Algebra

Learning Objectives

• Master set theory, complex numbers, quadratic equations, and sequences for NDA Paper 1
• Apply shortcut techniques to solve algebra MCQs in under 45 seconds
• Recognise NDA-specific question patterns and elimination strategies
• Build speed through practice with 15+ topic-wise solved examples and 12 practice MCQs

1. Sets & Relations

Sets form the foundational language of mathematics. In NDA exams, set theory is tested through questions on set operations, Venn diagrams, cardinality of unions and intersections, and De Morgan’s laws. Relations and functions appear as interconnected topics with moderate weightage.

Set Theory Basics

Definitions & Terminology

A set is a well-defined collection of distinct objects called elements (or members). Sets are denoted by capital letters (A, B, C, …) and elements by lowercase letters. The notation a ∈ A means a belongs to set A. Sets can be described in roster form (listing elements within curly braces) or set-builder form (stating the property that defines membership).

Types of Sets � Complete Classification

Type	Definition	Example	NDA Note
Empty / Null Set	Set with no elements	φ or { }	φ ≠ {0}
Singleton Set	Exactly one element	{5}	Cardinality = 1
Finite Set	Countable number of elements	{1, 2, 3, 4}	|A| = n
Infinite Set	Uncountable elements	ℕ = {1, 2, 3, …}	Natural numbers
Equal Sets	Exactly the same elements	{1,2} = {2,1}	Order irrelevant
Equivalent Sets	Same cardinality	{a,b} ~ {1,2}	n(A) = n(B)
Subset	Every element of A is in B	{1} ⊆ {1,2}	∅ ⊆ every set
Proper Subset	A ⊆ B and A ≠ B	{1} ⊂ {1,2}	Strict subset
Power Set	Set of all subsets of A	P({a}) = {φ, {a}}	|P(A)| = 2n
Universal Set	All elements under discussion	U	Context-dependent

Subsets, Power Set & Cardinality Formulas

If |A| = n, then:
• Number of subsets = 2ⁿ
• Number of proper subsets = 2ⁿ − 1
• Number of non-empty subsets = 2ⁿ − 1
• Number of non-empty proper subsets = 2ⁿ − 2

Venn Diagrams & Set Operations

Venn diagrams use overlapping circles to visually represent relationships between sets. They are particularly useful for solving NDA word problems involving surveys, preferences, and group memberships.

Operation	Symbol	Set-Builder	Venn Region
Union	A ∪ B	{x | x ∈ A or x ∈ B}	Entire shaded area
Intersection	A ∩ B	{x | x ∈ A and x ∈ B}	Overlapping region
Difference	A − B	{x | x ∈ A and x ∉ B}	A only
Complement	A′ or Ac	{x | x ∈ U and x ∉ A}	Outside circle A
Symmetric Diff.	A Δ B	(A − B) ∪ (B − A)	Non-overlapping parts

Key Cardinality Formulas

Two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C)

De Morgan’s Laws

• (A ∪ B)′ = A′ ∩ B′
• (A ∩ B)′ = A′ ∪ B′

Relations & Functions � NDA Essentials

A relation from set A to set B is a subset of A × B (Cartesian product). A function f: A → B is a special relation where each element of A has exactly one image in B. Key function types tested in NDA:

• One-one (injective): f(x₁) = f(x₂) ⇒ x₁ = x₂
• Onto (surjective): For every y ∈ B, there exists x ∈ A with f(x) = y
• Bijective: Both one-one and onto
• Even function: f(−x) = f(x) � symmetric about y-axis
• Odd function: f(−x) = −f(x) � symmetric about origin

2. Complex Numbers

Complex numbers extend the real number system to include solutions of equations like x² + 1 = 0. A complex number is written as z = a + ib, where i = √(−1). The real part is Re(z) = a and the imaginary part is Im(z) = b. NDA consistently tests modulus, argument, polar form, De Moivre’s theorem, and cube roots of unity.

Argand Plane & Modulus

Geometric Representation

Every complex number z = x + iy corresponds uniquely to a point (x, y) in the Argand plane (also called the complex plane). The x-axis is the real axis and the y-axis is the imaginary axis.

Modulus, Argument & Conjugate

• Modulus: |z| = √(x² + y²) � distance from origin
• Argument: arg(z) = tan⁻¹(y/x) � angle with positive real axis, range (−π, π]
• Conjugate: z̅ = x − iy � reflection across the real axis

Three Representations of Complex Numbers

Form	Expression	Best For
Cartesian (Rectangular)	z = x + iy	Addition, subtraction, conjugate
Polar (Trigonometric)	z = r(cos θ + i sin θ)	Multiplication, division, powers
Exponential (Euler)	z = reiθ	De Moivre, roots of unity

Properties of Modulus and Conjugate

• |z₁ z₂| = |z₁| · |z₂|
• |z₁ / z₂| = |z₁| / |z₂| (z₂ ≠ 0)
• |z₁ + z₂| ≤ |z₁| + |z₂| (Triangle inequality)
• |z₁ − z₂| ≥ ||z₁| − |z₂||
• zz̅ = |z|²
• z₁ + z₂ = z̅₁ + z̅₂
• z₁ z₂ = z̅₁ · z̅₂

Polar Form & De Moivre’s Theorem

To convert to polar form: z = r(cos θ + i sin θ), where r = |z| and θ = arg(z).

De Moivre’s Theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for integer n.

To find n^th roots: z^1/n = r^1/n[cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], for k = 0, 1, …, n−1.

Cube Roots of Unity

The cube roots of unity are the three solutions of z³ = 1: 1, ω, ω², where ω = (−1 + i√3)/2 and ω² = (−1 − i√3)/2.

Essential Properties (memorise for NDA):

• 1 + ω + ω² = 0
• ω³ = 1 and (ω²)³ = 1
• ω^3k = 1, ω^3k+1 = ω, ω^3k+2 = ω²
• ω and ω² are complex conjugates

3. Quadratic Equations

A quadratic equation in x is of the form ax² + bx + c = 0, where a, b, c are real and a ≠ 0. NDA tests nature of roots, sum and product relations, formation of equations, and common root conditions.

Standard Form & Quadratic Formula

The roots are given by: x = [−b ± √(b² − 4ac)] / 2a. The discriminant D = b² − 4ac determines the nature.

Nature of Roots � Complete Classification

Discriminant	Nature of Roots	Example
D > 0, perfect square	Real, distinct, rational	x2 − 5x + 6 = 0
D > 0, not perfect square	Real, distinct, irrational	x2 − 4x + 2 = 0
D = 0	Real and equal (repeated)	x2 − 4x + 4 = 0
D < 0	Complex conjugate pair	x2 + x + 1 = 0

Sum & Product of Roots

If α and β are roots, then:
Sum: α + β = −b/a
Product: αβ = c/a
Form equation: x² − (α+β)x + αβ = 0

Useful Expressions

• α² + β² = (α + β)² − 2αβ
• α³ + β³ = (α + β)³ − 3αβ(α + β)
• |α − β| = √D / |a|

Common Roots

One common root: Condition: (c₁a₂ − c₂a₁)² = (a₁b₂ − a₂b₁)(b₁c₂ − b₂c₁). Both roots common: a₁/a₂ = b₁/b₂ = c₁/c₂.

Graphical Interpretation

The parabola y = ax² + bx + c opens upward if a > 0, downward if a < 0. Vertex at x = −b/2a. If a > 0 and D < 0, expression is always positive; if a < 0 and D < 0, always negative.

4. Sequences & Series

A sequence is an ordered list of numbers following a pattern (e.g., 2, 4, 6, 8, …). A series is the sum of terms. NDA tests AP, GP, HP, and standard summation formulas.

Arithmetic Progression (AP)

General form: a, a+d, a+2d, … Common difference = d.

• n^th term: T_n = a + (n − 1)d
• Sum: S_n = n/2 [2a + (n − 1)d] = n/2 (a + l)
• Arithmetic Mean: AM of a, b = (a + b)/2

Geometric Progression (GP)

General form: a, ar, ar², … Common ratio = r.

• n^th term: T_n = arⁿ⁻¹
• Sum (n terms): S_n = a(rⁿ − 1)/(r − 1), r ≠ 1
• Sum (∞): S_∞ = a/(1 − r), |r| < 1
• Geometric Mean: GM of a, b = √(ab)

Harmonic Progression (HP)

Reciprocals of terms form an AP. HM of a, b = 2ab/(a+b). Key relationship: AM ≥ GM ≥ HM (equality when numbers are equal).

Sum of Special Series

Series	Sum of First n Terms	Memory Aid
1 + 2 + 3 + … + n	n(n + 1)/2	“n n plus one by two”
12 + 22 + … + n2	n(n+1)(2n+1)/6	“n n+1 2n+1 by six”
13 + 23 + … + n3	[n(n+1)/2]2	“Square of the sum”
1 + 3 + 5 + … + (2n−1)	n2	“Sum of odds = n2”

5. Permutations & Combinations

Permutations count arrangements (order matters), combinations count selections (order does not). High-weightage topic in NDA with 4–6 questions on word arrangements, committee formation, and geometric counting.

Fundamental Principle of Counting

Multiplication: If event 1 in m ways and event 2 in n ways, both occur together in m × n ways.
Addition: If events are mutually exclusive, total = m + n.

nPr & nCr Formulas

Concept	Formula	Example
Permutation	nPr = n!/(n−r)!	5P3 = 60
Combination	nCr = n!/[r!(n−r)!]	5C3 = 10
All distinct	n!	5! = 120
With repetitions	n!/(p!q!r!…)	MISSISSIPPI: 11!/(4!4!2!)
Combination identity	nCr = nCn−r	7C4 = 7C3 = 35

Circular Permutations

• Distinct objects around circle: (n − 1)!
• Necklace/garland (clockwise = anticlockwise): (n − 1)! / 2

6. Binomial Theorem

The binomial theorem expands (x + y)ⁿ for positive integer n. NDA tests general term, middle term, term independent of x, and coefficient properties.

Statement & Expansion

(x + y)ⁿ = Σ ⁿC_r x^n−r y^r, where r = 0 to n. The expansion has (n+1) terms.

General Term & Middle Term

Term Type	Formula	Condition
General term (r+1)th	Tr+1 = nCr xn−r yr	0 ≤ r ≤ n
Middle term (n even)	Tn/2+1	One middle term
Middle terms (n odd)	T(n+1)/2 and T(n+3)/2	Two middle terms
Independent of x	Set exponent of x = 0	Solve for r

Binomial Coefficients Properties

• ⁿC₀ + ⁿC₁ + … + ⁿC_n = 2ⁿ
• ⁿC₀ − ⁿC₁ + … + (−1)^{n n}C_n = 0
• Sum of even-indexed coefficients = Sum of odd-indexed coefficients = 2ⁿ⁻¹
• ⁿC_r + ⁿC_r−1 = ⁿ⁺¹C_r (Pascal’s rule)

7. Matrices & Determinants

A matrix is a rectangular array of numbers (rows × columns). Determinants are scalars for square matrices. NDA tests operations, determinant properties, and Cramer’s rule.

Matrix Operations

Operation	Rule	Note
Addition	Element-wise (same order)	Commutative
Scalar Multiplication	Each element × scalar	Distributive
Multiplication	Row × Column (dot product)	NOT commutative
Transpose	Swap rows & columns	(AB)T = BTAT
Inverse (2×2)	A−1 = (1/det A) adj(A)	det A ≠ 0

Matrix Properties

• AB ≠ BA (non-commutative)
• (AB)C = A(BC) (associative)
• A(B+C) = AB + AC (distributive)
• (AB)⁻¹ = B⁻¹A⁻¹

Determinant Properties

• det(kA) = kⁿ det(A) for n×n matrix
• det(AB) = det(A) · det(B)
• det(A^T) = det(A)
• Two identical rows/columns ⇒ det = 0
• Row interchange changes sign of det
• det(A⁻¹) = 1 / det(A)
• Triangular matrix: det = product of diagonal entries

Cramer’s Rule

For a₁x + b₁y = c₁, a₂x + b₂y = c₂:
D = det[[a₁,b₁],[a₂,b₂]], D_x = det[[c₁,b₁],[c₂,b₂]], D_y = det[[a₁,c₁],[a₂,c₂]]
x = D_x/D, y = D_y/D (provided D ≠ 0).

8. Linear Inequalities

Linear inequalities involve >, ≥, <, ≤. NDA tests solving, interval representation, and sign reversal rules.

Interval Notation

• (a, b) → a < x < b
• [a, b] → a ≤ x ≤ b
• (a, ∞) → x > a
• (−∞, b] → x ≤ b

Solving Rules

• Adding/subtracting preserves inequality
• Multiplying/dividing by positive preserves inequality
• Multiplying/dividing by negative reverses the inequality
• Never multiply/divide by a variable with unknown sign

9. Probability

Probability ranges from 0 to 1. NDA tests classical probability, conditional probability, and Bayes’ theorem at a basic level.

Basic Terminology

• Experiment: Process with well-defined outcomes (rolling a die)
• Sample Space (S): All possible outcomes; n(S) = total count
• Event (E): Subset of S; n(E) = favourable count
• Probability: P(E) = n(E)/n(S), 0 ≤ P(E) ≤ 1

Key Formulas

Concept	Formula
Union	P(A∪B) = P(A) + P(B) − P(A∩B)
Complement	P(A′) = 1 − P(A)
Conditional	P(A|B) = P(A∩B)/P(B)
Independent	P(A∩B) = P(A)·P(B)
Mutually Exclusive	P(A∩B) = 0

10. Statistics (basic)

Statistics covers collection, organisation, and analysis of data. NDA tests mean, median, mode, variance, and standard deviation.

Measures of Central Tendency

• Mean (𝑥): For ungrouped: (Σx_i)/n. For grouped: (Σf_ix_i)/(Σf_i)
• Median: Middle value when sorted. n odd: (n+1)/2^th term. n even: average of n/2^th and (n/2+1)^th terms
• Mode: Most frequent value

Measures of Dispersion

Measure	Definition	Formula
Range	Max − Min	—
Variance (σ2)	Avg of squared deviations	Σ(xi−𝑥)2/n
Std Dev (σ)	√(variance)	√[Σ(xi−𝑥)2/n]
Coeff of Variation	Relative dispersion	(σ/𝑥) × 100%

Computational Formula for Variance

σ² = [Σx_i²/n] − (𝑥)². This avoids individual deviations and is much faster for NDA.