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

• Mathematical Logic 
  • Statements - Introduction in Logic 
  • Sentences and Statement in Logic 
  • Open Sentences in Logic 
  • Compound Statement in Logic 
  • Examples Related to Real Life and Mathematics 

    Examples related to real life and mathematics

  • Difference Between Converse, Contrapositive, Contradiction 
• Truth Value of Statement 
• Quantifier and Quantified Statements in Logic 
  • Universal quantifier (∀)
  • Existential quantifier (∃)
• Logical Connective, Simple and Compound Statements 
  • Concept of Statements

  1. Conjunction (∧)
  2. Disjunction (∨)
  3. Conditional statement (Implication) (→)
  4. Biconditional (Double implication) (↔) or (⇔)
  5. Negation (∼)
• Statement Patterns and Logical Equivalence 
• Algebra of Statements 
  • Idempotent law
  • Associative law
  • Commutative law
  • Distributive law
  • Identity law
  • Complement law
  • Involution law
  • DeMorgan’s laws
• Application of Logic to Switching Circuits 
  • Two switches in series
  • Two switches in parallel

• Concept of Statements 
• Truth Value of Statement 
• Logical Connective, Simple and Compound Statements 
  • Concept of Statements

  1. Conjunction (∧)
  2. Disjunction (∨)
  3. Conditional statement (Implication) (→)
  4. Biconditional (Double implication) (↔) or (⇔)
  5. Negation (∼)
• Statement Patterns and Logical Equivalence 
• Tautology, Contradiction, and Contingency 
• Duality 
• Quantifier and Quantified Statements in Logic 
  • Universal quantifier (∀)
  • Existential quantifier (∃)
• 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
  • Associative law
  • Commutative law
  • Distributive law
  • Identity law
  • Complement law
  • Involution law
  • DeMorgan’s laws
• Application of Logic to Switching Circuits 
  • Two switches in series
  • Two switches in parallel

• Elementry Transformations 
• Inverse of Matrix 
  •  Inverse of a nonsingular matrix by elementary transformation
  •  Inverse of a square matrix by adjoint method
• Application of Matrices 
  • Method of Inversion
  • Method of Reduction
• Applications of Determinants and Matrices 
  • Consistent System
  • Inconsistent System
  • Solution of a system of linear equations using the inverse of a matrix

• Trigonometric Equations and Their Solutions 
  • Trigonometric equation
  • Solution of Trigonometric equation
  • Principal Solutions
  • The General Solution
• Solutions of Triangle 
  • Polar co-ordinates
  • Relation between the polar co-ordinates and the Cartesian co-ordinates
  • Solving a Triangle
  • The Sine 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

• Combined Equation of a Pair Lines 
• Homogeneous Equation of Degree Two 
  • Degree of a term
  • Homogeneous Equation
• Angle between lines represented by ax2 + 2hxy + by2 = 0 
• General Second Degree Equation in x and y 
  • The necessary conditions for a general second degree equation
    ax² + 2hxy + by² + 2gx + 2fy + c = 0

  1. abc + 2fgh - af² - bg² - ch² = 0
  2. h² - ab ≥ 0
• Equation of a Line in Space 
  • Equation of a line through a given point and parallel to a given vector `vec b`
  • Equation of a line passing through two given points

• 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 
  • Addition of Two Vectors
    - Parallelogram Law
    - Triangle Law of addition of two vectors
  • Subtraction of two vectors
  • Scalar multiplication of a vector
• 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
• Scalar Product of Vectors (Dot) 
  • Finding angle between two vectors
  • Projections
  • Direction Angles and Direction Cosine
  • Direction ratios
  • Relation between direction ratios and direction cosines
• Vector Product of Vectors (Cross) 
  • Angle between two vectors
  • Geometrical meaning of vector product
• Scalar Triple Product of Vectors 
• Vector Triple Product 
• Addition of Vectors 

• Vector and Cartesian Equations of a Line 
  • Equation of a line passing through a given point and parallel to given vector
  • Equation of a line passing through given two points
• Distance of a Point from a Line 
  • Introduction of Distance of a Point from a Line
  • Distance between two parallel lines
• Distance Between Skew Lines and Parallel Lines 
  • Distance between skew lines
  • Distance between parallel lines
• Equation of a Plane 
  • Passing through a point and perpendicular to a vector
  • Passing through a point and parallel to two vectors
  • Passing through three non-collinear points
  • In normal form
  • Passing through the intersection of two planes
• Angle Between Planes 
• Coplanarity of Two Lines 
• Distance of a Point from a Plane 

• 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 a linear programming problem
  • Familiarize with terms related to Linear Programming Problem
• Lines of Regression of X on Y and Y on X Or Equation of Line of Regression 
• Graphical Method of Solving Linear Programming Problems 
  • Graphical method of solution for problems in two variables
  • Feasible and infeasible regions and bounded regions
  • Feasible and infeasible solutions
  • Optimum feasible solution
• Linear Programming Problem in Management Mathematics 

• Elementary Transformations 
  • Interchange of any two rows or any two columns
  • Multiplication of the elements of any row or column by a non-zero scalar
  • Adding the scalar multiples of all the elements of any row (column) to corresponding elements of any other row (column)
• Matrices 
  • Inverse by Elementary Transformation 
  • Elementary Transformation of a Matrix Revision of Cofactor and Minor 
  • Inverse of a Matrix Existance 
  • Solving System of Linear Equations in Two Or Three Variables Using Reduction of a Matrix Or Reduction Method 
• Determinants 
  • Adjoint Method 
• Operations on Matrices 
  • Addition of Matrices 
• Solution of System of Linear Equations by – Inversion Method 

• 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.

• Differentiation 
  • Rule of Differentiation
  • Introduction
• Derivatives of Composite Functions - Chain Rule 
• Geometrical Meaning of Derivative 
• Derivatives of Inverse Functions 
• Logarithmic Differentiation 
• Derivatives of Implicit Functions 
• Derivatives of Parametric Functions 
• Higher Order Derivatives 

• Applications of Derivatives in Geometry 
• Derivatives as a Rate Measure 
  • Velocity
  • Acceleration
  • Jerk
• Approximations 
• Rolle's Theorem 
• Lagrange's Mean Value Theorem (Lmvt) 
• Increasing and Decreasing Functions 
• Maxima and Minima 
  • First and Second Derivative test
  • Determine critical points of the function
  • Find the point(s) of local maxima and local minima and corresponding local maximum and local minimum values
  • Find the absolute maximum and absolute minimum value of a function

• Indefinite Integration 
• Methods of Integration: Integration by Substitution 
  • ∫ tan x dx = log | sec x |  + C
  • ∫ cot x dx = log | sin x | + C
  • ∫ sec x dx = log | sec x + tan x | + C
  • ∫ cosec x dx = log | cosec x – cot x | + C
• Methods of Integration: Integration by Parts 
  • `int(u.v) dx = u intv dx - int((du)/(dx)).(intvdx) dx`
  • Integral of the type ∫ e^x [ f(x) + f'(x)] dx = e^xf(x) + C
  • Integrals of some more types

  1. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
  2. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
  3. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
• Methods of Integration: Integration Using Partial Fractions 

  No	From of the rational function	Form of the partial fraction
  1	`(px + q )/((x-a)(x-b))`a ≠ b	`A/(x-a) + B/(x-b)`
  2	`(px+q)/(x-a)^2`	`A/(x-a) + B/(x-a)^2`
  3	`((px)^2 + qx +r)/((x-a)(x-b)(x-c))`	`A/(x-a)+B/(x-b) + C /(x-c)`
  4	`((px)^2 + qx + r)/((x-a)^2 (x-b))`	` A/(x-a) + B/(x-a)^2 +C/(x-b)`
  5	`((px)^2 + qx +r)/((x-a)(x^2 + bx +c))`	`A/(x-a) + (Bx + C)/ (x^2 + bx +c)`,

• Definite Integral as Limit of Sum 
• Fundamental Theorem of Integral Calculus 

  If ∫ f(x) dx = g(x) + c, then

  `int_a^b f(x) dx = [g(x) + c]_a^b = g(b) - g(a)`.

• Methods of Evaluation and Properties of Definite Integral 

• Application of Definite Integration 
• Area Bounded by the Curve, Axis and Line 
• Area Between Two Curves 

• Differential Equations 
• Order and Degree of a Differential Equation 
• Formation of Differential Equations 
  • Formation of Differential equations from Physical Situations
  • Formation of Differential Equations from Geometrical Problems
• Methods of Solving First Order, First Degree Differential Equations 
  • Homogeneous Differential Equations 
  • Linear Differential Equations 
• Application of Differential Equations 
  • Population Growth and Growth of bacteria
  • Ratio active Decay
  • Newton's Law of Cooling
  • Surface Area

• Random Variables and Its Probability Distributions 
  • Probability distribution of a random variable
• Types of Random Variables 
  • Discrete random variable
  • Continuous random variable
  • Probability Mass Function
  • Cumulative Distribution Function or Distribution Function
  • Cumulative Distribution Function from Probability Mass function
  • Probability Mass Function from Cumulative Distribution Function 
• Probability Distribution of Discrete Random Variables 
• Probability Distribution of a Continuous Random Variable 
  • Probability density function
  • Cumulative distribution function
• Variance of a Random Variable 

• Bernoulli Trial 
• Binomial Distribution 
• Mean of Binomial Distribution (P.M.F.) 
• Variance of Binomial Distribution (P.M.F.) 
• Bernoulli Trials and Binomial Distribution 

• 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

  • Hero’s Formula in Trigonometric Functions 
  • Napier Analogues in Trigonometric Functions 
• Solutions of Triangle 
  • Polar co-ordinates
  • Relation between the polar co-ordinates and the Cartesian co-ordinates
  • Solving a Triangle
  • The Sine rule
  • The Projection rule
  • Applications of the Sine rule, the Cosine rule and the Projection rule
• Basic Concepts of Inverse Trigonometric Functions 
  • sine, cosine, tangent, cotangent, secant, cosecant function
• Inverse Trigonometric Functions 
  • Inverse Trigonometric Functions - Principal Value Branch 
  • Graphs of Inverse Trigonometric Functions 
• Properties of Inverse Trigonometric Functions 

  Inverse of Sin, Inverse of cosin, Inverse of tan, Inverse of cot, Inverse of Sec, Inverse of Cosec

• 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.

• Pair of Straight Lines 
  • Pair of Lines Passing Through Origin - Combined Equation 
  • Pair of Lines Passing Through Origin - Homogenous Equation 
  • Theorem - the Joint Equation of a Pair of Lines Passing Through Origin and Its Converse 
  • Condition for Parallel Lines 
  • Condition for Perpendicular Lines 
  • Pair of Lines Not Passing Through Origin-combined Equation of Any Two Lines 
  • Point of Intersection of Two Lines 
• Acute Angle Between the Lines 

  Acute Angle Between the Lines represented by ax²+2hxy+by²=0

• 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.

• Circle 
  • Tangent of a Circle - Equation of a Tangent at a Point to Standard Circle 
  • Tangent of a Circle - Equation of a Tangent at a Point to 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 to Circle 
  • Normal to a Circle - Equation of Normal at a Point 

• 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.

• 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

• 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).

• Vector and Cartesian Equations of a Line 
  • Vectors Revision 
  • Linear Combination of Vectors 
  • Condition of collinearity of two vectors 
  • Conditions of Coplanarity of Three Vectors 
  • Midpoint Formula for Vector 
  • Centroid Formula for Vector 
  • 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 
• Collinearity and Coplanarity of Vectors 
• Section Formula 
  • Section formula for internal division
  • Midpoint formula
  • Section formula for external division
• Basic Concepts of Vector Algebra 
  • Position Vector
  • Direction Cosines and Direction Ratios of a Vector
• Scalar Triple Product of Vectors 
• Geometrical Interpretation of Scalar Triple Product 

• 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.

• Introduction of Three Dimensional Geometry 
• Direction Cosines and Direction Ratios of a Line 
  • Relation between the direction cosines of a line
  • Direction cosines of a line passing through two points
  • Direction cosines/ratios of a line joining two points
• Angle Between Two Lines 
• Three - Dimensional Geometry 
  • Condition for Perpendicular Lines 
• Relation Between Direction Ratio and Direction Cosines 
• Three Dimensional Geometry - Problems 

• 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.

• Concept of Line 
  • Equation of Line Passing Through Given Point and Parallel to Given Vector 
  • Distance Between Two Skew Lines 
• Distance of a Point from a Line 
  • Introduction of Distance of a Point from a Line
  • Distance between two parallel lines
• Shortest Distance Between Two Lines 
  • Coplanar
  • Skew lines
  • Distance between two skew lines
  • Distance between parallel lines
• Equation of a Line in Space 
  • Equation of a line through a given point and parallel to a given vector `vec b`
  • Equation of a line passing through two given points

• 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).

• 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 a Plane Passing Through Three Non Collinear Points 
  • Equation of Plane Passing Through the Intersection of Two Given Planes 
• Vector and Cartesian Equation of a Plane 
• Angle Between Two Planes 
• Angle Between Line and a Plane 
• Coplanarity of Two Lines 
• Distance of a Point from a 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).

• Introduction of Linear Programming 
  • Definition of related terminology such as constraints, objective function, optimization.
• Mathematical Formulation of Linear Programming Problem 
• Linear Programming 
  • Constraint Equations 
  • Non Negativity Restrictions 
• Different Types of Linear Programming Problems 
  • Different types of linear programming (L.P.) problems

  1. Manufacturing problem
  2. Diet Problem
  3. Transportation problem
• Graphical Method of Solving Linear Programming Problems 
  • Graphical method of solution for problems in two variables
  • Feasible and infeasible regions and bounded regions
  • Feasible and infeasible solutions
  • Optimum feasible solution

• 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.

• Introduction of Continuity 
• Definition of Continuity 
  • Continuity of a Function at a Point 

    left hand limit, right hand limit

    • Condition 1: If f (x) is to be continuous at x = a then f (a) must be defined.
    • Condition 2: If f(x) is to be continuous at  x = a then limxa→f (x) must exist.
    • Condition 3: If f(x) is to be continuous at  x = a then limxa→f (x) = f (a).
  • Defination of Continuity of a Function at a Point 
  • Discontinuity of a Function 
  • Types of Discontinuity 
    • Jump Discontinuity
    • Removable Discontinuity
    • Infinite Discontinuity
  • Continuity in Interval - Definition 
    • The intermediate value theorem for continuous functions
• Concept of Continuity 
• Algebra of Continuous Functions 
• Exponential and Logarithmic Functions 
• Continuity of Some Standard Functions - Polynomial Function 
• Continuity of Some Standard Functions - Rational Function 
• Continuity of Some Standard Functions - Trigonometric Function 
• Continuity - Problems 

• 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.

• The Concept of Derivative 
  • Revision of Derivative 
  • Every Differentiable Function is Continuous but Converse is Not True 
  • Derivative of Inverse Function 
  • Derivative of Functions in Product of Function Form 

    Derivative of Functions Which Are Expressed in Product of Function Form

  • Derivative of Functions in Quotient of Functions Form 

    Derivative of Functions Which Are Expressed in Quotient of Function Form

• Relationship Between Continuity and Differentiability 

  left hand derivative and right hand derivative (need and concept)

• Derivatives of Composite Functions - Chain Rule 
• Derivatives of Inverse Trigonometric Functions 
• Derivatives of Implicit Functions 
• Exponential and Logarithmic Functions 
• Derivatives of Functions in Parametric Forms 
• Higher Order Derivative 
  • Derivative of Functions Which Expressed in Higher Order Derivative Form
• Second Order Derivative 

• 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)]

• Mean Value Theorem 
  • Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations
• Rate of Change of Bodies or Quantities 
• Increasing and Decreasing Functions 
• Tangents and Normals 
• Approximations 
• Maxima and Minima - Introduction of Extrema and Extreme Values 
• Maxima and Minima in Closed Interval 
• Maxima and Minima 
  • First and Second Derivative test
  • Determine critical points of the function
  • Find the point(s) of local maxima and local minima and corresponding local maximum and local minimum values
  • Find the absolute maximum and absolute minimum value of a function

• 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.

• Methods of Integration: Integration by Substitution 
  • ∫ tan x dx = log | sec x |  + C
  • ∫ cot x dx = log | sin x | + C
  • ∫ sec x dx = log | sec x + tan x | + C
  • ∫ cosec x dx = log | cosec x – cot x | + C
• Methods of Integration: Integration Using Partial Fractions 

  No	From of the rational function	Form of the partial fraction
  1	`(px + q )/((x-a)(x-b))`a ≠ b	`A/(x-a) + B/(x-b)`
  2	`(px+q)/(x-a)^2`	`A/(x-a) + B/(x-a)^2`
  3	`((px)^2 + qx +r)/((x-a)(x-b)(x-c))`	`A/(x-a)+B/(x-b) + C /(x-c)`
  4	`((px)^2 + qx + r)/((x-a)^2 (x-b))`	` A/(x-a) + B/(x-a)^2 +C/(x-b)`
  5	`((px)^2 + qx +r)/((x-a)(x^2 + bx +c))`	`A/(x-a) + (Bx + C)/ (x^2 + bx +c)`,

• Methods of Integration: Integration by Parts 
  • `int(u.v) dx = u intv dx - int((du)/(dx)).(intvdx) dx`
  • Integral of the type ∫ e^x [ f(x) + f'(x)] dx = e^xf(x) + C
  • Integrals of some more types

  1. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
  2. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
  3. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
• Definite Integral as the Limit of a Sum 
• Fundamental Theorem of Calculus 

  Area function, First fundamental theorem of integral calculus and Second fundamental theorem of integral calculus

• Properties of Definite Integrals 
  1. `int_a^a f(x) dx = 0`
  2. `int_a^b f(x) dx = - int_b^a f(x) dx`
  3. `int_a^b f(x) dx = int_a^b f(t) dt`
  4. `int_a^b f(x) dx = int_a^c f(x) dx + int_c^b f(x) dx`
     where a < c < b, i.e., c ∈ [a, b]
  5. `int_a^b f(x) dx = int_a^b f(a + b - x) dx`
  6. `int_0^a f(x) dx = int_0^a f(a - x) dx`
  7. `int_0^(2a) f(x) dx = int_0^a f(x) dx + int_0^a f(2a - x) dx`
  8. `int_(-a)^a f(x) dx = 2. int_0^a f(x) dx`, if f(x) even function
     = 0,  if f(x) is odd function
• Evaluation of Definite Integrals by Substitution 
• Integration 
  • Integration by Non-repeated Quadratic Factors 

• 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.

• Area of the Region Bounded by a Curve and a Line 
  • Circle-line, elipse-line, parabola-line
• Area Between Two Curves 
• Applications of Integrations 
  • Volume of Solid of Revolution 

    volume of solid obtained by revolving the area under the curve about the axis (simple problems)

• 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).

• Basic Concepts of Differential Equation 
• Order and Degree of a Differential Equation 
• General and Particular Solutions of a Differential Equation 
• Formation of Differential Equation by Eliminating Arbitary Constant 
• Methods of Solving First Order, First Degree Differential Equations 
  • Differential Equations with Variables Separable Method 
  • Homogeneous Differential Equations 
• Differential Equations 
  • Linear Differential Equation 
  • Applications of Differential Equation 

    Population growth, Bacterial colony growth, Surface area, Newton’s laws of cooling, Radioactive decay

• 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.

• Statistics (Entrance Exam) 
  • Bivariate Frequency Distribution 

    bivariate data, tabulation of bivariate data

  • Scatter Diagram 

    (I) a) Perfect positive correlation, b) Positive correlation with high degree, c) Positive correlation with low degree
    (II) a) Perfect negative correlation, b) Negative correlation with high degree, c) Negative correlation with low degree
    (III) No correlation (Zero correlation)

  • Covariance of Ungrouped Data 
  • Covariance for Bivariate Frequency Distribution 
  • Karl Pearson’s Coefficient of Correlation 

• 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.

• Conditional Probability 
• Random Variables and Its Probability Distributions 
  • Probability distribution of a random variable
• Probability Distribution 
  • Discrete and Continuous Random Variable 
  • Probability Mass Function (P.M.F.) 
  • Cumulative Probability Distribution of a Discrete Random Variable 
  • Expected Value, Variance and Standard Deviation of a Discrete Random Variable 
    • Apply arithmetic mean of frequency distribution to find the expected value of a random variable 
    • Calculate the Variance and S.D. of a random variable
  • Probability Density Function (P.D.F.) 
  • Distribution Function of a Continuous Random Variable 
• Probability Distribution of a Discrete Random Variable 

• 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.

• Bernoulli Trials and Binomial Distribution 
• Bernoulli Trial 
  • Conditions for Binomial Distribution 
  • Calculation of Probabilities 
  • Normal Distribution (P.D.F) 

    mean, variance and standard deviation, standard normal variable, simple problems

• Mean of Binomial Distribution (P.M.F.) 
• Variance of Binomial Distribution (P.M.F.) 
• Standard Deviation of Binomial Distribution (P.M.F.) 

• 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).