# UPSC: SCRA Examination Syllabus (Paper - 3)

## Union Public Service Commission

**Special Class Railway Apprentices
Examination**

## SYLLABUS (Paper - 3)

**MATHEMATICS:**__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- fficients. 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- limination method, Cramers 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.
Solution 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 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 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.

**Personality Test: **Each candidate will be interviewed
by a Board who will have before them a record of his career both academic and
extramural. They will be asked questions on matters of general interest. Special
attention will be paid to assessing their potential qualities of leadership,
initiative and intellectual curiosity, tact and other social qualities, mental
and physical energy, power of practical application and integrity of character.