Maths MID-TERM MARATHON - Class 11 ISC Mathematics

Sets, Relations, Functions, Trigonometric and Complex Numbers: Key Concepts and Exam Methods

Definition and Properties of Sets

A set is defined as a well-defined collection of objects, where "well-defined" means the membership of an element in the set is unambiguous and the same for everyone. Sets are only those collections where the answer to "is this element in the set?" is the same for all.

• Examples of sets: Set of vowels in English alphabet, set of even natural numbers less than 10, set of prime numbers less than 20.
• Non-examples: Set of beautiful girls in a…

Sets, Relations, Functions, Trigonometric and Complex Numbers: Key Concepts and Exam Methods

Definition and Properties of Sets

A set is defined as a well-defined collection of objects, where "well-defined" means the membership of an element in the set is unambiguous and the same for everyone. Sets are only those collections where the answer to "is this element in the set?" is the same for all.

• Examples of sets: Set of vowels in English alphabet, set of even natural numbers less than 10, set of prime numbers less than 20.
• Non-examples: Set of beautiful girls in a school, set of favorite cricketers—these are subjective and vary from person to person.
• Representation: Sets can be represented in two forms:
  • Roster (Tabular) Form: Listing all elements, e.g., $\{2, 4, 6, 8\}$.
  • Set Builder Form: Describing the property, e.g., $\{x : x$ is an even natural number less than $10\}$.

Types of Sets and Set Operations

Sets can be classified based on their elements and relationships with other sets. Operations on sets are fundamental for problem-solving.

• Singleton Set: A set with exactly one element, e.g., $\{2\}$.
• Empty (Null) Set: A set with no elements, denoted by $\{\}$ or $\varnothing$.
• Finite and Infinite Sets: Finite sets have countable elements; infinite sets do not end, e.g., set of natural numbers.
• Equivalent Sets: Sets with the same number of elements (cardinal number), regardless of what the elements are.
• Equal Sets: Sets with exactly the same elements.
• Subset: $A$ is a subset of $B$ if every element of $A$ is in $B$.
• Proper Subset: Subset that is not equal to the original set.
• Power Set: The set of all subsets of a set.
• Universal Set: A set that contains all elements under consideration.
• Cardinal Number: The number of unique elements in a set, denoted $n(A)$.

Set Operations and Venn Diagrams

Set operations allow combining or comparing sets, often visualized with Venn diagrams.

• Union ($A \cup B$): All elements in $A$ or $B$.
• Intersection ($A \cap B$): Elements common to both $A$ and $B$.
• Difference ($A - B$): Elements in $A$ but not in $B$.
• Complement ($A'$): Elements in the universal set not in $A$.
• Symmetric Difference ($A \triangle B$): Elements in $A$ or $B$ but not both.

Key properties:

• $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
• $n(A - B) = n(A) - n(A \cap B)$

Intervals and Notation

Intervals describe subsets of real numbers using inequalities and bracket notation.

• Open Interval $(a, b)$: $a < x < b$, both endpoints excluded.
• Closed Interval $[a, b]$: $a \leq x \leq b$, both endpoints included.
• Half-open Intervals: $(a, b]$ or $[a, b)$, one endpoint included.
• Infinity: Always paired with a round bracket, e.g., $(-\infty, a)$.

Bracket choice:

• Round bracket: Strict inequality.
• Square bracket: Includes endpoint (equality allowed).

Relations and Functions

A relation is a rule that connects elements of one set to another, often represented as ordered pairs.

• Ordered Pair: $(a, b)$, where order matters.
• Cartesian Product $A \times B$: Set of all ordered pairs $(a, b)$ with $a \in A$, $b \in B$.
• Number of Relations: If $|A| = m$, $|B| = n$, then $2^{mn}$ possible relations from $A$ to $B$.
• Domain: Set of all first elements ($x$ values) in the relation.
• Range: Set of all second elements ($y$ values) in the relation.
• Co-domain: The entire set $B$ in a relation from $A$ to $B$.

A function is a special type of relation where every element of the domain is associated with exactly one element of the co-domain.

• Criteria for a function:
  • Every $x$ in domain has exactly one $y$ in co-domain.
  • Two $x$ can have the same $y$ (many-to-one allowed).
  • No $x$ left unmapped; no $x$ maps to more than one $y$.

Domain and Range of Functions

The domain of a function is the set of all permissible input values ($x$) for which the function is defined. The range is the set of all possible output values ($y$).

• For rational functions: Denominator $\neq 0$.
• For square roots: Expression under root $\geq 0$.
• For logarithms: Argument $> 0$.
• For modulus: Output is always non-negative.

Examples:

• $f(x) = \frac{1}{x+2}$: Domain is all real numbers except $x = -2$.
• $f(x) = \sqrt{3-2x}$: Domain is $x \leq \frac{3}{2}$.
• $f(x) = \log(4x-3)$: Domain is $x > \frac{3}{4}$.

Trigonometric Functions and Identities

Trigonometric functions relate angles to ratios of sides in a right triangle and extend to all real numbers using the unit circle.

• Basic ratios:
  • $\sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$
  • $\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}}$
  • $\tan \theta = \frac{\text{Perpendicular}}{\text{Base}}$
  • $\csc \theta = \frac{\text{Hypotenuse}}{\text{Perpendicular}}$
  • $\sec \theta = \frac{\text{Hypotenuse}}{\text{Base}}$
  • $\cot \theta = \frac{\text{Base}}{\text{Perpendicular}}$
• Key identities:
  • $\sin^2 \theta + \cos^2 \theta = 1$
  • $1 + \tan^2 \theta = \sec^2 \theta$
  • $1 + \cot^2 \theta = \csc^2 \theta$
• Reciprocal and quotient relations:
  • $\sin \theta \cdot \csc \theta = 1$
  • $\cos \theta \cdot \sec \theta = 1$
  • $\tan \theta = \frac{\sin \theta}{\cos \theta}$
  • $\cot \theta = \frac{\cos \theta}{\sin \theta}$

Trigonometric Angles, Quadrants, and Periodicity

Angles can be measured in degrees or radians. The sign and value of trigonometric functions depend on the quadrant.

• Conversion: $180^\circ = \pi$ radians; to convert degrees to radians, multiply by $\frac{\pi}{180}$.
• Quadrants:
  • I: All positive
  • II: $\sin$, $\csc$ positive
  • III: $\tan$, $\cot$ positive
  • IV: $\cos$, $\sec$ positive
• Periodicity:
  • $\sin x$ and $\cos x$ repeat every $2\pi$.
  • $\tan x$ and $\cot x$ repeat every $\pi$.
• Reference angle: The acute angle made with the $x$-axis; used to find function values for any angle.

Trigonometric Transformations and Ranges

Transformations involve shifting, scaling, or reflecting trigonometric functions.

• Function change at $\frac{\pi}{2}$ and $\frac{3\pi}{2}$: $\sin$ becomes $\cos$, $\tan$ becomes $\cot$, etc.
• No function change at $\pi$ and $2\pi$: Function remains the same.
• Range of $a\sin x + b\cos x$: Maximum value is $\sqrt{a^2 + b^2}$, minimum is $-\sqrt{a^2 + b^2}$.
• For $f(x) = 1 + 3\cos 2x$: Range is $-2$ to $4$.

Complex Numbers: Definition, Operations, and Properties

A complex number is of the form $a + i b$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$.

• Real part: $a$
• Imaginary part: $b$
• Addition/Subtraction: Add/subtract real and imaginary parts separately.
• Multiplication: Expand as binomials, use $i^2 = -1$.
• Division: Rationalize denominator using conjugate.
• Conjugate: Change sign of imaginary part, e.g., conjugate of $a + ib$ is $a - ib$.
• Modulus: $|z| = \sqrt{a^2 + b^2}$

Polar Representation and Cube Roots of Unity

Complex numbers can be represented in polar form as $r(\cos \theta + i \sin \theta)$, where $r$ is modulus and $\theta$ is argument.

• Argument ($\theta$): $\tan \theta = \frac{y}{x}$, adjust for quadrant.
• Cube roots of unity: Solutions to $x^3 = 1$ are $1$, $\omega = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$, $\omega^2 = -\frac{1}{2} - \frac{\sqrt{3}}{2}i$.
• Properties:
  • $1 + \omega + \omega^2 = 0$
  • $\omega^3 = 1$
  • Product of all three roots is $1$

Quadratic Equations and Inequalities

Quadratic equations are of the form $ax^2 + bx + c = 0$.

• Roots: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Nature of roots:
  • $D = b^2 - 4ac > 0$: Real and distinct
  • $D = 0$: Real and equal
  • $D < 0$: Complex conjugate roots
• Sum and product of roots:
  • Sum: $-\frac{b}{a}$
  • Product: $\frac{c}{a}$
• Forming quadratic from roots $\alpha, \beta$: $x^2 - (\alpha + \beta)x + \alpha\beta = 0$
• Inequalities:
  • When multiplying/dividing by a negative, reverse the inequality sign.
  • For modulus inequalities $|x + a| < b$, solution is $-b < x + a < b$.

Key Exam Takeaways

• Always check for well-definedness in set questions.
• Use set builder and roster forms as required.
• For functions, ensure every $x$ maps to only one $y$.
• For trigonometric identities, memorize key formulas and quadrant rules.
• In complex numbers, use $i^2 = -1$ and properties of conjugate and modulus.
• For quadratic equations, use discriminant to determine root nature and apply sum/product relations.
• For inequalities, apply sign rules and modulus properties carefully.