1. ARITHMETIC

Number System

Natural numbers: N = {1, 2, 3, …}
Whole numbers: W = {0, 1, 2, 3, …}
Integers: Z = {…, −2, −1, 0, 1, 2, …}
Rational number:

p/q, q ≠ 0

HCF and LCM

For two positive integers a, b:

HCF × LCM = a × b

Euclid's division lemma:

a = bq + r, 0 ≤ r < b

Fractions

a/b + c/d = (ad + bc)/(bd)

a/b − c/d = (ad − bc)/(bd)

a/b × c/d = ac/(bd)

a/b ÷ c/d = a/b × d/c = ad/(bc)

Reciprocal:

a/b → b/a

2. PERCENTAGE

Percentage = Part / Whole × 100

x% of y = x/100 × y

Percentage increase:

% Increase = Increase / Original × 100

Percentage decrease:

% Decrease = Decrease / Original × 100

New value after x% increase:

New = Original × (1 + x/100)

After x% decrease:

New = Original × (1 − x/100)

3. PROFIT AND LOSS

Profit = SP − CP

Loss = CP − SP

Profit% = Profit / CP × 100

Loss% = Loss / CP × 100

Selling price with profit:

SP = CP((100 + Profit%) / 100)

Selling price with loss:

SP = CP((100 − Loss%) / 100)

Cost price:

CP = (SP × 100) / (100 + Profit%)

or

CP = (SP × 100) / (100 − Loss%)

4. DISCOUNT

Discount = MP − SP

Discount% = Discount / MP × 100

SP = MP(1 − d/100)

5. SIMPLE INTEREST

SI = PRT/100

where:

P = Principal
R = Rate %
T = Time

Amount:

A = P + SI

6. COMPOUND INTEREST

Annual compounding:

A = P(1 + R/100)ⁿ

CI = A − P

Half-yearly:

A = P(1 + (R/2)/100)²ⁿ

Quarterly:

A = P(1 + (R/4)/100)⁴ⁿ

Depreciation:

A = P(1 − R/100)ⁿ

7. RATIO AND PROPORTION

a : b = a/b

If

a/b = c/d

then

ad = bc

Continued proportion:

a : b = b : c

then

b² = ac

Direct proportion:

y ∝ x

y = kx

Inverse proportion:

y ∝ 1/x

xy = k

8. AVERAGE

Average = Sum of observations / Number of observations

Sum:

Sum = Average × Number

Weighted average:

x̄ = Σwx / Σw

9. TIME, SPEED AND DISTANCE

Speed = Distance / Time

Distance = Speed × Time

Time = Distance / Speed

Average speed:

Average Speed = Total Distance / Total Time

Conversion:

1 m/s = 18/5 km/h

1 km/h = 5/18 m/s

10. BOATS AND STREAMS

Let:

b = speed of boat in still water
s = speed of stream

Downstream:

D = b + s

Upstream:

U = b − s

Boat speed:

b = (D + U)/2

Stream speed:

s = (D − U)/2

11. TIME AND WORK

If A completes work in x days:

A's one-day work = 1/x

If B takes y days:

1/y

Together:

1/T = 1/x + 1/y

So:

T = xy/(x + y)

Work:

Work = Efficiency × Time

12. ALGEBRA — BASIC IDENTITIES

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

(a + b)(a − b) = a² − b²

(x + a)(x + b) = x² + (a + b)x + ab

(x − a)(x − b) = x² − (a + b)x + ab

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

13. CUBE IDENTITIES

(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)

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

If

a + b + c = 0

then

a³ + b³ + c³ = 3abc

14. EXPONENTS / INDICES

a^m × aⁿ = a^m+n

a^m/aⁿ = a^m−n

(a^m)ⁿ = a^mn

(ab)ⁿ = aⁿbⁿ

(a/b)ⁿ = aⁿ/bⁿ

a⁰ = 1

a⁻ⁿ = 1/aⁿ

a^1/n = ⁿ√a

15. LINEAR EQUATION

ax + b = 0

x = −b/a

General equation in two variables:

ax + by + c = 0

16. QUADRATIC EQUATION

Standard form:

ax² + bx + c = 0

Quadratic formula:

x = (−b ± √(b² − 4ac))/(2a)

Discriminant:

D = b² − 4ac

If:

D > 0: two real roots
D = 0: equal roots
D < 0: no real roots

Sum of roots:

α + β = −b/a

Product:

αβ = c/a

Equation from roots:

x² − (α + β)x + αβ = 0

17. SIMULTANEOUS EQUATIONS

a₁x + b₁y = c₁

a₂x + b₂y = c₂

Cross multiplication:

x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁)

18. POLYNOMIALS

Remainder theorem:

If p(x) is divided by (x − a),

Remainder = p(a)

Factor theorem:

If

p(a) = 0

then

(x − a) is a factor

19. BINOMIAL THEOREM

(a + b)ⁿ = Σ_r=0ⁿ ⁿC_ra^n−rb^r

where:

ⁿC_r = n!/(r!(n − r)!)

General term:

T_r+1 = ⁿC_ra^n−rb^r

20. GEOMETRY — ANGLE RULES

Right angle:

90°

Straight angle:

180°

Complete angle:

360°

Complementary angles:

A + B = 90°

Supplementary angles:

A + B = 180°

Angles around a point:

A + B + C + ⋯ = 360°

Linear pair:

A + B = 180°

Vertically opposite angles:

∠1 = ∠3

∠2 = ∠4

21. PARALLEL LINES CUT BY A TRANSVERSAL

Corresponding angles

Corresponding angles are equal

Alternate interior angles

Alternate interior angles are equal

Alternate exterior angles

Alternate exterior angles are equal

Same-side / co-interior angles

A + B = 180°

These four rules solve most parallel-line angle questions.

22. TRIANGLE ANGLE RULES

Interior angles:

A + B + C = 180°

Exterior angle:

Exterior angle = sum of two opposite interior angles

For example:

x = A + B

23. ISOSCELES TRIANGLE

If two sides are equal:

AB = AC

then opposite angles are equal:

∠B = ∠C

Converse:

If

∠B = ∠C

then

AB = AC

24. EQUILATERAL TRIANGLE

All sides equal:

AB = BC = CA

Every angle:

60°

25. PYTHAGORAS THEOREM

For a right triangle:

c² = a² + b²

Hypotenuse:

c = √(a² + b²)

Other side:

a = √(c² − b²)

26. TRIANGLE CONGRUENCE RULES

Two triangles are congruent by:

SSS — Side-Side-Side
SAS — Side-Angle-Side
ASA — Angle-Side-Angle
AAS — Angle-Angle-Side
RHS — Right angle-Hypotenuse-Side

For congruent triangles:

Corresponding sides and angles are equal

27. SIMILAR TRIANGLES

Criteria:

AA
SAS
SSS

If:

△ABC ∼ △DEF

then

AB/DE = BC/EF = AC/DF

Corresponding angles:

∠A = ∠D

∠B = ∠E

∠C = ∠F

Area ratio:

Area of △ABC / Area of △DEF = (AB/DE)²

28. POLYGONS

Sum of interior angles:

(n − 2) × 180°

Each interior angle of a regular polygon:

(n − 2)180°/n

Sum of exterior angles:

360°

Each exterior angle of regular polygon:

360°/n

Number of diagonals:

n(n − 3)/2

29. QUADRILATERALS

All interior angles:

360°

Parallelogram

Opposite sides equal
Opposite sides parallel
Opposite angles equal
Adjacent angles supplementary

A + C = 180°

for adjacent angles appropriately:

A + B = 180°

Diagonals bisect each other.

Rectangle

All angles:

90°

Diagonals are equal and bisect each other.

Square

All sides equal.

All angles:

90°

Diagonals are equal, perpendicular and bisect each other.

Rhombus

All sides equal.

Diagonals are perpendicular bisectors.

30. CIRCLE THEOREMS

Central angle:

Angle at center = 2 × angle at circumference

Angle in a semicircle:

90°

Angles in the same segment:

Equal

Opposite angles of cyclic quadrilateral:

A + C = 180°

B + D = 180°

Radius and tangent:

Radius ⊥ Tangent

Tangents from the same external point:

PA = PB

31. COORDINATE GEOMETRY

Distance between:

(x₁, y₁), (x₂, y₂)

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Midpoint:

((x₁ + x₂)/2, (y₁ + y₂)/2)

Slope:

m = (y₂ − y₁)/(x₂ − x₁)

Equation of straight line:

y = mx + c

Point-slope form:

y − y₁ = m(x − x₁)

Section formula for ratio m : n:

((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))

32. MENSURATION — TRIANGLE

Perimeter:

P = a + b + c

Area:

A = ½bh

Heron's formula:

A = √(s(s − a)(s − b)(s − c))

where:

s = (a + b + c)/2

Equilateral triangle area:

A = (√3/4)a²

33. SQUARE

P = 4a

A = a²

Diagonal:

d = a√2

34. RECTANGLE

P = 2(l + b)

A = lb

Diagonal:

d = √(l² + b²)

35. PARALLELOGRAM

P = 2(a + b)

A = bh

36. RHOMBUS

P = 4a

A = ½d₁d₂

37. TRAPEZIUM / TRAPEZOID

P = a + b + c + d

A = ½(a + b)h

where a, b are parallel sides.

38. CIRCLE

Diameter:

d = 2r

Circumference:

C = 2πr = πd

Area:

A = πr²

39. SEMICIRCLE

Area:

A = ½πr²

Perimeter:

P = πr + 2r

40. ARC AND SECTOR

Arc length:

L = θ/360° × 2πr

Sector area:

A = θ/360° × πr²

41. CUBE

LSA:

4a²

TSA:

6a²

Volume:

a³

Diagonal:

a√3

42. CUBOID

LSA:

2h(l + b)

TSA:

2(lb + bh + hl)

Volume:

lbh

Diagonal:

√(l² + b² + h²)

43. CYLINDER

CSA:

2πrh

TSA:

2πr(r + h)

Volume:

πr²h

44. CONE

Slant height:

l = √(r² + h²)

CSA:

πrl

TSA:

πr(l + r)

Volume:

⅓πr²h

45. SPHERE

Surface area:

4πr²

Volume:

(4/3)πr³

46. HEMISPHERE

CSA:

2πr²

TSA:

3πr²

Volume:

⅔πr³

47. TRIGONOMETRY BASICS

For a right triangle:

sin θ = Opposite / Hypotenuse

cos θ = Adjacent / Hypotenuse

tan θ = Opposite / Adjacent

Reciprocals:

csc θ = 1/(sin θ)

sec θ = 1/(cos θ)

cot θ = 1/(tan θ)

Important identities:

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

Common values

Angle: 0°, 30°, 45°, 60°, 90°

sin θ: 0, ½, 1/√2, √3/2, 1

cos θ: 1, √3/2, 1/√2, ½, 0

tan θ: 0, 1/√3, 1, √3, Undefined

Very Important Geometry Problem-Solving Rules

When solving an unknown angle, check these first:

Straight line → 180°
Around a point → 360°
Right angle → 90°
Triangle → 180°
Vertically opposite angles → equal
Corresponding angles → equal when lines are parallel
Alternate angles → equal when lines are parallel
Same-side interior angles → 180°
Isosceles triangle → base angles equal
Exterior triangle angle → sum of two opposite interior angles
Quadrilateral → 360°
Angle in semicircle → 90°
Cyclic quadrilateral opposite angles → 180°
Pythagoras → c² = a² + b² for a right triangle
Radius to tangent → 90°

https://goodone.club/prashnaya/papers/PAP-NP-00000018

MASTER MATHEMATICS FORMULA SHEET

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