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: 7 minutes
- Modulus of z
- Argument of z
- Argument of z in different quadrants/axes - Properties of modulus of complex numbers, Properties of arguments
- Polar & Exponential form of C.N.
Maharashtra State Board: Class 12
Key Points: Argand Diagram or Complex Plane
1. Representation
- z = a + ib → point (a, b)
- X-axis → Real part (Re)
- Y-axis → Imaginary part (Im)
2. Modulus
- |z| = distance from origin
- |z| = √(a² + b²)
3. Argument (θ)
- Angle made with +X-axis (anticlockwise)
- θ = tan⁻¹(b/a)
| z = a + ib | Quadrant / Axis | θ = arg z |
|---|---|---|
| a > 0, b = 0 | On the positive real (X) axis | θ = 0 |
| a > 0, b > 0 | Quadrant I | \[\Theta=\tan^{-1}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)\] |
| a = 0, b > 0 | On the positive imaginary (Y) axis | θ = π/2 |
| a < 0, b > 0 | Quadrant II | \[\theta=\pi-\tan^{-1}\left|\frac{\mathrm{b}}{\mathrm{a}}\right|\] |
| a < 0, b = 0 | On the negative real (X) axis | θ = π |
| a < 0, b < 0 | Quadrant III | \[\Theta=\pi+\tan^{-1}\left|\frac{\mathrm{b}}{\mathrm{a}}\right|\] \[\theta=\tan^{-1}\left|\frac{\mathrm{b}}{\mathrm{a}}\right|-\pi\] |
| a = 0, b < 0 | On the negative imaginary (Y) axis | θ = 3π/2 |
| a > 0, b < 0 | Quadrant IV | \[\Theta=2\pi-\tan^{-1}\left|\frac{\mathrm{b}}{\mathrm{a}}\right|\] |
Maharashtra State Board: Class 1
Definition: Polar Form of a Complex Number
The polar form of a complex number z = x + iy is
z = r(cos θ + i sin θ), where x = r cos θ, y = r sin θ and r = \[r=\sqrt{x^{2}+y^{2}}\].
Maharashtra State Board: Class 12
Definition: Exponential Form or Euler’s Form
∴ z = a + ib = r(cos θ + i sin θ) = reiθ,
where r = |z| and θ = arg z is called an exponential form of a complex number.
