NERIST NEE II 2015 Mathematics Syllabus:
Algebra: Sets: Sets and their representations; finite and infinite sets; subsets; empty or null set; universal set; equal sets; power set and complement of a set; union and intersection of sets and their algebraic properties; difference of sets; Venn diagrams; applications of sets.
Relations and functions: Ordered pairs; Cartesian product of sets; relations; domain; co-domain and range into and onto functions; one one into and one one onto functions; constant functions; identity function; composition of functions; invertible functions; Binary operations.
Complex Numbers: Complex number in the form (a+ib); representation of complex numbers by points in plane; Argand diagram; algebra of complex numbers; real and imaginary parts of a complex number; triangle inequality; modulus and argument (or amplitude) of a complex number; conjugate; square root of a complex number; cube root of unity; polar representation of a complex number.
Theory of quadratic equation: Solution of quadratic equation in the complex number system by (i) factorization (ii) using formula, relation between roots & coefficients, the nature of roots, formation of quadratic equations with given roots; Symmetric functions of roots; Equations reducible to quadratic forms.
Sequences and Series: Sequence and examples of finite and infinite sequence; Arithmetic progression (A P) – first term; common difference and nth term; sum to n terms of an A.P; Arithmetic mean (AM) insertion of the AM between any two given numbers; Geometric progression (GP) first terms; common ratio and nth term; Sum to n terms and infinite number of terms of a G.P. recurring decimal numbers as G.P. Geometric Mean (G.M) insertion of G.M and H.M. between two numbers; Harmonic progression; Harmonic mean (HM) relationship among A.M. and G.M. and H.M. arithmetic geometric series; special cases of exponential series concept of e as the sum of an infinite series; proof of function; as the infinite series; logarithmic series- infinite series for loge and related problems.
Permutations and Combinations: Fundamental principles of counting; the factorial notation; Permutation as an arrangement; meaning of P (n,r) combination, meaning of C (n,r) application of permutations & combinations.
Mathematical Induction: the principle of mathematical induction; simple applications
Binomial theorems: Statement of binomial theorem; proof of the binomial theorem for positive integral exponent using the principle of mathematical induction; general and middle terms in binomial expansions; binomial theorem for any index (without proof); applications of binomial theorem for approximation and properties of binomial coefficients.
Mathematical logic: Mathematical logic statement; Venn diagrams; negation; basic logical connectives and compound statement including the negations; truth tables; duality algebra of statements and applications of logic in solving simple problems.
Matrices and determinants: Types of matrices; Equality of matrices; operations of addition; scalar multiplication and multiplication of matrices; statements of important results on operations of matrices and their verification by numerical problems only; linear equations in matrix notation; determinants; determinants of a square matrix; properties of determinants; minors & cofactors of determinants; applicants of determinants in (i) finding area of a triangle (ii) solving a system of linear equations; Cramer’s rule; transpose; adjoint and inverse of a matrix; consistency and inconsistency of system of linear equations; applications of matrices in solving simultaneous linear equations in two or three variables.
Boolean Algebra: Boolean algebra as an algebraic structure; principle of duality; Boolean function; conditional and bi conditional statements; valid arguments; switching circuits; application of Boolean algebra to switching circuits.
Trigonometry: Trigonometric function of sum and difference of numbers; Trigonometric functions of multiples and sub multiples of numbers; conditional identities for the angles of a triangle; solution of trigonometric equations; solutions of triangles; concept of inverse trigonometric functions and their uses to reduce expression to simplest form. Vectors: Vectors & scalars; Magnitude and direction of a vector; types of vector; position vector of a point dividing a line segment in a given ratio; components of a vector; addition of vectors; multiplication of a vector by a scalar; scalar (dot) product of vectors; projection of a vector on a line; Vector (cross) product of two vector; application of dot & cross products in (i) finding area of a triangle and a parallelogram (ii) problems of plane geometry and trigonometry (iii) finding work done by a force (iv) vector moment of a vector about a point; scalar triple product and its applications; Moment of a vector about a line; co planarity of three vectors or four points using scalar triple product; vector triple product.
Coordinate Geometry: Two Dimension: (i) Area of a triangle; condition for the collinearity of three points; centroid and in – centre of a triangle; locus and its equation. The straight line and pair of straight lines- Various forms of equation of a line; intersection of lines; angles between two lines; condition for concurrency of three lines; distance of a point from a line; coordinates of orthocenter and Circum centre of a triangle; equation of family of lines; passing through the point of inter-section of two lines; homogenenous equation of second degree in x & y ; angle between pair of lines through the origin; combined equation of the bisectors of the angles between a pair of lines; condition for the general second degree equation to represent a pair of lines; point of inter-section and angles between two lines represented by S=0 and the factors of S.
Circles: Standard form of the equation of a circle; general form of the equation of a circle; its radius and centre; equation of a circle in the parametric form; equation of a circle when the end points of a diameter are given; points of inter-section of a line and a circle in the centre at the origin and condition of a line to be tangent to the circle; length of tangent; equation of the tangent; equation of a family of circles through the inter section of two circles; condition for two intersecting circles to be orthogonal. Conic sections- sections of cones; equations of conic sections (parabola, ellipse, hyperbola) in standard forms; conditions for y=mx+c to be a tangent and points of tangency.
Geometry of Three Dimension (3D): Coordinate axes; planes in three dimensional space; coordinates of a point in space; distance between two points; section formula; d.c’s and d.r’s of a line joining two points; projection of the join of two points on a given line; angle between two lines whose d.r’s are given; Cartesian and vector equation of a line through (i)a point and parallel to a given vector (ii) through two points; co linearity of three points; coplanar & skew line; shortest distance between two lines; condition for the intersection of two lines; Cartesian & vector equation of a plane (i) when the normal vector and the distance of the plane from the origin is given (ii) passing through a point and perpendicular to given vector (iii) passing through a point and parallel to two given lines through the intersection of two other planes (iv) containing two lines (v) passing through three points; Angle between two lines (ii) two planes (iii) a line and a plane. Condition of co-planarity of two lines in vector and Cartesian form; Length of perpendicular of a point from a plane by both vector and Cartesian methods; vector and Cartesian equation of a sphere, its centre and radius; diameter from of the equation of a sphere
Calculus: Function; limits and Continuity: Concept of real function; its domain and range; types of function; limit of a function, meaning and related notation; left and right hand limits; fundamental theorems on limits; limit at finite and infinite limits; continuity of a function (i) at a point (ii) over an open/closed intervals; Sum, product and quotient of continous functions, continuity of special functions; polynomial trigonometric, exponential, logarithmic, inverse trigonometric functions.
Differentiation: Derivative of a function, its geometrical and physical significance; relationship between continuity and differentiability ; derivative of some simple functions from first principle; derivative of sum; difference; product and quotient of functions; derivative of polynomial; trigonometric; exponential; logarithmic and implicit functions; derivative of functions; expressed in parametric form; chain rule and differentiation by substitution; derivatives of second order.
Application of Derivatives: Rate of change of quantities; tangents and normals; increasing and decreasing functions and sign of the derivatives; maxima & minima; greatest and least values; Rolle’s theorem and mean value theorem (without proof), curve sketching of simple curves.
Indefinite integrals: Integration as inverse of differentiation; properties of integrals; integration by substitution by parts; partial functions; and their use in integration; integration of rational functions; integration of trigonometric functions of the type.
Definite Integrals: Definite integral as limit of a sum; fundamental theorems of integral calculus; evaluation of definite integrals by (i) substitution (ii) using properties of definite integrals; application of definite integrals in finding the areas bounded by a curve, circle; parabola and ellipse in standard form between two ordinates and x-axis; area between two curves (line and circle; line & parabola, line & ellipse)
Differential Equations: Definition; order and degree; general and particular solution; formation of a differential equation whose general solution is given; solution of differential equation by the method of separation of variables; homogeneous differential equations; linear differential equation of the type.
Statics and Dynamics: Elementary Statics – Introduction, basic concepts, laws of mechanics; force; resultant of forces; acting at a point; parallelogram law of forces; resolved parts of a force; equilibrium of a particle under three concurrent forces; triangle law of forces and its converse; Lami’s theorem and its converse; two parallel; like and unlike parallel forces; couple and its moment.
Elementary Dynamics: Basic concept like displacement; speed; velocity; average; speed; instantaneous speed; acceleration and retardation; resultant of two velocities; motion of a particle along a line when moving with constant acceleration; motion of a particle under gravity; projectile motion; the path of a projectile; its horizontal range; velocity at any instant; greatest height and time of flight.
Probability: Random experiment and associated sample space event as subsets of sample space; occurrence of an event; impossible events; sure events; combination of events through the operation “and” “or” “not” and their set representation; meaning of equally likely outcomes to the total number of outcomes; equally likely events; addition rule of mutually exclusively events; Conditional probability; independent events; independent experiments P (A or B); P (A and B); Baye’s theorem and its application; recall of concept of random variables and its probability distribution; mean and variance of random variables; Binomial and Poisson’s distributions; their mean; variance and application of these distributions in commerce and industry.