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: 3 minutes
- Passing through a point and perpendicular to a vector
- Passing through a point and parallel to two vectors
- Passing through three non-collinear points
- In normal form
- Passing through the intersection of two planes
Maharashtra State Board: Class 12
Key Points: Equation of a Plane
| Case | Vector Form | Cartesian Form |
|---|---|---|
| 1. Normal form (given normal vector) | \[\overline{\mathbf{r}}.\hat{\mathbf{n}}=\mathbf{p}\] | ax + by + cz + d = 0 |
| 2. Through a point (x₁, y₁, z₁) | \[\begin{bmatrix} \mathbf{\overline{r}}-\mathbf{\overline{a}} \end{bmatrix}.\mathbf{\overline{n}}=\mathbf{0}\] | a(x−x₁) + b(y−y₁) + c(z−z₁) = 0 |
| 3. Through point + parallel to two vectors | \[\begin{bmatrix} \overline{\mathbf{r}}\overline{\mathbf{b}}\overline{\mathbf{c}} \end{bmatrix}= \begin{bmatrix} \overline{\mathbf{a}}\overline{\mathbf{b}}\overline{\mathbf{c}} \end{bmatrix}\] | \[\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ \mathbf{b}_1 & \mathbf{b}_2 & \mathbf{b}_3 \\ \mathbf{c}_1 & \mathbf{c}_2 & \mathbf{c}_3 \end{vmatrix}=0\] |
| 4. Through three non-collinear points | \[(\mathbf{r-a})\cdot[(\mathbf{b-a})\times(\mathbf{c-a})]=0\] | \[\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix}=0\] |
| 5. Through the intersection of two planes | \[\left(\overline{\mathbf{r}}.\overline{\mathbf{n}}_1-\mathbf{d}_1\right)+\lambda\left(\overline{\mathbf{r}}.\overline{\mathbf{n}}_2-\mathbf{d}_2\right)=0\] | (a₁x + b₁y + c₁z + d₁) + λ(a₂x + b₂y + c₂z + d₂) = 0 |
Equation of a Plane in Intercept form:
\[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\]
Distance of the Plane from Origin is
\[d=\frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}}}\]
