HSC Arts 12th Board ExamMaharashtra State Board
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# Mathematics and Statistics 12th Board Exam HSC Arts Maharashtra State Board Topics and Syllabus

## Topics with syllabus and resources

100.00 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, biconditional
• 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, 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).
200.00 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
• Application - solution of system of linear equations by – reduction method, inversion method.
300.00 Trigonometric Functions
• Trigonometric equations -
• General solution of trigonometric equation of the type : sinθ, = 0, cosθ = 0, tanθ = 0, sinθ = sinα, cosθ = cosα, tanθ = tanα, sin2 θ = sin2 α, cos2 θ = cos2 α, tan2 θ = tan2 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.
400.00 Pair of Straight Lines
• Pair of lines passing through origin - Combined equation, Homogenous equation
• Theorem - the joint equation of a pair of lines passing through origin and its converse
• Acute angle between the lines represented by ax2 +2hxy+by2 =0
• Condition for parallel lines
• Condition for perpendicular lines
• Pair of lines not passing through origin-combined 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.
500.00 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.
600.00 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).
700.00 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.
800.00 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.
900.00 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).
1000.00 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 noncollinear points
• Equation of plane passing through the intersection of two given planes
• Angle between two planes
• Angle between line and plane
• Condition for the coplanarity of two lines
• Distance of a point from a plane (vector approach).
1100.00 Linear Programming Problems
• Introduction of L.P.P.
• Definition of constraints
• Objective function
• Optimization
• Constraint equations
• Nonnegativity 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.
1200.00 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.
1300.00 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 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)]
1400.00 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)
• Maxima and minima-introduction of extrema and extreme values
• Maxima and minima in a closed interval
• First derivative test
• Second derivative test.
1500.00 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
• 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.
1600.00 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).
1700.00 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
1800.00 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.
1900.00 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.
2000.00 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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