Key Formulas:
- Total Relations from A to B = 2^(m × n)
- Total Functions = n^m
- Total One-One Functions (Injective) = nPm (if n ≥ m)
- Fractional Part {x} = x - [x], Domain = R, Range = [0, 1)
Context 1: Functional Equations via Variable Substitution
Example 3: Check if R = {(a, b) : a - b is divisible by 3} is Equivalence
Reflexive: a - a = 0 = 3(0) -> True.
Symmetric: If a - b = 3k, then b - a = -3k = 3(-k) -> True.
Transitive: If a - b = 3k and b - c = 3m, then (a - b) + (b - c) = a - c = 3(k + m) -> True.
Final Result: R is an Equivalence Relation
Homework Problems
1. Number of onto functions from A={1,2,3} to B={a,b}
Answer: 2³ - 2 = 6
2. Domain of 1 / sqrt(x² - 16)
Answer: (-∞, -4) ∪ (4, ∞)
3. Period of f(x) = |sin x| + |cos x|
Answer: π/2
4. Range of f(x) = x / (1 + x²)
Answer: [-1/2, 1/2]
2. Complex Numbers & Locus Problems
Key Formulas:
- Cube Roots of Unity: 1 + w + w² = 0, w³ = 1
- De Moivre's Theorem: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)
- Locus |z - z1| / |z - z2| = k: Straight line if k = 1, Circle if k ≠ 1
Context 1: Cube Roots of Unity Simplifications
Example 1: Simplify (1 + w - w²)⁵ + (1 - w + w²)⁵
Step 1: Use 1 + w = -w² and 1 + w² = -w.
Step 2: First term: (-w² - w²)⁵ = (-2w²)⁵ = -32 w^10 = -32 w.
Final Result: Circle with Center (0, -5) and Radius 4
Homework Problems
1. Value of (1 + i)^8
Answer: 16
2. Amplitude of -1 - i sqrt(3)
Answer: -2π/3
3. Value of w^2026 + w^2027
Answer: -1
4. Locus of |z - 1| = |z + 1|
Answer: Y-axis (x = 0)
3. Quadratic Equations & Newton's Method
Key Formulas:
- Newton's Theorem: If S_n = α^n + β^n, then a S_n + b S_(n-1) + c S_(n-2) = 0
- Location of Roots: Both roots > k -> D ≥ 0, -b/2a > k, a f(k) > 0
Context 1: Newton's Power Sum Recurrence
Example 1: Find (a10 - 2 a8) / (2 a9) if x² - 6x - 2 = 0
Context 2: Location of Roots relative to Real Constants
Example 2: Find range of 'm' if both roots of x² - mx + 4 = 0 are real and > 1
Condition 1 (D ≥ 0): m² - 16 ≥ 0 -> m ≤ -4 or m ≥ 4.
Condition 2 (-b/2a > 1): m/2 > 1 -> m > 2.
Condition 3 (f(1) > 0): 1 - m + 4 > 0 -> m < 5.
Intersection: m ∈ [4, 5)
Final Result: [4, 5)
Homework Problems
1. Number of real roots of x² - 5|x| + 6 = 0
Answer: 4
2. Min value of 2x² - 8x + 11
Answer: 3
3. Condition for roots of ax² + bx + c = 0 to be reciprocal
Answer: a = c
4. Product of roots of x² + 3x - 10 = 0
Answer: -10
4. Matrices & Determinants
Key Formulas:
- |adj A| = |A|^(n-1)
- |adj(adj A)| = |A|^((n-1)²)
- Cramer's Rule: Infinite Solutions -> D = D1 = D2 = D3 = 0. No Solution -> D = 0 and at least one Di ≠ 0.
Context 1: Matrix Adjoint Determinant Properties
Example 1: If |A| = 4 for order 3, find |adj(3A)|
Step 1: Let B = 3A. Since order n = 3, |B| = |3A| = 3³ |A| = 27 × 4 = 108.
4. Determinant of skew-symmetric matrix of order 3
Answer: 0
5. Permutations & Combinations
Key Formulas:
- Derangement D_n = n! [1/2! - 1/3! + ... + (-1)^n / n!]
- Gap Method: Used when no two items can be together.
- String/Block Method: Used when specific items must always be together.
Context 1: Derangements (Complete Mismatch)
Example 1: Number of ways to put 4 letters into 4 wrong envelopes