Here you will find SCRA Mathematics Syllabus 2015.
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, 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 the-
orem, 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 nor-
mal 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.
SCRA General Knowledge Syllabus
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