✓ Mathematics Master Notes

Number System · Arithmetic · Algebra · Geometry · Data | RRB NTPC CBT-1 & CBT-2

📚 6000+ Words 📌 Formulas & Shortcuts ✅ 30–35 Questions in Exam

🔢 NUMBER SYSTEM

A. Types of Numbers

Type	Definition	Example
Natural Numbers (N)	Counting numbers starting from 1	1, 2, 3, 4 ...
Whole Numbers (W)	Natural numbers including 0	0, 1, 2, 3 ...
Integers (Z)	Positive, negative, and 0	...-3, -2, -1, 0, 1, 2 ...
Rational Numbers (Q)	Can be expressed as p/q (q≠0)	1/2, 0.5, -3/4
Irrational Numbers	Cannot be expressed as p/q	√2, π, e
Prime Numbers	Only 2 factors: 1 and itself	2, 3, 5, 7, 11, 13 ...
Co-prime Numbers	HCF = 1; no common factor other than 1	(4, 9), (5, 7)

B. Divisibility Rules

Divisor	Rule	Example
2	Last digit is even (0,2,4,6,8)	124 ÷ 2 ✓
3	Sum of digits divisible by 3	123: 1+2+3=6 ✓
4	Last 2 digits divisible by 4	1216: 16÷4=4 ✓
5	Last digit is 0 or 5	135, 240 ✓
6	Divisible by both 2 and 3	12 ✓
7	Double last digit, subtract from rest; result divisible by 7	161: 16 − 2 = 14 ✓
8	Last 3 digits divisible by 8	1208: 208÷8=26 ✓
9	Sum of digits divisible by 9	729: 7+2+9=18 ✓
11	Alternating digit sum difference divisible by 11	2728: (2+2)−(7+8)=−11 ✓

C. HCF & LCM

HCF × LCM = Product of two numbers HCF of fractions = HCF of numerators / LCM of denominators LCM of fractions = LCM of numerators / HCF of denominators

💡 Shortcut: Finding LCM

For large numbers: express each as prime factors. LCM = product of highest powers of all prime factors.

Example: LCM(12, 18) = LCM(2²×3, 2×3²) = 2²×3² = 36

PYQ Pattern: Three bells ring at intervals 4, 6, 12 minutes. When do they ring together again? → LCM(4,6,12) = 12 minutes

D. Powers, Squares, and Cubes to Memorise

Squares (1–25)

1²=1 · 2²=4 · 3²=9 · 4²=16 · 5²=25
6²=36 · 7²=49 · 8²=64 · 9²=81 · 10²=100
11²=121 · 12²=144 · 13²=169 · 14²=196 · 15²=225
16²=256 · 17²=289 · 18²=324 · 19²=361 · 20²=400
21²=441 · 22²=484 · 23²=529 · 24²=576 · 25²=625

Cubes (1–15)

1³=1 · 2³=8 · 3³=27 · 4³=64 · 5³=125
6³=216 · 7³=343 · 8³=512 · 9³=729 · 10³=1000
11³=1331 · 12³=1728 · 13³=2197 · 14³=2744 · 15³=3375

📈 PERCENTAGE & PROFIT/LOSS

A. Percentage

% = (Part / Whole) × 100 % increase = [(New − Old) / Old] × 100 % decrease = [(Old − New) / Old] × 100 Successive discount x% then y%: Net discount = x + y − (xy/100)

💡 Fraction ↔ Percentage Conversions (Must Memorise)

Fraction	%	Fraction	%
1/2	50%	1/3	33.33%
1/4	25%	1/5	20%
1/6	16.67%	1/7	14.28%
1/8	12.5%	1/9	11.11%
2/3	66.67%	3/4	75%

B. Profit & Loss

Profit = SP − CP (when SP > CP) Loss = CP − SP (when CP > SP) Profit % = (Profit / CP) × 100 Loss % = (Loss / CP) × 100 SP = CP × (100 + P%) / 100 [for profit] SP = CP × (100 − L%) / 100 [for loss] CP = SP × 100 / (100 + P%) [when profit given] CP = SP × 100 / (100 − L%) [when loss given]

Mark-up & Discount

MP = Marked Price (listed price); SP = Selling Price after discount
Discount = MP − SP; Discount % = (Discount / MP) × 100
SP = MP × (100 − D%) / 100

📌 PYQ Shortcut — Dishonest Trader

A trader uses a 900g weight instead of 1000g and claims no profit/loss.

Actual profit % = (Error / True value − Error) × 100 = (100/900) × 100 = 11.11%

⚖ RATIO, PROPORTION, AVERAGES & MIXTURES

A. Ratio & Proportion

a : b = a/b If a:b = c:d, then ad = bc (cross multiplication) Componendo: (a+b)/(a-b) = (c+d)/(c-d) Invertendo: b/a = d/c Alternendo: a/c = b/d

Compound Ratio & Partnership

Compound ratio: a:b and c:d → ac:bd
Duplicate ratio: a:b → a²:b²
Sub-duplicate ratio: a:b → √a:√b
Partnership profit sharing: Proportional to (Capital × Time)

B. Average

Average = Sum of all observations / Number of observations Sum = Average × Number of observations Average of first n natural numbers = (n+1)/2 Average of first n even numbers = n+1 Average of first n odd numbers = n Average of consecutive numbers = (First + Last) / 2

Example: If average of 5 numbers is 10, and a 6th number (15) is added, new average = (50+15)/6 = 65/6 = 10.83

C. Mixture & Alligation

Alligation Rule: Cheaper quantity : Dearer quantity = (d − m) : (m − c) Where: c = cheaper price, d = dearer price, m = mean/mixture price For mixing two mixtures of different concentrations: Final concentration = (C₁V₁ + C₂V₂) / (V₁ + V₂)

Example: Mix Rs 20/kg rice with Rs 30/kg rice to get Rs 25/kg. Ratio = (30-25):(25-20) = 5:5 = 1:1

💵 SIMPLE & COMPOUND INTEREST

A. Simple Interest

SI = (P × R × T) / 100 Amount = P + SI = P (1 + RT/100) P = 100 × SI / (R × T) R = 100 × SI / (P × T) T = 100 × SI / (P × R)

B. Compound Interest

A = P (1 + R/100)ⁿ [compounded annually] CI = A − P Quarterly: A = P (1 + R/400)⁴ⁿ Half-yearly: A = P (1 + R/200)²ⁿ Difference (CI − SI) for 2 years = P(R/100)² Difference (CI − SI) for 3 years = P(R/100)²(R/100 + 3)

💡 Short trick for CI calculations

For 2 years at R% CI, effective rate = R + R + R²/100
For 10% CI, 2 years: 10 + 10 + 1 = 21% (on principal)
For 20% CI, 2 years: 20 + 20 + 4 = 44% (on principal)

📌 PYQ Pattern

A sum doubles in 6 years at SI. In how many years will it triple?

SI rate = 100/6 %; For triple: 200% interest needed → T = 200/(100/6) = 12 years

⏰ TIME, WORK & SPEED

A. Time & Work

If A can do work in 'a' days, A's 1-day work = 1/a If A and B together finish in 'n' days: n = ab/(a+b) [for 2 persons] Work = Efficiency × Time If A is twice as fast as B, and B takes 20 days → A takes 10 days Together: 10×20/(10+20) = 200/30 = 6.67 days

Pipes & Cisterns

Inlet pipe fills tank in A hours → in 1 hour it fills 1/A of tank
Outlet pipe empties tank in B hours → in 1 hour it empties 1/B of tank
Net filling per hour = 1/A − 1/B
Time to fill = AB/(B−A) when A < B

B. Time, Speed & Distance

Speed = Distance / Time Distance = Speed × Time Time = Distance / Speed Average Speed = Total Distance / Total Time (NOT average of speeds!) If equal distances: Average Speed = 2xy/(x+y) [harmonic mean] Conversion: 1 km/h = 5/18 m/s; 1 m/s = 18/5 km/h

C. Train Problems

Time to cross a pole = Train length / Speed Time to cross a platform = (Train length + Platform length) / Speed Two trains same direction: Relative speed = S₁ − S₂ Two trains opposite direction: Relative speed = S₁ + S₂ Time for trains to cross each other = (L₁ + L₂) / Relative speed

D. Boat & Stream

Downstream speed = u + v (u = boat speed, v = stream speed) Upstream speed = u − v Boat speed = (Downstream + Upstream) / 2 Stream speed = (Downstream − Upstream) / 2 Time downstream / Time upstream = (u − v) / (u + v)

💡 Memory Trick for Boat Problems

DSUM (Downstream = Sum), UDIFF (Upstream = Difference)

Boat speed = Average of downstream and upstream speeds

🔬 ALGEBRA

A. Important Algebraic Identities

(a+b)² = a² + 2ab + b² (a−b)² = a² − 2ab + b² (a+b)(a−b) = a² − b² (a+b)³ = a³ + 3a²b + 3ab² + b³ (a−b)³ = a³ − 3a²b + 3ab² − b³ a³ + b³ = (a+b)(a² − ab + b²) a³ − b³ = (a−b)(a² + ab + b²) (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ca a³ + b³ + c³ − 3abc = (a+b+c)(a² + b² + c² − ab − bc − ca) If a + b + c = 0 → a³ + b³ + c³ = 3abc

💡 Key Pattern: If x + 1/x = k

x² + 1/x² = k² − 2
x³ + 1/x³ = k³ − 3k = k(k²−3)
x − 1/x = √(k² − 4)

B. Linear Equations

For two equations: ax + by = c and dx + ey = f x = (ce − bf) / (ae − bd) y = (af − cd) / (ae − bd) For quadratic ax² + bx + c = 0: x = [−b ± √(b² − 4ac)] / 2a Sum of roots = −b/a Product of roots = c/a Discriminant (D) = b² − 4ac D > 0: 2 real roots; D = 0: equal roots; D < 0: imaginary roots

C. Arithmetic & Geometric Progressions

AP: a, a+d, a+2d, ... (d = common difference) nth term = a + (n−1)d Sum of n terms Sₙ = n/2 × [2a + (n−1)d] = n/2 × (first + last) GP: a, ar, ar², ... (r = common ratio) nth term = a × r^(n−1) Sum of n terms Sₙ = a(rⁿ−1)/(r−1) for r ≠ 1 Sum to infinity S∞ = a/(1−r) for |r| < 1 Sum of first n natural numbers = n(n+1)/2 Sum of first n squares = n(n+1)(2n+1)/6 Sum of first n cubes = [n(n+1)/2]²

⬜ GEOMETRY & MENSURATION

A. Triangles

Area = (1/2) × base × height Heron's formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2 For right triangle: a² + b² = c² (Pythagorean theorem) Common Pythagorean triplets: (3,4,5) (5,12,13) (8,15,17) (7,24,25) Equilateral triangle area = (√3/4) × a² Equilateral triangle height = (√3/2) × a Sum of angles of a triangle = 180°

B. Quadrilaterals

Shape	Area	Perimeter	Key Properties
Square	a²	4a	Diagonal = a√2
Rectangle	l × b	2(l+b)	Diagonal = √(l²+b²)
Parallelogram	base × height	2(a+b)	Opposite sides equal & parallel
Rhombus	(d₁ × d₂)/2	4a	Diagonals bisect at 90°
Trapezium	(1/2)(a+b)×h	Sum of all sides	One pair of parallel sides

C. Circle

Area = πr² Circumference = 2πr Arc length = (θ/360) × 2πr Sector area = (θ/360) × πr² Semi-circle area = πr²/2 Semi-circle perimeter = πr + 2r π ≈ 22/7 ≈ 3.14159

Circle Theorems (PYQ)

Angle in a semicircle = 90° (Thales' theorem)
Central angle = 2 × inscribed angle (for same arc)
Angles in the same segment are equal
Tangent to circle is perpendicular to radius at point of contact
Length of tangent from external point: T = √(d²−r²) where d = distance from point to centre

D. 3D Mensuration

Shape	Volume	Curved Surface Area	Total Surface Area
Cube (side a)	a³	4a²	6a²
Cuboid (l×b×h)	lbh	2h(l+b)	2(lb+bh+hl)
Cylinder (r,h)	πr²h	2πrh	2πr(r+h)
Cone (r,h,l)	(1/3)πr²h	πrl	πr(r+l)
Sphere (r)	(4/3)πr³	4πr²	4πr²
Hemisphere (r)	(2/3)πr³	2πr²	3πr²

📌 Common PYQ: Cone melted into spheres

Number of spheres = Volume of cone / Volume of one sphere

= [(1/3)πR²H] / [(4/3)πr³] = R²H / (4r³)

📦 TRIGONOMETRY

A. Trigonometric Ratios

sin θ = Opposite/Hypotenuse = P/H cos θ = Adjacent/Hypotenuse = B/H tan θ = Opposite/Adjacent = P/B = sin θ/cos θ cot θ = B/P = cos θ/sin θ = 1/tan θ sec θ = H/B = 1/cos θ cosec θ = H/P = 1/sin θ

B. Standard Values Table

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

💡 Memory Trick for sin values: "0, 1, √2, √3, 2 divided by 2"

sin 0°=0/2; sin 30°=1/2; sin 45°=√2/2; sin 60°=√3/2; sin 90°=2/2=1

C. Important Identities

sin²θ + cos²θ = 1 → sin²θ = 1 − cos²θ; cos²θ = 1 − sin²θ 1 + tan²θ = sec²θ → sec²θ − tan²θ = 1 1 + cot²θ = cosec²θ → cosec²θ − cot²θ = 1 sin(90°−θ) = cosθ; cos(90°−θ) = sinθ tan(90°−θ) = cotθ; cot(90°−θ) = tanθ

D. Height & Distance

Angle of Elevation: θ = arctan(height/horizontal distance) tan θ = Height / Base distance → Height = Base × tan θ If angle of elevation from distance d₁ is α and d₂ is β: Height = d₁ × d₂ × (tan α × tan β) / (d₁ tan β − d₂ tan α) [general case]

Example: A tower 30m high. Angle of elevation from 30m away? tan θ = 30/30 = 1 → θ = 45°

📊 DATA INTERPRETATION & STATISTICS

A. Measures of Central Tendency

Mean = Sum of all values / Number of values (Arithmetic Mean) Weighted Mean = Σ(wᵢxᵢ) / Σwᵢ Median = Middle value when data is arranged in order For odd n: Median = [(n+1)/2]th term For even n: Median = average of (n/2)th and (n/2+1)th terms Mode = Most frequently occurring value Empirical relationship: Mode = 3(Median) − 2(Mean)

B. Measures of Dispersion

Range = Maximum − Minimum Variance = Σ(xᵢ − mean)² / n Standard Deviation (σ) = √Variance For grouped data: Mean = Σfx / Σf Variance = Σf(x − mean)² / Σf

C. Probability

P(E) = Number of favourable outcomes / Total outcomes 0 ≤ P(E) ≤ 1 P(E) + P(E') = 1 P(A or B) = P(A) + P(B) − P(A and B) P(A and B) = P(A) × P(B) [for independent events] For a pack of 52 cards: 4 suits × 13 cards each Face cards (J, Q, K): 12 total Aces: 4 total Red cards: 26; Black cards: 26

D. Permutation & Combination

nPr = n! / (n−r)! [Permutation: order matters] nCr = n! / [r!(n−r)!] [Combination: order doesn't matter] nCr = nCₙ₋ᵣ 0! = 1; 1! = 1; 2! = 2; 3! = 6; 4! = 24; 5! = 120; 6! = 720; 7! = 5040 nC0 = nCn = 1 nC1 = n

E. Reading Data — DI Strategies

For Bar Charts & Pie Charts

Read the scale carefully before calculating
For pie chart: Value = (Angle/360) × Total or (Percentage/100) × Total
For bar chart: differences between bars = absolute change
% change = (New − Old)/Old × 100
In table DI: check row totals and column totals first

📚 MASTER FORMULA TABLE — Quick Reference

#	Topic	Key Formula
1	LCM × HCF	Product of two numbers
2	Percentage	(Part/Whole) × 100
3	Profit %	(Profit/CP) × 100
4	SP when Profit	CP × (100+P%)/100
5	SI	PRT/100
6	CI	P[(1+R/100)ⁿ − 1]
7	Average Speed	2xy/(x+y) for equal distances
8	Two persons work	ab/(a+b) days
9	Train crossing platform	(Train+Platform)/Speed
10	Downstream	u + v (boat + stream)
11	Circle area	πr²
12	Cylinder volume	πr²h
13	Cone volume	(1/3)πr²h
14	Sphere volume	(4/3)πr³
15	sin²θ + cos²θ	= 1
16	tan 45°	= 1
17	Sum AP	n/2 × (first + last)
18	Sum of first n naturals	n(n+1)/2
19	Quadratic roots sum	−b/a
20	nCr	n!/[r!(n−r)!]
21	Probability	Favourable/Total outcomes
22	Alligation	(d−m) : (m−c)
23	% increase	(New−Old)/Old × 100
24	Successive discount	x+y−xy/100
25	Area of triangle (Heron)	√[s(s-a)(s-b)(s-c)]

✅ Mathematics Exam Strategy

Number system and arithmetic (Percentage, P&L, SI/CI) carry highest weightage
Time, Work, Speed questions appear in every paper — master the shortcuts
For geometry: memorise standard formulae and Pythagorean triplets
DI (Data Interpretation) questions can be fast if you read the data format first
Practice calculation speed: squaring numbers quickly saves 30+ seconds per question
If stuck on a question, eliminate obviously wrong options and estimate