Mathematics — Defence Exams Study Material

Algebra

Quadratic Equations

Standard Form & Discriminant

ax² + bx + c = 0 x = [-b ± √(b²-4ac)] / 2a

D > 0 → two distinct real roots
D = 0 → two equal real roots
D < 0 → no real roots (complex)

⚔️ CDS/NDA Focus

Sum of roots = −b/a; Product of roots = c/a. Nature of roots questions appear every year. Memorise the relationship between coefficients and roots.

Progressions (AP & GP)

Arithmetic Progression (AP)

nth term: aₙ = a + (n-1)d Sum: Sₙ = n/2 [2a + (n-1)d]

Geometric Progression (GP)

nth term: aₙ = arⁿ⁻¹ Sum (finite): Sₙ = a(rⁿ-1)/(r-1), r≠1 Sum (infinite, |r|<1): S∞ = a/(1-r)

Logarithms

Key Properties

log(ab) = log a + log b
log(a/b) = log a − log b
log(aⁿ) = n·log a
log_b(a) = log a / log b (change of base)
log 1 = 0; log 10 = 1 (base 10)

Binomial Theorem

General Term

(x + y)ⁿ = Σ ⁿCᵣ · xⁿ⁻ʳ · yʳ (r = 0 to n)

General term: T(r+1) = ⁿCᵣ · xⁿ⁻ʳ · yʳ

💡 Quick Tip

For NDA, focus on finding the middle term and coefficient of a specific term. The sum of coefficients is obtained by putting x = y = 1: sum = 2ⁿ.

Trigonometry

Trigonometric Ratios & Identities

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	∞

Fundamental Identities

sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = cosec²θ

Compound Angle Formulas

Addition/Subtraction

sin(A±B) = sinA·cosB ± cosA·sinB cos(A±B) = cosA·cosB ∓ sinA·sinB tan(A±B) = (tanA ± tanB)/(1 ∓ tanA·tanB)

Double Angle

sin2A = 2sinA·cosA cos2A = cos²A - sin²A = 1 - 2sin²A = 2cos²A - 1 tan2A = 2tanA/(1 - tan²A)

Heights & Distances

⚔️ High-Value Topic for CDS

Angle of Elevation: Looking UP at an object — use tan θ = height/base
Angle of Depression: Looking DOWN at an object — alternate interior angles
Draw a clear diagram for every H&D problem
Common patterns: tower on a hill, two poles, shadow problems

Matrices & Determinants

Matrix Operations

Types of Matrices

Type	Property
Row Matrix	1 × n (single row)
Column Matrix	m × 1 (single column)
Square Matrix	m = n
Diagonal Matrix	Non-diagonal elements = 0
Identity Matrix (I)	Diagonal = 1, rest = 0; A·I = A
Symmetric Matrix	A = Aᵀ (transpose = original)
Skew-Symmetric	A = −Aᵀ; diagonal elements = 0

Determinants (2×2 and 3×3)

2×2 Determinant

|A| = ad - bc for [[a,b],[c,d]]

Properties of Determinants

If any row/column = 0, determinant = 0
If two rows/columns are identical, det = 0
Swapping two rows changes sign of determinant
|AB| = |A|·|B|; |Aᵀ| = |A|

⚔️ Inverse Matrix

A⁻¹ exists only if |A| ≠ 0 (non-singular matrix).
A⁻¹ = adj(A) / |A|, where adj(A) is the adjugate (transpose of cofactor matrix).

Differential & Integral Calculus

Differentiation

Standard Derivatives

Function	Derivative
xⁿ	nxⁿ⁻¹
eˣ	eˣ
ln x	1/x
sin x	cos x
cos x	−sin x
tan x	sec²x

Chain, Product, Quotient Rules

Chain: d/dx[f(g(x))] = f'(g(x))·g'(x) Product: d/dx[uv] = u'v + uv' Quotient: d/dx[u/v] = (u'v - uv')/v²

Applications of Derivatives

💡 Maxima & Minima

f'(x) = 0 at critical points. If f''(x) < 0 → local maximum. If f''(x) > 0 → local minimum. Rate of change, increasing/decreasing functions tested in NDA.

Integration

Standard Integrals

Function	Integral
xⁿ	xⁿ⁺¹/(n+1) + C, n≠−1
1/x	ln\|x\| + C
eˣ	eˣ + C
sin x	−cos x + C
cos x	sin x + C
sec²x	tan x + C

Definite Integral (Area)

∫[a to b] f(x)dx = F(b) − F(a)

Area under a curve between x = a and x = b.

Coordinate Geometry

Straight Lines

Important Formulas

Formula	Expression
Distance PQ	√[(x₂−x₁)² + (y₂−y₁)²]
Mid-point	((x₁+x₂)/2, (y₁+y₂)/2)
Slope (m)	(y₂−y₁)/(x₂−x₁)
Slope-intercept form	y = mx + c
Point-slope form	y − y₁ = m(x − x₁)
Two-point form	(y−y₁)/(y₂−y₁) = (x−x₁)/(x₂−x₁)

Circles

Standard Equations

Centre (0,0): x² + y² = r² Centre (h,k): (x−h)² + (y−k)² = r² General: x² + y² + 2gx + 2fy + c = 0 → centre (−g,−f), r = √(g²+f²−c)

Conic Sections

Conic	Standard Equation	Key Property
Parabola	y² = 4ax	Focus (a,0); Directrix x = −a
Ellipse	x²/a² + y²/b² = 1 (a>b)	Foci (±c,0); c² = a²−b²
Hyperbola	x²/a² − y²/b² = 1	Foci (±c,0); c² = a²+b²

Statistics & Probability

Measures of Central Tendency

Mean, Median, Mode

Measure	Definition	Formula
Mean (x̄)	Sum ÷ Count	Σxᵢ/n
Median	Middle value (sorted)	For even n: avg of n/2 and (n/2+1)
Mode	Most frequent value	—

Measures of Dispersion

Range = Maximum − Minimum Variance (σ²) = Σ(xᵢ − x̄)²/n Standard Deviation (σ) = √Variance

Probability

Basic Probability

P(E) = Favourable outcomes / Total outcomes 0 ≤ P(E) ≤ 1; P(E) + P(E') = 1

Addition & Multiplication Rules

P(A∪B) = P(A) + P(B) − P(A∩B) P(A∩B) = P(A)·P(B) if A and B are independent

⚔️ Conditional Probability (NDA)

P(A|B) = P(A∩B) / P(B)

Bayes' theorem questions and problems on balls-in-urns are common. Practice systematically listing sample spaces.

Mensuration & Number System

2D & 3D Mensuration

Shape	Area / Volume	Perimeter / Surface Area
Circle	πr²	2πr
Triangle	½ × b × h (or Heron's)	a+b+c
Rectangle	l × b	2(l+b)
Cube	a³	6a²
Cylinder	πr²h	2πr(r+h)
Cone	⅓πr²h	πr(r+l), l=√(r²+h²)
Sphere	4/3 πr³	4πr²

Number System

Divisibility Rules

Divisor	Rule
2	Last digit even
3	Sum of digits divisible by 3
4	Last two digits divisible by 4
5	Ends in 0 or 5
9	Sum of digits divisible by 9
11	(Sum of odd-position digits) − (Sum of even-position digits) divisible by 11

💡 HCF & LCM

HCF × LCM = Product of two numbers

For CDS: ratio problems, time & work, speed & distance often reduce to HCF/LCM. Practice identifying the right approach quickly.

Quick Revision Summary

Quadratics: discriminant determines nature of roots; sum = −b/a, product = c/a
AP sum formula: Sₙ = n/2[2a+(n-1)d]; GP sum: a(rⁿ−1)/(r−1)
Trig identities: sin²+cos²=1; derive sec and cosec identities from it
Heights & Distances: always draw diagram; angle of elevation = atan(height/base)
Matrices: inverse exists iff |A| ≠ 0; A⁻¹ = adj(A)/|A|
Calculus: d/dx(xⁿ)=nxⁿ⁻¹; ∫xⁿ=xⁿ⁺¹/(n+1)+C
Probability: P(A∪B) = P(A)+P(B)−P(A∩B); conditional P(A|B)=P(A∩B)/P(B)
Mensuration: Cylinder V=πr²h; Cone V=⅓πr²h; Sphere V=4/3 πr³