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

## Syllabus

1 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).
1.1 Mathematical Logic

Statement and its truth value

Logical connective, simple and compound statements

Truth tables of statements and compound statements

• Conjecture
• Disjunction
• Conditional (Implication)
• Biconditional (Double implication)
• Negation of a statement

Statement pattern, logical equivalence

Quantifiers and quantified statements

Duality

Negations of compound statements

• Negation of conjunction
• Negation of disjunction
• Negation of implication
• Negation of biconditional

Converse, inverse, and contrapositive

Algebra of statements

• Idempotent Law
• Commutative Law
• Associative Law
• Distributive Law
• De Morgan's Law
• Identity Law
• Complement Law
• Absorption Law
• Conditional Law
• Biconditional Law

Application of logic to switching circuits, switching table.

• Two switches in series
• Two switches in parallel
1.2 Matrics

Elementry transformations

1. Interchange of any two rows or any two columns
2. Multiplication of the elements of any row or column by a non-zero scalar
3. Adding the scalar multiples of all the elements of any row (column) to corresponding elements of any other row (column)

Inverse of a matrix

•  Inverse of a nonsingular matrix by elementary transformation
•  Inverse of a square matrix by the adjoint method

Application of matrices

• Method of Inversion
• Method of Reduction
1.3 Trigonometric Functions

Trigonometric Equations and their solutions

Solutions of triangle

• Polar co-ordinates
• Relation between the polar co-ordinates and the Cartesian co-ordinates
• Solving a Triangle
• The Sine rule
• The Cosine rule
• The Projection rule
• Applications of the Sine rule, the Cosine rule, and the Projection rule.

Inverse Trigonometric Functions

• Inverse sine function
• Inverse cosine function
• Inverse tangent function
• Inverse cosecant function
• Inverse secant function
• Inverse cotangent function
• Principal Values of Inverse Trigonometric Functions
• Properties of inverse trigonometric functions
1.4 Pair of Straight Lines

Combined equation of a pair lines

Homogeneous equation of degree two

• Degree of a term
• Homogeneous Equation

Angle between lines

• Angle between lines represented by ax2 + 2hxy + by2 = 0

General second degree equation in x and y

1.5 Vectors

Representation of Vector

• Magnitude of a Vector

Vectors and their types

• Zero Vector
• Unit Vector
• Co-initial and Co-terminus Vectors
• Equal Vectors
• Negative of a Vector
• Collinear Vectors
• Free Vectors
• Localised Vectors

Algebra of Vectors

• Scalar Multiplication
• Subtraction of two vectors

Coplanar Vectors

Vector in Two Dimensions (2-D)

Three Dimensional (3-D) Coordinate System

• Co-ordinates of a point in space
• Co-ordinates of points on co-ordinate axes
• Co-ordinates of points on co-ordinate planes
• Distance of P(x, y, z) from co-ordinate planes
• Distance of any point from origin
• Distance between any two points in space
• Distance of a point P(x, y, z) from coordinate axes

Components of Vector

Position vector of a point P(x, y, z) in space

Component form of a position vector

Vector joining two points

Section formula

• Section formula for internal division
• Midpoint formula
• Section formula for external division

Dot/Scalar Product of Vectors

• Finding angle between two vectors
• Projections
• Direction Angles and Direction Cosine
• Direction ratios
• Relation between direction ratios and direction cosines

Cross/Vector Product of Vectors

• Angle between two vectors
• Geometrical meaning of vector product

Scalar Triple Product

Vector Triple Product

1.6 Line and Plane

Vector and Cartesian equations of a line

• Equation of a line passing through a given point and parallel to a given vector
•  Equation of a line passing through given two points

Distance of a point from a line

Skew lines

• Distance between skew lines
• Distance between parallel lines.

Equations of Plane

•  Equation of plane passing through a point and perpendicular to a vector
• Passing through a point and parallel to two vectors.
• The vector equation of the plane passing through three non-collinear points
• The normal form of the equation of a plane
• Equation of plane passing through the intersection of two planes

Angle between planes

• Angle between two planes.
• Angle between a line and a plane.

Coplanarity of two lines

Distance of a point from a plane

1.7 Linear Programming

Linear Inequations in two variables

• Convex Sets.
• Graphical representation of linear inequations in two variables.
• Graphical solution of linear inequation.

Linear Programming Problem (L.P.P.)

• Meaning of Linear Programming Problem.
• Mathematical formulation of L.P. P.
• Formal definitions related to L.P.P
• Solution of L. P. P. by graphical methods.
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
• Application - solution of system of linear equations by – reduction method, inversion method.
2.1 Differentiation
2.2 Applications of Derivatives
2.4 Definite Integration
2.5 Application of Definite Integration
2.7 Probability Distributions
3 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.
4 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.
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
• 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.
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
• Distance 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 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).
11 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.
12 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.
13 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)]
14 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.
15 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.
16 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).
17 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
18 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.
19 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.
20 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).