# NERIST 2012 Mathematics Syllabus

The mathematics is the longest section in NEE II examination. It consists of 60 marks. The syllabus of mathematics is given in this section.

**Algebra **:Set : Algebra of sets, Cartesian product, Function, Domain and Range of a function etc. Function from AxA->A, Associativity and Commutativity of binary operations, Inverse of a function.

**Complex Numbers: **Square root of a complex number, cube root of unity, Triangle inequality.

**Quadratic Equation : **Identity and equations, solution of a quadratic equation in complex number system, Equations reducible to quadratic equations, Relation between roots and coefficients, Nature of the roots, Formation of quadratic equation with given roots, Symmetric functions of roots of quadratic equation.

**Permutations and Combinations :** Standard results on nPr and nCr, Applications including circular permutations.

**Binomial Theorem **: Binomial theorem for any index, General and particular terms, Application of binomial theorem for approximation, properties of binomial coefficients.

**Matrices and Determinants :** Definition of matrix, types of matrices, Algebra of matrices, singular and non singular matrices, Linear equations in matrix notation. Determinants, Minors and co-factors of determinants, Expansion of a determinant, properties and elementary transformation of determinants, Application of determinants in solution of equations, Cramers rule, Adjoint and Inverse of matrices and its properties, Inverse of a matrix using elementary row transformations, Application of matrices in solving simultaneous linear equations in three variables.

**Exponential and Logarithmic Series :** The infinite series for ex , Infinite series for log (1-x), log (1+x) etc and related problems.

**Trigonometry :** Relation between the sides and the trigonometrical ratios of the angles of a triangle, Simple problems on solutions of triangle, Inverse trigonometric functions. Principal value branches of inverse trigonometric functions.

**Coordinate Geometry : **Orthocentre, circumcentre and centroid of a triangle, Family of lines, parametric equation of a circle, Equation of tangent and its length, system of circles, Conic sections, Parabola, ellipse and hyperbola - standard forms.

**Vector and its Applications to Geometry : **Vector as direct line segment, Magnitude and direction of a vector, Equal vectors, unit vector, Zero vector, position vector of a point, Components of a vector, Vectors in two and three dimensions; addition of vectors, multiplication of a vector by a scalar, position vector of the point dividing a line segment in a given ratio, application of dot and cross products in (i) finding areas of triangle and parallelogram (ii) finding work done by a force (iii) vector moment of a vector about a point (iv) problems of plane geometry and trigonometry, scalar triple product and its application, co planarity of three vectors .

**Three Dimensional Geometry : **Decomposition of a vector into three non-coplanar directions as base in 3-D, direction ratios and direction cosines for any vector, Angle between two vectors where direction cosines are given, Distance between two points, Condition for the intersection of two lines, Shortest distance between two lines, equation of a plane containing a given point and normal to a given direction, angle between two planes, Angles between a line and a plane, Distance of a point from the plane, Equation of any plane passing through the intersection of the two planes, equation of a sphere in (i) central form (ii) general form (iii) diameter form.

**Differential Calculus :** Differentiation of Logarithmic and Implicit functions, Differentiation of parametric functions, Application of Derivatives, Rolles theorem, Mean Value theorems, tangent and normal, increasing and decreasing functions, maxima and minima, curvature, asymptotes, tracing of curves.

**Integral Calculus : **Simple integration, Integration by parts, integration by substitution, Integration of rational and irrational functions, Integration of Trigonometric functions of the type ò dx / (a + b cosx) , ò dx / (a + b sin x) ; and ò sin mx cosn x dx.

Definite integrals, Evaluation of definite integrals, application of definite integrals in finding (i) the areas of simple curves, (ii) volumes and surfaces of solids of revolution.

**Differential Equations :** Order and Degree of a differential equation, solution of differential equation by the method of variables separable, homogeneous and their solution, linear equation of the first order.

**Statistics : **Classification, tabulation and graphical representation of data. Measures of central tendency such as mean and median. Standard deviation, variance, Mean deviation and mean deviation from median as a measure of location.

Bivariate frequency distribution, marginal and conditional frequency distribution, correlation coefficient. Use of scatter diagram in interpreting the values of correlation coefficient, calculation of regression coefficient and the two lines of regression by the method of least squares.

Random experiment and associated sample space. Event, Algebra of events, classical probability. Law of addition of probabilities. Multiplication law of probability and conditional probability . Independent events, Bayes Theorem.

Random variables, Probability mass function, different random experiments giving rise to random variables with the binomial distribution.

**Correlation and Regression : **Bivariate frequency distributions as arising from observations of two variables on the same unit of observation, Marginal and conditional frequency distributions, derived from a bivariable frequency distribution, The concept of relationship between variables introduced as the dependence of conditional distribution on the values of the conditioning variable, Distinction between relationship and functional relationship.

Correlation analysis as the measurement of the strength of relationship between two quantitative variables and regression analysis as the method of predicting the values of one quantitative variable from those of the other quantitative variable.

Definition and calculation of the correlation coefficient, positive and negative correlation, perfect correlation, Use of the scatter diagram in interpreting the values of the correlation coefficient.

Calculation of the regression coefficient and the two lines of regression by the method of least squares, Use of the lines of regression for production.

**Probability :** Random experiment and associated sample space (i.e. set of all outcomes), Events as subject of the sample space, occurrence of an event, sure event, impossible event, mutually exclusive events, elementary events, equally likely elementary events, Definition of probability of an even as the ratio of the number of favourable equally likely events, to the total number of equally likely events, Addition rule for mutually exclusive events.

Combination of events through the operations GT and Not, and their set representation, Probability of the events Aor B, Not A. Conditional probability Independent events; Independent experiments, Calculation of probabilities of events associated with independent experiments, Applications of Bayes theorem.

Random variables as a function on a sample space, (only random variables taking units number of values to be considered).

Distribution of a random variable derived from the probabilities of events of the sample space on which the random variable is defined.

**Binomial/Distribution : **Examples of different random experiments giving rise to random variables with the binomial distribution.