The Union Public Service Commission (UPSC) holds a competitive examination named Special Class Railway Apprentice (SCRA) Examination for posts like Commercial Apprentice, Traffic Apprentice, Assistant Station Master, Clerk Grade I, guards etc. This examination is held once a year, generally in the month of July. Blank application forms and other particulars are published in Employment News, generally in the month of February.

## UPSC SCRA Mathematics Syllabus

### 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..

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, 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.