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Mathematics and Statistics 12th Board Exam HSC Science (Computer Science) Maharashtra State Board Topics and Syllabus

Academic year:

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.
2.7 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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