*Mathematics & Statistics HSC Syllabus – Maharashtra HSC Board*

*Introduction *

* Mathematics is the language of all sciences and is perhaps the only subject which merits this distinction. Mathematics is the backbone of all sciences and it is an inseparable part of human life. *

* Higher Secondary is a launching stage from where students would go to either for academic education in Mathematics or professional courses like Engineering and Computer Technology, Physical and Biological Sciences. Hence to fulfil the needs of students, it is utmost important to make the study of Mathematics more meaningful by acquainting the student with many branches of mathematics. This will help them in developing Mathematical tools to be used in the professional education. Apart from motivating topics from real life situations and other subject areas, major thrust is also on application of various concepts. *

* The proposed syllabus has been designed in accordance with National Curriculum Framework 2005 and as per guidelines given in Focus Group on Teaching of Mathematics 2005 which is to meet the emerging needs of all categories of students.*

*Objectives*

*To enable the students *

*1) to acquire knowledge and critical understanding, particularly by way of motivation and visualization of basic concepts, terms, principles, symbols and mastering the underlying processes and skills. *

*2) to apply the knowledge and skills in Mathematics and related problems from other subjects, by more than one method. *

*3) to develop positive attitude to think, analyze and articulate logically.*

*4) to develop interest in Mathematics by participating in various related competitions and self-learning. *

*5) to acquaint students with different aspects of Mathematics used in real life. *

*6) to develop an interest in students to study Mathematics as a discipline. *

*7) to develop awareness of the need for national integration, protection of an environment, removal of social barriers, elimination of sex biases and observance of small family norm. *

*8) to develop reverence and respect towards great mathematicians for their contribution to the field of Mathematics. *

*9) to develop interest in the subject by participating in related competitions.*

* Mathematics HSC Syllabus *

**Std. 12th : PART – I**

*Mathematical Logic*

*Statements – Introduction, sentences and statement, truth value of statement, open sentences, compound statement, quantifier and quantified statements, logical connectives : conjunction, disjunction, negation, implication/ conditional, bi conditional, truth tables of compound statements, examples related to real life and mathematics, statement patterns and logical equivalence – tautology, contradiction, contingency, duality, negation of compound statement, contra positive, converse, inverse, algebra of statements-idempotent law, associative law, commutative law, distributive law, identity law, complement law, involution law, DeMorgan’s laws, difference between converse, contra positive, contradiction, application-introduction to switching circuits (simple examples).*

*Matrices*

*Elementary transformation of a matrix revision of co-factor and minor, elementary row transformation, elementary column transformation, inverse of a matrix existence and uniqueness of inverse of a matrix, inverse by elementary transformation, adjoint method, application-solution of system of linear equations by – reduction method, inversion method.*

*Trigonometric functions*

*Trigonometric equations-general solution of trigonometric equation of the type : sinθ, = 0, cosθ = 0, tanθ = 0, sinθ = sinα, cosθ = cosα, tanθ = tanα, sin ^{2} θ = sin^{2} α, cos^{2} θ = cos^{2} α, tan^{2} θ = tan^{2} α, acosθ + bsinθ = C solution of a triangle : polar coordinates, sine rule, cosine rule, projection rule, area of a triangle, application, Hero’s formula, Napier Analogues, inverse trigonometric functions-definitions, domain, range, principle values, graphs of inverse trigonometric function, properties of inverse functions.*

*Pair of straight lines*

*Pair of lines passing through origin combined equation, homo-genous equation, theorem-the joint equation of a pair of lines passing through origin and its converse, acute angle between the lines represented by ax ^{2} +2hxy+by^{2} =0, condition for parallel lines, condition for perpendicular lines, pair of lines nt passing through origin-combined equation of any two lines, condition that the equation ax^{2} +2hxy+by^{2} +2gx+2fy+c=0 should represent a pair of lines (without proof), acute angle between the lines (without proof), condition of parallel and perpendicular lines, point of intersection of two lines.*

*Circle*

*Tangent of a circle-equation of a tangent at a point to *

*1) standard circle,*

*2) general circle, condition of tangency only for line y = mx + c to the circle x ^{2} + y^{2} = a^{2} , tangents to a circle from a point outside the circle, director circle, length of tangent segments, normal to a circle-equation of normal at a point.*

*Conics*

*Tangents and normal-equations of tangent and normal at a point for parabola, ellipse, hyperbola; condition of tangency for parabola; ellipse, hyperbola; tangents in terms of slope for parabola, ellipse, hyperbola, tangents from a point outside conics, locus of points from which two tangents are mutually perpendicular, properties of tangents and normal to conics (without proof).*

*Vectors*

*Revision, Collinearity and coplanarity of vectors : linear combination of vectors, condition of collinearity of two vectors, conditions of coplanarity of three vectors, section formula : section formula for internal and external division, midpoint formula, centroid formula, scaler triple product : definition, formula, properties, geometrical interpretation of scalar triple product, application of vectors to geometry medians of a triangle are concurrent, altitudes of a triangle are concurrent, angle bisectors of a triangle are concurrent, diagonals of a parallelogram bisect each other and converse, median of trapezium is parallel to the parallel sides and its length is half the sum of parallel sides, angle subtended on a semicircle is right angle.*

*Three dimensional geometry*

*Direction cosines and direction ratios: direction angles, direction cosines, direction ratios, relation between direction ratio and direction cosines, angle between two lines, condition of perpendicular lines.*

*Line*

*Equation of line passing through given point and parallel to given vector, equation of line passing through two given points, distance of a point from a line, distance between two skew lines, distance between two parallel lines (vector approach).*

*Plane*

*Equation of plane in normal form, equation of plane passing through the given point and perpendicular to given vector, equation of plane passing through the given point and parallel to two given vectors, equation of plane passing through three non-collinear points, equation of plane passing through the intersection of two given planes, angle between two planes, angle between line and plane, condition for the co-planarity of two lines, distance of a point from a plane (vector approach) *

*Linear programming problems*

*Introduction of L.P.P. definition of constraints, objective function, optimization, constraint equations, non negativity restrictions, feasible and in feasible region, feasible solutions, Mathematical formulation-mathematical formulation of L.P.P. different types of L.P.P. problems, graphical solutions for problem in two variables, optimum feasible solution.*

*Mathematics HSC Syllabus*

**Std. 12th : PART – II**

*Continuity*

*Continuity of a function at a point : left hand limit, right hand limit, definition of continuity of a function at a point, discontinuity of a function, types of discontinuity, algebra of continuous functions, continuity in interval-definition, continuity of some standard functions polynomial, rational, trigonometric, exponential and logarithmic function.*

*Differentiation*

*Revision- revision of derivative, relationship between continuity and different iability-left hand derivative and right hand derivative (need and concept), every different iable function is continuous but converse is not true, Derivative of composite function-chain rule, derivative of inverse function, derivative of inverse trigonometric function : Derivative of implicit function definition and examples, derivative of parametric function – definition of parametric function , exponential and logarithmic function derivative of functions which are expressed in one of the following form *

*a) product of functions, *

*b) quotient of functions, *

*c) higher order derivative, second order derivative d) [f _{(x)} ] ^{[g(x)]} *

*Applications of derivative*

*Geometrical application-tangent and normal at a point, Rolle’s theorem, and Mean value theorem and their geometrical interpretation (without proof), derivative as a rate measure-introduction, increasing and decreasing function, approximation (without proof), Maxim and minim introduction of extrema and extreme values, maxim and minim in a closed interval, first derivative test, second derivative test.*

*Integration*

*Indefinite integrals-methods of integration, substitution method, integrals of the various types, integration by parts (reduction formulae are not expected), integration by partial fraction-factors involving repeated and non-repeated linear factors, non-repeated quadratic factors, definite integral-definite integral as a limit of sum, fundamental theorem of integral calculus (without proof), evaluation of definite integral *

*1) by substitution, *

*2) integration by parts, properties of definite integrals.*

*Applications of definite integral*

*Area under the curve : area bounded by curve and axis (simple problems), area bounded by two curves, volume of solid of revolution-volume of solid obtained by revolving the area under the curve about the axis (simple problems).*

*Differential equation*

*Definition-differential equation, order, degree, general solution, particular solution of differential equation, formation of differential equation-formation of differential equation by eliminating arbitary constants (at most two constants), solution of first order and first degree differential equation-variable separable method, homogeneous differential equation (equation reducible to homogeneous form are not expected), Linear differential equation, applications : population growth, bacterial colony growth, surface area, Newton’s laws of cooling, radioactive decay.*

*Statistics*

*Bivariate frequency distribution – bivariate data, tabulation of bivariate data, scatter diagram, covariance of ungrouped data, covariance for bivariate frequency distribution, Karl Pearson’s coefficient of correlation.*

*Probability distribution*

*Probability distribution of a random variable-definition of a random variable, discrete and continuous random variable, probability mass function (p.m.f.), probability distribution of a discrete random variable, cumulative probability distribution of a discrete random variable, expected value, variance and standard deviation of a discrete random variable, probability density function (p.d.f.), distribution function of a continuous random variable.*

*Bernoulli trials and Binomial distribution*

*Definition of Bernoulli trial, conditions for Binomial distribution, binomial distribution (p.m.f.), mean, variance and standard deviation, calculation of probabilities (without proof), Normal distribution: p.d.f., mean, variance and standard deviation, standard normal variable, simple problems (without proof).*

*List of Practical’s : 12th std Mathematics Syllabus Maharashtra HSC Board*

*Applications of logic.**Inverse of a matrix by adjoint method and hence solution of system of linear equations.**Inverse of a matrix by elementary transformation and hence solution of system of linear equations.**Solutions of a triangle.**Tracing of tangents and normal for circle and parabola.**Tracing of tangents and normal for ellipse and hyperbola.**Applications of scalar triple product of vectors.**Three dimensional geometry – line.**Three dimensional geometry – plane.**Formations and solutions of LPP.**Applications of derivatives (Geometric applications).**Applications of derivatives – Rate measure.**Applications of derivatives – Maxim and minim**Applications of definite integrals – Limit of a sum.**Applications of definite integrals – Area.**Applications of definite integrals – volume.**Applications of differential equations.**Bivariate frequency distribution.**Expected value, variance and S.D of a random variable.**Binomial distribution.*

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