Topics
Angle and Its Measurement
- Directed Angle
- Angles of Different Measurements
- Angles in Standard Position
- Measures of Angles with Various Systems
- Area of a Sector
- Length of an Arc
Trigonometry - 1
- Trigonometric Ratios
- Trigonometric Functions with the Help of a Circle
- Signs of Trigonometric Functions in Different Quadrants
- Range of Cosθ and Sinθ
- Trigonometric Functions of Specific Angles
- Trigonometric Functions of Negative Angles
- Important Identities and Standard Results
- Periodicity of Trigonometric Functions
- Domain and Range of Trigonometric Functions
- Graphs of Trigonometric Functions
- Polar Co-ordinate System
Trigonometry - 2
Determinants and Matrices
- Definition and Expansion of Determinants
- Minors and Cofactors of Elements of Determinants
- Properties of Determinants
- Application of Determinants
- Determinant Method (Cramer’s Rule)
- Consistency of Three Equations in Two Variables
- Area of Triangle and Collinearity of Three Points
- Concept of Matrices
- Types of Matrices
- Operation on Matrices
- Properties of Matrix Multiplication
- Transpose of a Matrix
Straight Line
- Locus of a Points in a Co-ordinate Plane
- Equations of Line in Different Forms
- Family & Concurrent Lines
Circle
Conic Sections
Measures of Dispersion
- Meaning and Definition of Dispersion
- Measures of Dispersion
- Quartiles and Range in Statistics
- Variance
- Standard Deviation
- Change of Origin and Scale of Variance and Standard Deviation
- Standard Deviation for Combined Data
- Coefficient of Variation
Probability
Complex Numbers
Sequences and Series
- Sequence, Series, and Progression
- Arithmetic Progression (A.P.)
- Geometric Progression (G. P.)
- Harmonic Progression (H. P.)
- Arithmetico Geometric Series
- Power Series
Permutations and Combination
Methods of Induction and Binomial Theorem
- Principle of Mathematical Induction
- Binomial Theorem for Positive Integral Index
- General Term in Expansion of (a + b)n
- Middle term(s) in the expansion of (a + b)n
- Binomial Theorem for Negative Index Or Fraction
- Binomial Coefficients
Sets and Relations
- Sets and Their Representations
- Classification of Sets
- Fundamental Concepts of Ordered Pairs and Relations
- Intervals
Functions
Limits
- Concept of Limits
- Methods to Find Limit of Rational Function>Factorization Method
- Methods to Find Limit of Rational Function> Rationalization Method
- Limits of Trigonometric Functions
- Substitution Method
- Limits of Exponential and Logarithmic Functions
- Limit at Infinity
Continuity
Differentiation
- Definition of Derivative and Differentiability
- Rules of Differentiation (Without Proof)
- Derivative of Algebraic Functions
- Derivatives of Inverse Trigonometric Functions
- Derivative of Logarithmic Functions
- Derivatives of Exponential Functions
- L' Hospital'S Theorem
Estimated time: 4 minutes
Maharashtra State Board: Class 12
Key Points: Equation of a Circle in Different Forms
| Form | Condition / Description | Equation |
|---|---|---|
| Standard form | Centre (0, 0), radius r | x² + y² = r² |
| Centre-radius form | Centre (h, k), radius r | (x − h)² + (y − k)² = r² |
| General form | General equation of a circle | x² + y² + 2gx + 2fy + c = 0 Centre: (−g, −f) Radius: √(g² + f² − c) |
| Diameter form | Ends of diameter (x₁, y₁), (x₂, y₂) | (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0 |
| Parametric (centre at origin) | — | x = r cosθ, y = r sinθ |
| Parametric (centre (h, k)) | — | x = h + r cosθ, y = k + r sinθ |
| Intercepts on Axes |
\[\mathrm{X-axis}=2\sqrt{g^2-c}\]\[\mathrm{Y-axis}=2\sqrt{f^2-c}\] |
Condition for Radius:
| Condition | Result |
|---|---|
| g² + f² − c > 0 | Radius is real → Circle is real |
| g² + f² − c = 0 | Radius = 0 → Point circle |
| g² + f² − c < 0 | Radius imaginary → No real circle |
