Topics
Mathematical Logic
- Statements and Truth Values in Mathematical Logic
- Logical Connectives
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Logical Equivalence
- Tautology, Contradiction, and Contingency
- Quantifier, Quantified and Duality Statements in Logic
- Algebra of Statements
- Application of Logic to Switching Circuits
Trigonometry - II
- Trigonometric Functions of Allied Angels
- Trigonometric Functions of Compound Angles
- Trigonometric Functions of Sum and Difference of Three Angles
- Trigonometric Functions of Multiple Angles
- Trigonometric Functions of Sub-Multiple Angles
- Conversion Formulae in Trigonometry
- Trigonometric Functions of Angles of a Triangle
- Important Identities and Standard Results
11th Std
Straight Line
Matrices
12th Std
Trigonometric Functions
Circle
Conic Sections
Pair of Straight Lines
Measures of Dispersion
- Range, Variance and Standard Deviation
- Coefficient of Variation
- Standard Deviation for Combined Data
- 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
- Mean Deviation
Vectors
- Vector Algebra: Representation and Types
- Algebra of Vectors
- Collinearity and Coplanarity of Vectors
- Vector in Two Dimensions (2-D)
- Three Dimensional (3-D) Coordinate System
- 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 in Vectors
- Product of Vector in Algebra (Dot Product)
- Direction Ratios, Direction Cosine & Direction Angles
- Vector Product of two vectors in Algebra (Cross Product)
- Scalar Triple Product
- Vector Triple Product
Probability
Line and Plane
Complex Numbers
- Concept of Complex Numbers
- Algebraic Operations of Complex Numbers
- Equality of Two Complex Numbers
- Conjugate of a Complex Number
- Square Root of a Complex Number
- Fundamental Theorem of Algebra
- Argand Diagram or Complex Plane
- Modulus of a Complex Number
- Argument of a Complex Number
- DeMoivre's Theorem
- Cube Root of Unity
- Set of Points in Complex Plane
Linear Programming
Permutations and Combinations
Differentiation
- Introduction & Derivatives of Some Standard Functions
- Algebra of Differentiation
- Derivative of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- Logarithmic Differentiation
- Derivative of Implicit Functions
- Derivative of Parametric Functions
- Higher Order Derivatives
- Successive Differentiation
Functions
Limits
- Concept of Limits
- Methods to Find Limit of Rational Function>Factorization Method
- Algebra of Limits
- Methods to Find Limit of Rational Function> Rationalization Method
- Methods to Find Limit of Rational Function> Substitution Method
- Limits of Exponential and Logarithmic Functions
- Limits of Trigonometric Functions
- Limits of Algebraic Functions
Applications of Derivatives
Continuity
Indefinite Integration
Definite Integration
- Definite Integral as Limit of Sum
- Fundamental Theorem of Integral Calculus
- Properties of Definite Integrals
- Method for Finding Definite Intergrals
Conics
Sets and Relation
- Concept of Sets
- Classification of Sets
Application of Definite Integration
Sequences and Series
- Sequence, Series, and Progression
- nth Term of A.G.P
- Expressing Recurring Decimals as Rational Numbers
- Arithmetic Progression (A.P.)
- Geometric Progression (G. P.)
- Sum to' n' Terms of a Geometric Progression
- Harmonic Progression (H. P.)
- Types of Means
- Arithmetico Geometric Series
- Power Series
Differential Equations
Methods of Induction and Binomial Theorem
- Principle of Mathematical Induction
- General Term in Expansion of (a + b)n
- Middle term(s) in the expansion of (a + b)n
Probability Distribution
- Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Mass Function (P.M.F.)
- Cumulative Distribution Function (c. d. f. )
- Expected Value and Variance of a Random Variable
- Probability Distribution of a Continuous Random Variable
- Probability Density Function (P.D.F.)
- Cumulative Distribution Functions (c. d. f.)
Binomial Distribution
Determinants and Matrices
- Minors and Cofactors of Elements of Determinants
- Properties of Matrix Multiplication
Estimated time: 3 minutes
Maharashtra State Board: Class 12
Formula: Trigonometric Functions of Sub-Multiple Angles
| Function | Formula |
|---|---|
| sin θ | =\[2\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)\] =\[\frac{2\tan\left(\frac{\theta}{2}\right)}{1+\tan^2\left(\frac{\theta}{2}\right)}\] |
| cos θ | = \[\cos^{2}\left(\frac{\theta}{2}\right)-\sin^{2}\left(\frac{\theta}{2}\right)\] \[=1-2\sin^2\left(\frac{\theta}{2}\right)\] \[=2\cos^2\left(\frac{\theta}{2}\right)-1\] \[=\frac{1-\tan^2\left(\frac{\theta}{2}\right)}{1+\tan^2\left(\frac{\theta}{2}\right)}\] |
| tan θ | \[=\frac{2\tan\left(\frac{\theta}{2}\right)}{1-\tan^{2}\left(\frac{\theta}{2}\right)}\] |
| cot θ | \[=\frac{\cot^2\frac{θ}{2}-1}{2\cot\frac{θ}{2}}\] |
| sin θ | \[=3\sin\left(\frac{ θ }{3}\right)-4\sin^3\left(\frac{ θ }{3}\right)\] |
| cos θ | \[4\cos^3\left(\frac{θ}{3}\right)-3\cos\left(\frac{θ}{3}\right)\] |
| tan θ | \[=\frac{3\tan\left(\frac{θ}{3}\right)-\tan^{3}\left(\frac{θ}{3}\right)}{1-3\tan^{2}\left(\frac{θ}{3}\right)}\] |
