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# Syllabus of Paper-III for SCRA Examination

## 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-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 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 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. 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 (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 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- 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 - 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 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.