📚 CLASS 10 MATHEMATICS — FORMULA VAULT

1. REAL NUMBERS

Euclid's Division Lemma:

a = bq + r,   0 ≤ r < b

For two positive integers:

HCF(a, b) × LCM(a, b) = a × b

Decimal expansion:

If a rational number is in lowest terms and its positive denominator is of the form

2^m5ⁿ,

with non-negative integers m and n, then its decimal expansion is terminating.

2. POLYNOMIALS

For quadratic polynomial:

p(x) = ax² + bx + c, a ≠ 0

Sum of zeroes:

α + β = −b/a

Product of zeroes:

αβ = c/a

Polynomial in terms of zeroes:

p(x) = a[x² − (α + β)x + αβ]

3. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

General form:

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

Unique solution:

a₁/a₂ ≠ b₁/b₂

No solution:

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Infinitely many solutions:

a₁/a₂ = b₁/b₂ = c₁/c₂

Cross-multiplication:

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

The ratio tests above assume their denominators are nonzero. Cross-multiplication gives a unique solution when a₁b₂ − a₂b₁ ≠ 0.

4. QUADRATIC EQUATIONS

Standard form:

ax² + bx + c = 0,   a ≠ 0

Quadratic formula:

x = (−b ± √(b² − 4ac))/(2a)

Discriminant:

D = b² − 4ac

D > 0 → two distinct real roots

D = 0 → two equal real roots

D < 0 → no real roots

Sum of roots:

α + β = −b/a

Product of roots:

αβ = c/a

5. ARITHMETIC PROGRESSIONS

Common difference:

d = a₂ − a₁

n^th term:

a_n = a + (n − 1)d

Sum of first n terms:

S_n = n/2 [2a + (n − 1)d]

Also,

S_n = n/2(a + l)

where l is the last term.

6. TRIANGLES

Basic Proportionality Theorem:

If DE ∥ BC in ΔABC:

AD/DB = AE/EC

Similarity criterion:

AA, SAS, SSS

Areas of similar triangles:

ar(ΔABC)/ar(ΔDEF) = (AB/DE)²

Pythagoras theorem:

Hypotenuse² = Base² + Perpendicular²

c² = a² + b²

7. COORDINATE GEOMETRY

Distance formula:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Section formula:

If P divides A(x₁, y₁) and B(x₂, y₂) internally in ratio m:n:

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

Midpoint formula:

P = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Area of triangle:

Area = 1/2 |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

8. INTRODUCTION TO TRIGONOMETRY

sin θ = Perpendicular/Hypotenuse

cos θ = Base/Hypotenuse

tan θ = Perpendicular/Base

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

Identities:

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = cosec²θ

Important relation:

tan θ = sin θ/cos θ

9. SOME APPLICATIONS OF TRIGONOMETRY

Angle of elevation:

Angle formed when the object is above the observer's horizontal line.

Angle of depression:

Angle formed when the object is below the observer's horizontal line.

Most problems use:

tan θ = Height/Base

Therefore:

Height = Base × tan θ

10. CIRCLES

Tangent theorem:

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Radius ⟂ Tangent

Tangents from an external point:

PA = PB

11. AREAS RELATED TO CIRCLES

Circumference:

C = 2πr

Area of circle:

A = πr²

Area of sector:

A = (θ/360°)πr²

Length of arc:

l = (θ/360°) × 2πr

Area of segment:

Area of minor segment = Area of minor sector − Area of triangle

12. SURFACE AREAS AND VOLUMES

🔹 Cuboid

Volume = lbh

Total surface area = 2(lb + bh + hl)

Area of four walls = 2h(l + b)

Diagonal = √(l² + b² + h²)

🔹 Cube

Volume = a³

Total surface area = 6a²

Lateral surface area = 4a²

Diagonal = a√3

🔹 Cylinder

Curved surface area = 2πrh

Total surface area = 2πr(h + r)

Volume = πr²h

🔹 Cone

Slant height:

l = √(r² + h²)

Curved surface area = πrl

Total surface area = πr(l + r)

Volume = 1/3 πr²h

🔹 Sphere

Surface area = 4πr²

Volume = 4/3 πr³

🔹 Hemisphere

Curved surface area = 2πr²

Total surface area = 3πr²

Volume = 2/3 πr³

13. STATISTICS

Mean:

Mean = Σx/n

Mean for grouped data:

Mean = Σf_ix_i/Σf_i

Class mark:

x_i = (Upper limit + Lower limit)/2

Assumed mean method:

x̄ = a + Σf_id_i/Σf_i

Step-deviation method:

x̄ = a + h(Σf_iu_i/Σf_i)

Median:

Median = l + [(n/2 − cf)/f] × h

Mode:

Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h

Empirical relation:

Mode ≈ 3 Median − 2 Mean (empirical approximation)

14. PROBABILITY

Classical probability:

P(E) = Number of favourable outcomes / Total number of equally likely outcomes

Range:

0 ≤ P(E) ≤ 1

Impossible event:

P(E) = 0

Sure event:

P(E) = 1

Complementary event:

P(not E) = 1 − P(E)

Therefore:

P(E) + P(not E) = 1

⭐ IMPORTANT ALGEBRAIC IDENTITIES

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

a² − b² = (a − b)(a + b)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a − b)³ = a³ − b³ − 3ab(a − b)

a³ + b³ = (a + b)(a² − ab + b²)

a³ − b³ = (a − b)(a² + ab + b²)

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

📌 TRIGONOMETRIC VALUES

🚀 SIMPOMATHS FORMULA VAULT

Class 10 Mathematics — Quick Revision Reference

Use these formulas for revision and practice. Always check the conditions and definitions before applying a formula.