Topics
Mathematical Logic
- Statements and Truth Values in Mathematical Logic
- Logical Connectives
- Tautology, Contradiction, and Contingency
- Quantifier, Quantified and Duality 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
Trigonometric Functions
Pair of Straight Lines
Vectors
Line and Plane
Linear Programming
Differentiation
- Introduction & Derivatives of Some Standard Functions
- Derivative of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- 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 Two Curves
- Overview of Application of Definite Integration
Differential Equations
Probability Distributions
- 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
Estimated time: 5 minutes
Maharashtra State Board: Class 12
Key Points: Area Under the Curve
| Case | Formula |
|---|---|
| Area under y = f(x) | \[\int_a^bf(x)dx\] |
| Area between curves | \[\int_a^b[f(x)-g(x)]dx\] |
| Area w.r.t. y-axis x = g(y) |
\[\int_c^dg(y)dy\] |
| Even function | \[2\int_0^af(x)dx\] |
| Odd function | 0 |
If the area A lies below the X-axis, then A is negative, and in this case, we take | A |.
Maharashtra State Board: Class 12
Key Points: Symmetry of Curve
| Type of Symmetry | Condition | Replacement Rule | Result |
|---|---|---|---|
| About the X–axis | (x, y) ∈ C ⇔ (x, -y) ∈ C | Replace y by -y | Curve is symmetric about the X–axis |
| About the Y–axis | (x, y) ∈ C ⇔ (-x, y) ∈ C | Replace x by -x | Curve is symmetric about the Y–axis |
| About Origin | Equation unchanged when both signs change | Replace x → -x, y → -y | The curve is symmetric about Origin |
Maharashtra State Board: Class 12
Key Points: Standard Curves
Parabola:
-
y2 = 4ax → opens right
-
y2 = −4ax → opens left
-
x2 = 4ay → opens upward
-
x2 = −4ay → opens downward
Ellipse:
-
\[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\quad(a>b)\]
-
\[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\quad(a<b)\]
