Senior Secondary School Mathematics For Class - 12

1. Relations Defined: Reflexivity, Symmetry, and Transitivity

A relation R in a set A is a subset of A x A.

Understanding relations. A relation is essentially a way to describe how elements within a set are connected. It's a subset of the Cartesian product of a set with itself, meaning it's a collection of ordered pairs. If an element 'a' is related to 'b' under relation R, we write aRb.

Key properties. Relations can possess three crucial properties:

• Reflexivity: Every element is related to itself (aRa).
• Symmetry: If a is related to b, then b is related to a (aRb implies bRa).
• Transitivity: If a is related to b, and b is related to c, then a is related to c (aRb and bRc implies aRc).

Equivalence relations. A relation that is reflexive, symmetric, and transitive is called an equivalence relation. Equivalence relations partition a set into disjoint equivalence classes, grouping together elements that share a specific property. For example, congruence modulo n is an equivalence relation on integers.

2. Functions: Mapping Sets with Precision

Then, a rule f which associates to each element x € A, a unique element, denoted by f(x) of B, is called a function from A to B and we write, f: A— B.

Functions as mappings. A function is a rule that assigns each element from one set (the domain) to a unique element in another set (the codomain). The set of all actual output values is called the range.

Types of functions:

• One-to-one (injective): Distinct elements in the domain map to distinct elements in the codomain.
• Onto (surjective): Every element in the codomain has at least one pre-image in the domain.
• Bijective: Both one-to-one and onto, establishing a perfect pairing between elements of the domain and codomain.

Function composition. Functions can be combined through composition. If f: A -> B and g: B -> C, then g o f: A -> C is defined by (g o f)(x) = g(f(x)). The domain of g o f is A, and its range is a subset of C.

3. Binary Operations: Rules of Combination

An operation * on a nonempty set S, satisfying the closure property is known as a binary operation.

Combining elements. A binary operation is a rule that combines two elements from a set to produce another element within the same set. This property is known as closure.

Properties of binary operations:

• Associativity: (a * b) * c = a * (b * c) for all elements a, b, c in the set.
• Commutativity: a * b = b * a for all elements a, b in the set.
• Distributivity: One operation distributes over another if a * (b o c) = (a * b) o (a * c).

Identity and inverse. An identity element 'e' satisfies a * e = e * a = a for all elements a in the set. An inverse element 'b' for 'a' satisfies a * b = b * a = e.

4. Inverse Trigonometric Functions: Unveiling the Angles

Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

Undoing trigonometric functions. Inverse trigonometric functions "undo" the trigonometric functions, allowing us to find the angle that corresponds to a given trigonometric ratio.

Key aspects:

• Domain and range: Inverse trigonometric functions have restricted domains and ranges to ensure they are single-valued.
• Principal value branches: The principal value branch is a specific interval within the range that provides a unique output for each input.
• Properties: Inverse trigonometric functions have various properties, such as relationships between different inverse functions and identities involving their arguments.

Examples. sin⁻¹(x), cos⁻¹(x), and tan⁻¹(x) are the most common inverse trigonometric functions, each with its own specific domain and range.

5. Matrices: Organization and Operations

Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices.

Matrices as arrays. A matrix is a rectangular array of numbers arranged in rows and columns. The order of a matrix is defined by the number of rows and columns it contains (m x n).

Matrix operations:

• Addition: Matrices of the same order can be added by adding corresponding elements.
• Scalar multiplication: Multiplying a matrix by a scalar involves multiplying each element of the matrix by that scalar.
• Multiplication: Matrix multiplication is defined when the number of columns in the first matrix equals the number of rows in the second matrix.

Special matrices. There are several special types of matrices, including zero matrices, square matrices, diagonal matrices, scalar matrices, identity matrices, symmetric matrices, and skew-symmetric matrices.

6. Determinants: Properties and Applications

Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle.

Determinants as scalars. A determinant is a scalar value associated with a square matrix. It provides information about the matrix's properties and can be used in various applications.

Key properties:

• Interchanging rows or columns changes the sign of the determinant.
• Multiplying a row or column by a constant multiplies the determinant by that constant.
• If two rows or columns are identical, the determinant is zero.

Applications. Determinants are used to find the area of a triangle given its vertices, to determine the invertibility of a matrix, and to solve systems of linear equations.

7. Continuity and Differentiability: The Foundation of Calculus

Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function.

Continuity as unbrokenness. A function is continuous at a point if its limit exists at that point and equals the function's value. This means the graph of the function has no breaks or jumps at that point.

Differentiability as smoothness. A function is differentiable at a point if its derivative exists at that point. This implies the function has a well-defined tangent line at that point, indicating a smooth, gradual change.

Key theorems:

• Chain rule: Used to find the derivative of composite functions.
• Rolle's theorem: States that if a function is continuous on a closed interval, differentiable on the open interval, and has equal values at the endpoints, then there exists a point within the interval where the derivative is zero.
• Mean Value Theorem: Guarantees the existence of a point within an interval where the instantaneous rate of change (derivative) equals the average rate of change over the interval.

8. Applications of Derivatives: Rates, Maxima, and Minima

Applications of derivatives: rate of change, increasing/decreasing functions, tangents and normals, approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool).

Derivatives as rates of change. The derivative of a function represents its instantaneous rate of change. This concept is used to analyze how quantities change with respect to each other.

Increasing and decreasing functions. The sign of the first derivative indicates whether a function is increasing or decreasing. A positive derivative implies an increasing function, while a negative derivative implies a decreasing function.

Maxima and minima. Derivatives are used to find the maximum and minimum values of a function. Critical points (where the derivative is zero or undefined) are potential locations for local maxima or minima. The first and second derivative tests help determine the nature of these critical points.

9. Indefinite Integrals: The Reverse Process

Integration as inverse process of differentiation. Integration of a variaty of functions by substitution, by partial fractions and by parts, only simple integrals of the type

Integration as anti-differentiation. Indefinite integration is the reverse process of differentiation. It involves finding a function whose derivative is equal to a given function.

Key concepts:

• Integrand: The function being integrated.
• Constant of integration: An arbitrary constant added to the result of indefinite integration, reflecting the fact that the derivative of a constant is zero.

Basic integration formulas. There are several basic integration formulas that serve as building blocks for more complex integrals, such as the power rule, trigonometric integrals, and exponential integrals.

10. Methods of Integration: Substitution and Partial Fractions

Methods of Integration

Integration by substitution. This technique involves changing the variable of integration to simplify the integrand. The goal is to transform the integral into a form that can be evaluated using basic integration formulas.

Integration by partial fractions. This method is used to integrate rational functions (ratios of polynomials). It involves decomposing the rational function into simpler fractions that can be integrated separately.

Integration by parts. This technique is used to integrate products of functions. It involves choosing one function to differentiate and another to integrate, and then applying the formula: ∫ u dv = uv - ∫ v du.

11. Definite Integrals: Area Under the Curve

Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

Definite integrals as area. A definite integral represents the area under a curve between two specified limits. It is calculated by finding the difference between the antiderivative evaluated at the upper and lower limits.

Fundamental Theorem of Calculus. This theorem establishes the connection between differentiation and integration. It states that the derivative of the definite integral of a function is equal to the original function.

Properties of definite integrals. Definite integrals have several useful properties, such as linearity, additivity, and the ability to change the limits of integration.

12. Differential Equations: Modeling Change

Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given.

Differential equations as models. A differential equation is an equation that relates a function to its derivatives. They are used to model various phenomena involving rates of change.

Key concepts:

• Order: The highest order derivative in the equation.
• Degree: The power of the highest order derivative.
• General solution: A solution containing arbitrary constants.
• Particular solution: A solution obtained by assigning specific values to the arbitrary constants.

Methods of solving. Differential equations can be solved using various methods, such as separation of variables, homogeneous equations, and linear equations.

Last updated: April 3, 2025