Topics
Sets and Functions
Sets
- Sets and Their Representations
- Empty Set (Null or Void Set)
- Finite and Infinite Sets
- Equal Sets
- Subsets
- Power Set
- Universal Set
- Venn Diagrams
- Union of Sets
- Intersection of Sets
- Difference of Sets
- Complement of a Set
- Properties of Complement of Sets
- Problems on Union and Intersection of Two and Three Sets.
Relations and Functions
- Ordered Pairs
- Fundamental Concepts of Ordered Pairs and Relations
- Pictorial Diagrams
- Domain and Range of a Function
- Function as a Type of Mapping
- Types of Functions
- Some Functions and Their Graphs
- Exponential Function
- Sum, Difference, Product, Quotient of Functions
Trigonometry
- Magnitude of an Angle
- Angles and Their Measurement in Higher Mathematics
- Conversion from One Measure to Another
- Trigonometric Ratios
- Truth of the Identity
- Signs of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Expressing Sin (X±Y) and Cos (X±Y) in Terms of Sinx, Siny, Cosx and Cosy and Their Simple Applications
- Trigonometric Functions of Sum and Difference of Two Angles
- Trigonometric Equations
- Solution of Trigonometric Equations (Solution in the Specified Range)
- Graphs of Trigonometric Functions
- Trigonometric Functions of Multiple Angles
- Trigonometric Functions of Half Angles
- Convention of Sign of Angles
- The Relation S = rθ Where θ is in Radians
- Relation Among Trigonometric Ratios
- Periods of Trigonometric Functions
- Compound and Multiple Angles- Addition and Subtraction Formula
- Trigonometric Functions of All Angles
- Sum and Differences as Products
- Product to Sum Or Difference
- Trigonometric Equations
Algebra
Principle of Mathematical Induction
Complex Numbers
- Concept of Complex Numbers
- Conjugate of a Complex Number
- Properties of Conjugate, Modulus and Argument (or Amplitude) of Complex Numbers
- Argand Plane and Polar Representation
- Square Root of a Complex Number
- Cube Root of Unity
- Properties of Cube Roots of Unity
- Algebraic Properties of Complex Numbers
- Algebraic Operations of Complex Numbers
- Locus Questions on Complex Numbers.
- Triangle Inequality
Quadratic Equations
- Quadratic Equations
- Equations Reducible to Quadratic Equations
- Nature of Roots of a Quadratic Equation
- Quadratic Functions
- Sign of Quadratic
- Quadratic Inequalities
- Representation of Inequalities
- Graphical Solution of Linear Inequalities in Two Variables
- Solution of System of Linear Inequalities in Two Variables
Permutations and Combinations
- Permutations
- Fundamental Principles of Counting
- Derivation of Formulae and Their Connections
- Simple Applications of Permutations and Combinations
- Circular Permutations
- Restricted Permutation
- Permutation - Certain Things Always Occur Together
- Permutation - Certain Things Never Occur
- Permutation - Formation of Numbers with Digits
- Permutation - Permutation of Alike Things
- Permutation - Permutation of Repeated Things
- Permutation - Word Building
- Combination
Binomial Theorem
- Introduction of Binomial Theorem
- Binomial Theorem for Positive Integral Indices
- General and Middle Terms
- Simple Applications of Binomial Theorem
Sequence and Series
- Sequence, Series, and Progression
- Arithmetic Progression (A.P.)
- Three or More Terms in Arithmetic Progression (A.P.)
- Inserting Two Or More Arithmetic Means Between Any Two Numbers
- Geometric Progression (G. P.)
- Three Terms in Geometric Progression (G.P.)
- Four Terms Are in Geometric Progression (G.P.)
- Geometric Mean
- Relationship Between A.M. and G.M.
- Arithmetico Geometric Series
Coordinate Geometry
Straight Lines
- Brief Recall of Two Dimensional Geometry from Earlier Classes
- Shifting of Origin
- Concept of Slope (or, gradient)
- Various Forms of the Equation of a Line
- Equations of Line in Different Forms
- Equation of Family of Lines Passing Through the Point of Intersection of Two Lines
- Distance in Lines (Point & Parallel Lines)
- Equations of Bisectors of Angle Between Two Lines
- Family & Concurrent Lines
- Co-ordinate Geometry
- Locus
Circles
- Equations of a Circle in Standard Form
- Equations of a Circle in Diameter Form
- Equations of a Circle in General Form
- Equations of a Circle in Parametric Form
- Focus-directrix Property
- Given the Equation of a Circle, to Find the Centre and the Radius
- Finding the Equation of a Circle
- Equation of Tangent and Condition of Tangency
Calculus
Limits and Derivatives
- Derivative Introduced as Rate of Change Both as that of Distance Function and Geometrically
- Introduction of Limits
- Limits of Polynomials and Rational Functions
- Limits of Exponential Functions
- Limits of Logarithmic Functions
- Limits of Trigonometric Functions
- Limits of Algebraic Functions
- Fundamental Theorem on Limits
- Introduction of Derivatives
- Derivative of Slope of Tangent of the Curve
- Derivative of Algebraic Functions
- Differentiation Or Derivative Using First Principles
- Algebra of Derivative of Functions
- Derivative of Polynomials and Trigonometric Functions
Statistics and Probability
Statistics - 1
- Measures of Central Tendency for Different Data Types
- Mean Deviation
- Standard Deviation
- Standard Deviation - by Direct Method
- Standard Deviation - by Step Deviation Method"
- Introduction of Variance and Standard Deviation
- Comparison of Two Frequency Distributions with Same Mean
- Quartiles and Range in Statistics
Probability
- Concept of Probability
- Elementary Types of Events and Properties of Probability
- Addition Theorem – for Any Two Events a and B, Result on Complementary Events
- Occurrence of an Event
- Axiomatic Approach to Probability
- Probability of 'Not', 'And' and 'Or' Events
Conic Section
- Sections of a Cone
- Conics as a Section of a Cone
- Definition of Foci, Directrix, Latus Rectum
- Parabola and its types
- Latus Rectum
- Ellipse and its Types
- Latus Rectum
- Hyperbola and its Types
- Transverse and Conjugate Axes
- Coordinates of Vertices
- Foci and Centre
- Equations of the Directrices and the Axes
- General Second Degree Equation
- Equation of Tangent and Condition of Tangency
- Point of Contact and Locus Problems
Introduction to Three-dimensional Geometry
Mathematical Reasoning
- Mathematically Acceptable Statements
- New Statements from Old
- Special Words Or Phrases
- Consolidating the Understanding
- Introduction of Validating Statements
- Validation by Contradiction
- Difference Between Contradiction, Converse and Contrapositive
- Contrapositive and Converse
Statistics - 2
- Combined Mean and Standard Deviation
- Measures of Central Tendency for Different Data Types
- Deciles
- Percentiles
Correlation Analysis
- Definition and Meaning of Covariance
- Karl Pearson’s Coefficient of Correlation
- Rank Correlation by Spearman’s
Index Numbers and Moving Averages
Index Numbers
- Price Index Number
- Simple Aggregate Method
- Weighted Aggregate Method
- Simple Average of Price Relatives
- Weighted Average of Price Relatives
Moving Averages
- Meaning and Purpose of the Moving Averages
- Calculation of Moving Averages with the Given Periodicity and Plotting Them on a Graph
- If the Period is Even, Then the Centered Moving Average is to Be Found Out and Plotted
Estimated time: 4 minutes
Maharashtra State Board: Class 12
Formula: Algebraic Functions
1. \[\lim_{x\to\mathrm{a}}x=\mathrm{a}\]
2. \[\lim_{x\to\mathrm{a}}x^\mathrm{n}=\mathrm{a}^\mathrm{n}\]
3. \[\lim_{x\to a}\mathrm{k}=\mathrm{k},\] (where k is a constatnt)
4. \[\lim_{x\to a}\sqrt[r]{x}=\sqrt[r]{a}\]
5. If P(x) is a polynomial, then \[\lim_{x\to a}\mathrm{P}(x)=\mathrm{P}(\mathrm{a})\]
6. \[\lim_{x\to a}\frac{x^n-a^n}{x-a}=na^{n-1}\]
7. \[\lim_{x\to\infty}\frac{1}{x^{k}}=0,\mathrm{where~k}>0\]
