Topics
Mathematical Logic
- Concept of Statements
- Truth Value of Statement
- Logical Connective, Simple and Compound Statements
- Statement Patterns and Logical Equivalence
- Tautology, Contradiction, and Contingency
- Duality
- Quantifier and Quantified Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
- Overview of Mathematical Logic
Matrices
- Elementry Transformations
- Properties of Matrix Multiplication
- Application of Matrices
- Applications of Determinants and Matrices
- Overview of Matrices
Trigonometric Functions
- Trigonometric Equations and Their Solutions
- Solutions of Triangle
- Inverse Trigonometric Functions
- Overview of Trigonometric Functions
Pair of Straight Lines
- Combined Equation of a Pair Lines
- Homogeneous Equation of Degree Two
- Angle between lines represented by ax2 + 2hxy + by2 = 0
- General Second Degree Equation in x and y
- Equation of a Line in Space
- Overview of Pair of Straight Lines
Vectors
Line and Plane
- Vector and Cartesian Equations of a Line
- Distance of a Point from a Line
- Distance Between Skew Lines and Parallel Lines
- Equation of a Plane
- Angle Between Planes
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Overview of Line and Plane
Linear Programming
Differentiation
- Differentiation
- 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
- Overview of Differentiation
Applications of Derivatives
- Applications of Derivatives in Geometry
- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima
- Overview of Applications of Derivatives
Indefinite Integration
Definite Integration
- Definite Integral as Limit of Sum
- Integral Calculus
- Methods of Evaluation and Properties of Definite Integral
- Overview of Definite Integration
Application of Definite Integration
- Application of Definite Integration
- Area Bounded by the Curve, Axis and Line
- Area Between Two Curves
- Overview of Application of Definite Integration
Differential Equations
- Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Homogeneous Differential Equations
- Linear Differential Equations
- Application of Differential Equations
- Solution of a Differential Equation
- Overview of Differential Equations
Probability Distributions
- Random Variables and Its Probability Distributions
- Types of Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Distribution of a Continuous Random Variable
- Variance of a Random Variable
- Expected Value and Variance of a Random Variable
- Overview of Probability Distributions
Binomial Distribution
- Bernoulli Trial
- Binomial Distribution
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)
- Bernoulli Trials and Binomial Distribution
- Overview of Binomial Distribution
Estimated time: 12 minutes
Maharashtra State Board: Class 12
Definition: Combined Equation
An equation representing two lines together is called the combined (joint) equation of the lines.
Maharashtra State Board: Class 12
Definition: Homogeneous Equation
An equation in which the degree of every term is the same is called a homogeneous equation.
Homogeneous equation of degree 2:
\[ax^2+2hxy+by^2=0\]
Maharashtra State Board: Class 12
Definition: Degree of a Term
The sum of the indices of all variables in a term is called the degree of the term.
Maharashtra State Board: Class 12
Key Points: Nature of Lines
| Condition | Nature |
|---|---|
| \[h^2-ab>0\] | Distinct lines |
| \[h^2-ab=0\] | Coincident lines |
| \[h^2-ab<0\] | Not a pair of lines |
Maharashtra State Board: Class 12
Formula: Slopes of the Lines
If \[ax^2+2hxy+by^2=0\]
Then slopes are:
\[m_1=\frac{-h-\sqrt{h^2-ab}}{b}\]
\[m_2=\frac{-h+\sqrt{h^2-ab}}{b}\]
Their sum is m1 + m2 = \[-\frac{2h}{b}\]
product is m1 m2 = \[\frac{a}{b}\]
Maharashtra State Board: Class 12
Definition: Auxiliary Equation
For \[ax^2+2hxy+by^2=0\]
Slopes of lines are roots of: \[bm^2+2hm+a=0\]
This equation is called the Auxiliary Equation.
Maharashtra State Board: Class 12
Formula: Angle Between Lines
\[\tan\theta=\frac{2\sqrt{h^2-ab}}{a+b}\]
Maharashtra State Board: Class 12
Key Points: Conditions for Perpendicular and Parallel
| Sr. No. | Condition Type | Mathematical Condition | Additional Result |
|---|---|---|---|
| 1 | Perpendicular Lines | a + b = 0 | Lines are perpendicular |
| 2 | Parallel Lines | \[h^2-ab=0\] | Lines are parallel |
| 3 | Intersecting Lines | \[h^2-ab\geq0\] | Point of intersection is \[\left(\frac{hf-bg}{ab-h^2},\frac{gh-af}{ab-h^2}\right)\] |
Maharashtra State Board: Class 12
Definition: General Second Degree Equation
Equation of the form \[ax^2+2hxy+by^2+2gx+2fy+c=0\], where at least one of a,b,h is not zero, is called a general second degree equation in x and y.
The expression\[abc+2fgh-af^{2}-bg^{2}-ch^{2}\] is the expansion of the determinant \[\begin{vmatrix}
a & h & g \\
h & b & f \\
g & f & c
\end{vmatrix}\]
