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

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Maharashtra State Board Syllabus For 12th Board Exam Mathematics and Statistics: Knowing the Syllabus is very important for the students of 12th Board Exam. Shaalaa has also provided a list of topics that every student needs to understand.

The Maharashtra State Board 12th Board Exam Mathematics and Statistics syllabus for the academic year 2021-2022 is based on the Board's guidelines. Students should read the 12th Board Exam Mathematics and Statistics Syllabus to learn about the subject's subjects and subtopics.

Students will discover the unit names, chapters under each unit, and subtopics under each chapter in the Maharashtra State Board 12th Board Exam Mathematics and Statistics Syllabus pdf 2021-2022. They will also receive a complete practical syllabus for 12th Board Exam Mathematics and Statistics in addition to this.

Academic year:

## Maharashtra State Board 12th Board Exam Mathematics and Statistics Revised Syllabus

Maharashtra State Board 12th Board Exam Mathematics and Statistics and their Unit wise marks distribution

### Maharashtra State Board 12th Board Exam Mathematics and Statistics Course Structure 2021-2022 With Marking Scheme

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## 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

Tautology, contradiction, and contingency.

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
• Addition of Two Vectors
• 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
• Adjoint method
• 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
2.6 Differential Equations
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
• 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.
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
• Radioactive decay.
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).
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