# Maharashtra Board Math Syllabus for Class 12

## Maharashtra Board Syllabus for Class 12 Math

MATHEMATICS AND STATISTICS

STD. XII

PART – I

1. Mathematical Logic : Statements - Introduction, sentences and statement, truth value of

statement, open sentences and truth sets, compound statement, quantifier and quantified

statements, logical connective-conjunction, disjunction, negation, implication/ conditional,

biconditional, there exists, truth tables of compound statements, examples related to real life and

mathematics, statement patterns and logical equivalence-tautology, contradiction, contingency,

logical equivalence, logical equivalence, duality, negation of compound statement,

contrapositive, 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, contrapositive, contradiction, application-introduction to

switching circuits (simple examples).

2. Matrices : Elementary transformation of a matrix-revision of cofactor and minor, elementary

row transformation, elementary column transformation, inverse of a matrix-existance 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.

3. Trigonometric functions : Trigonometric equations-general solution of trigonometric equation of

the type, solution of Trigonometric equations, sine rule, cosine rule, projection rule, area of a

triangle, application, Hero‟s formula, Napier Analogues, inverse trigonometric functionsdefinitions, domain, range, principle values, graphs of inverse trigonometric function, properties

of inverse functions.

4. Pair of straight lines : Pair of lines passing through origin-conbined equation, homogenous

equation, theorem-the joint equation of a pair of lines passing through origin and its converse,

angle between the lines represented by ax

perpendicular lines, pair of lines not passing through origin-conbined equation of any two lines,

condition that the equation ax2+2hxy+by2+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.

5. 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 x2+y2=a2, 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.

6. Conics : Tangents and normals-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 normals to conics

(without proof).

7. Vectors : Revision, Collinearty and coplanerity of vectors-linear combination of vectors,

condition of collinearity of two vectors, conditions of coplanerity ofthree 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 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.

8. 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.

9. Line : Equation of line passing through given point and parallel to given vector, equation of line

passing through two given points, dist. of a point from a line, distance between two skew lines,

distance between two parallel lines (vector approach).

10. 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 coplanerity of two lines, distance of a point from a

plane (vector approach).

11. Linear programming problems: Introduction of L.P.P. definition of constraints, objective

function, optimization, constraint equations, non-negativity restrictions, feasible and infeasible

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

STD. XII - PART – II

1. Continuity : Continuity of a function at a point-left hand limit, right hand limit, definition of a

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 functionspolynomial, rational, trigonometric, exponential and logarithmic function.

2. Differentiation : Revision- revision of derivative, relationship between continuity and

differentiability-left hand derivative and right hand derivative (need and concept), every

differentiable function is continuous but converse is not true, Derivative of composite functionchain 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)

3. Application of derivative : Geometrical application-tangent and normal at a point, Rolls

theorem, and Mean value theorem and their geometrical interpretation (without proof), derivative

as rate measure-introduction, increasing and decreasing function, approximation (without proof),

Maxima and minima-introduction of extreme and extreme values, maxima and minima in a

closed interval, first derivative test, second derivative test.

4. Integration : Indefinite integrals-methods of integration, substitution method, integrals of the

type, integration by parts-integration by parts, integrals of type (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 properties of definite

integrals.

5. 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).

6. Differential equation : Definition-differential equation, order, degree, general solution,

particular solution of differential equation, formation of differential equation-formation of

differential equation by elimmating arbitary constants (at most two constants), solution of first

order and first degree differential equation-variable separable method, homogenous differential

equation (equation reducible to homogenous form are not expected), Linear differential equation,

applications-population growth, bacterial colony growth, surface area, Newton‟s laws of cooling,

radioactive decay.

7. Statistics: Bivariate frequency distribution - bivariate data, tabulation of bivariate data, scatter

diagram, covariance of or ungrouped data, covariance for bivariate frequency distribution, Karl

Pearson‟s co-efficient of correlation.

8. 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.

9. 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).

STD. XII

PART – I

1. Mathematical Logic : Statements - Introduction, sentences and statement, truth value of

statement, open sentences and truth sets, compound statement, quantifier and quantified

statements, logical connective-conjunction, disjunction, negation, implication/ conditional,

biconditional, there exists, truth tables of compound statements, examples related to real life and

mathematics, statement patterns and logical equivalence-tautology, contradiction, contingency,

logical equivalence, logical equivalence, duality, negation of compound statement,

contrapositive, 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, contrapositive, contradiction, application-introduction to

switching circuits (simple examples).

2. Matrices : Elementary transformation of a matrix-revision of cofactor and minor, elementary

row transformation, elementary column transformation, inverse of a matrix-existance 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.

3. Trigonometric functions : Trigonometric equations-general solution of trigonometric equation of

the type, solution of Trigonometric equations, sine rule, cosine rule, projection rule, area of a

triangle, application, Hero‟s formula, Napier Analogues, inverse trigonometric functionsdefinitions, domain, range, principle values, graphs of inverse trigonometric function, properties

of inverse functions.

4. Pair of straight lines : Pair of lines passing through origin-conbined equation, homogenous

equation, theorem-the joint equation of a pair of lines passing through origin and its converse,

angle between the lines represented by ax

^{2}+2hxy+by2=0, condition for parallel lines, condition forperpendicular lines, pair of lines not passing through origin-conbined equation of any two lines,

condition that the equation ax2+2hxy+by2+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.

5. 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 x2+y2=a2, 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.

6. Conics : Tangents and normals-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 normals to conics

(without proof).

7. Vectors : Revision, Collinearty and coplanerity of vectors-linear combination of vectors,

condition of collinearity of two vectors, conditions of coplanerity ofthree 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 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.

8. 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.

9. Line : Equation of line passing through given point and parallel to given vector, equation of line

passing through two given points, dist. of a point from a line, distance between two skew lines,

distance between two parallel lines (vector approach).

10. 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 coplanerity of two lines, distance of a point from a

plane (vector approach).

11. Linear programming problems: Introduction of L.P.P. definition of constraints, objective

function, optimization, constraint equations, non-negativity restrictions, feasible and infeasible

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

STD. XII - PART – II

1. Continuity : Continuity of a function at a point-left hand limit, right hand limit, definition of a

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 functionspolynomial, rational, trigonometric, exponential and logarithmic function.

2. Differentiation : Revision- revision of derivative, relationship between continuity and

differentiability-left hand derivative and right hand derivative (need and concept), every

differentiable function is continuous but converse is not true, Derivative of composite functionchain 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)

3. Application of derivative : Geometrical application-tangent and normal at a point, Rolls

theorem, and Mean value theorem and their geometrical interpretation (without proof), derivative

as rate measure-introduction, increasing and decreasing function, approximation (without proof),

Maxima and minima-introduction of extreme and extreme values, maxima and minima in a

closed interval, first derivative test, second derivative test.

4. Integration : Indefinite integrals-methods of integration, substitution method, integrals of the

type, integration by parts-integration by parts, integrals of type (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 properties of definite

integrals.

5. 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).

6. Differential equation : Definition-differential equation, order, degree, general solution,

particular solution of differential equation, formation of differential equation-formation of

differential equation by elimmating arbitary constants (at most two constants), solution of first

order and first degree differential equation-variable separable method, homogenous differential

equation (equation reducible to homogenous form are not expected), Linear differential equation,

applications-population growth, bacterial colony growth, surface area, Newton‟s laws of cooling,

radioactive decay.

7. Statistics: Bivariate frequency distribution - bivariate data, tabulation of bivariate data, scatter

diagram, covariance of or ungrouped data, covariance for bivariate frequency distribution, Karl

Pearson‟s co-efficient of correlation.

8. 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.

9. 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).

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