring formulae. Inverse
trigonometric functions - Domains, Ranges and Graphs. DeMoivre's theorem,
expansion of Sin n 0 and Cos n 0 in a series of multiples of Sines and Cosines.
Solution of simple trigonometric equations. Applications: Heights and Distance.
3. Analytic Geometry
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.
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 on continuous functions. Derivative of a
function at a point, geometrical and physical interpretation of a derivative -
applications. Derivative of sum, product and quotient of functions, derivative
of a function with respect to another function, derivative of a composite
function, 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 differentiation, 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-
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
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 - its properties, unit vector
perpendicular to two given 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
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