Class 10 Mathematics Formula Sheet

HCF–LCM product

HCF(a, b) × LCM(a, b) = a × b

Positive integers

Euclid’s division

a = bq + r, 0 ≤ r < b

Terminating decimal

Denominator of simplest form is 2^m × 5^n

Non-terminating repeating

Prime factors other than 2 and 5 in the denominator

Quadratic sum of zeros

α + β = −b / a

Quadratic product of zeros

αβ = c / a

Cubic sum of zeros

α + β + γ = −b / a

Cubic sum of products two at a time

αβ + βγ + γα = c / a

Cubic product of zeros

αβγ = −d / a

Quadratic polynomial from zeros

k[x² − (α+β)x + αβ]

From chapter list

α+β = −b/a; αβ = c/a

Unique solution

a1/a2 ≠ b1/b2

Intersecting lines, consistent

No solution

a1/a2 = b1/b2 ≠ c1/c2

Parallel, inconsistent

Infinitely many

a1/a2 = b1/b2 = c1/c2

Coincident

Cross-multiplication (x)

x = (b1c2 − b2c1) / (a1b2 − a2b1)

Cross-multiplication (y)

y = (c1a2 − c2a1) / (a1b2 − a2b1)

From chapter list

a1/a2, b1/b2, c1/c2 comparison

Standard form

ax² + bx + c = 0, a ≠ 0

Discriminant

D = b² − 4ac

Quadratic formula

x = [−b ± √(b² − 4ac)] / (2a)

Two distinct real roots

D > 0

Two equal real roots

D = 0, x = −b / (2a)

No real roots

D < 0

nth term

aₙ = a + (n − 1)d

Sum of n terms

Sₙ = n/2 [2a + (n − 1)d]

Sum using last term

Sₙ = n/2 (a + l)

Common difference

d = aₙ − aₙ₋₁

Number of terms

n = [(l − a) / d] + 1

From chapter list

an=a+(n−1)d; Sn=n/2[2a+(n−1)d]

Basic Proportionality (Thales)

If DE ∥ BC in ΔABC, then AD/DB = AE/EC

AA / SSS / SAS similarity

Corresponding angles equal ⇒ sides proportional

Areas of similar triangles

ar(Δ1) / ar(Δ2) = (side1 / side2)²

Pythagoras

In a right triangle, (hyp)² = (base)² + (perp)²

Converse of Pythagoras

If a² + b² = c², the angle opposite c is 90°

From chapter list

ΔABC ~ ΔDEF ⇒ ar(ABC)/ar(DEF)=(AB/DE)²

Distance

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Section formula (internal)

P = ( (m x₂ + n x₁)/(m+n) , (m y₂ + n y₁)/(m+n) )

Mid-point

M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )

Area of triangle

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

Collinear points

Area of triangle = 0

From chapter list

d=√[(x2−x1)²+(y2−y1)²]

Sine

sin θ = opposite / hypotenuse

Cosine

cos θ = adjacent / hypotenuse

Tangent

tan θ = opposite / adjacent = sin θ / cos θ

Reciprocals

csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ

Identity 1

sin²θ + cos²θ = 1

Identity 2

1 + tan²θ = sec²θ

Identity 3

1 + cot²θ = csc²θ

Standard angles

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

cos standard

cos 0°=1, 30°=√3/2, 45°=1/√2, 60°=1/2, 90°=0

tan standard

tan 0°=0, 30°=1/√3, 45°=1, 60°=√3, 90° undefined

From chapter list

sin²A+cos²A=1

Height from elevation

height = base × tan θ

θ = angle of elevation

Distance from depression

Use tan of angle of depression in the right triangle

Two elevations from same point

Draw two right triangles; subtract or add heights

Tangent ⊥ radius

Radius to the point of contact is perpendicular to the tangent

Two tangents from an external point

They are equal in length

Number of tangents

From a point: 0 (inside), 1 (on circle), 2 (outside)

Circumference

C = 2πr = πd

Area of circle

A = πr²

Length of arc

l = (θ/360°) × 2πr

Area of sector

A_sector = (θ/360°) × πr²

Area of segment

A_segment = area of sector − area of triangle

From chapter list

A=πr²; arc=(θ/360)×2πr

Cuboid volume

V = l × b × h

Cube

TSA = 6a², V = a³

Cylinder CSA

CSA = 2πrh

Cylinder TSA

TSA = 2πr(h + r)

Cylinder volume

V = πr²h

Cone CSA

CSA = πrl, l = √(r² + h²)

Cone TSA

TSA = πr(l + r)

Cone volume

V = (1/3) πr²h

Sphere CSA / TSA

4πr²

Sphere volume

V = (4/3) πr³

Hemisphere CSA (curved)

2πr²

Hemisphere TSA

3πr²

Hemisphere volume

V = (2/3) πr³

Recast solids

Volume remains constant when a solid is melted and recast

From chapter list

V_sphere=4/3πr³; V_cone=⅓πr²h

Mean (direct)

x̄ = Σ fᵢ xᵢ / Σ fᵢ

Mean (assumed mean)

x̄ = a + (Σ fᵢ dᵢ / Σ fᵢ), dᵢ = xᵢ − a

Mean (step-deviation)

x̄ = a + h (Σ fᵢ uᵢ / Σ fᵢ), uᵢ = (xᵢ − a)/h

Median (grouped)

Median = l + [(n/2 − cf) / f] × h

Mode (grouped)

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

Empirical relation

3 Median = Mode + 2 Mean

Approximate, for moderately skewed data

Classical probability

P(E) = n(E) / n(S)

Equally likely outcomes

Range

0 ≤ P(E) ≤ 1

Complement

P(not E) = 1 − P(E)

Sure event

P(S) = 1

Impossible event

P(∅) = 0