Class 10 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

Conservation of mass

A balanced equation has equal atoms of each element on both sides

Water formation

2H₂ + O₂ → 2H₂O

Combination

A + B → AB

Decomposition

AB → A + B

Displacement

A + BC → AC + B (A more reactive than B)

Double displacement

AB + CD → AD + CB

Neutralisation

Acid + Base → Salt + Water

Oxidation (oxygen)

Gain of oxygen or loss of hydrogen / electrons

From chapter list

e.g. 2H2 + O2 → 2H2O

pH

pH = −log₁₀ [H⁺] (idea as taught)

pH scale

pH < 7 acidic; pH = 7 neutral; pH > 7 basic

Water ion product (idea)

H₂O ⇌ H⁺ + OH⁻

Washing soda

Na₂CO₃ · 10H₂O

Baking soda

NaHCO₃

Bleaching powder

CaOCl₂

Plaster of Paris

CaSO₄ · ½H₂O

Gypsum

CaSO₄ · 2H₂O

From chapter list

pH = −log10[H+] (idea)

Ionic compound (idea)

Metal atom loses e⁻; non-metal gains e⁻

Rusting

4Fe + 3O₂ + x H₂O → 2Fe₂O₃ · x H₂O

Thermite (example)

Fe₂O₃ + 2Al → 2Fe + Al₂O₃ + heat

Aqua regia

conc. HCl + conc. HNO₃ (3 : 1 by volume)

Tetravalency

Carbon forms 4 covalent bonds

Homologous series

Successive members differ by −CH₂−

General alkane

CₙH₂ₙ₊₂

General alkene

CₙH₂ₙ

General alkyne

CₙH₂ₙ₋₂

Alcohol general

CₙH₂ₙ₊₁OH

Combustion (alkane idea)

CₓHᵧ + O₂ → CO₂ + H₂O + heat

Photosynthesis

6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂ (light, chlorophyll)

Aerobic respiration

C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + energy

Anaerobic (muscle)

Glucose → lactic acid + energy

Anaerobic (yeast)

Glucose → ethanol + CO₂ + energy

Reflex path

Receptor → sensory neuron → spinal cord → motor neuron → effector

Sexual outline

Gametes fuse → zygote → embryo

Monohybrid F2

Phenotypic ratio 3 : 1

Dihybrid F2 (if asked)

Phenotypic ratio 9 : 3 : 3 : 1

Human sex

XX female, XY male

From chapter list

F2 monohybrid 3:1

Laws of reflection

∠i = ∠r; incident ray, reflected ray and normal are coplanar

Mirror formula

1/v + 1/u = 1/f

Magnification (mirror)

m = h′/h = −v/u

Focal length (spherical mirror)

f = R/2

Snell’s law

n = sin i / sin r (= n₂/n₁)

Absolute refractive index

n = c / v

Lens formula

1/v − 1/u = 1/f

Magnification (lens)

m = h′/h = v/u

Power of a lens

P = 1/f (f in metres, P in dioptre)

Combination of thin lenses

P = P₁ + P₂ + …

From chapter list

1/v + 1/u = 1/f (mirror)

Power of accommodation

Ability of the eye lens to change focal length

Myopia correction

Concave (diverging) lens

Hypermetropia correction

Convex (converging) lens

Prism deviation (idea)

White light splits into VIBGYOR

Current

I = Q / t

Ohm’s law

V = I R (temperature constant)

Resistance (material)

R = ρ l / A

Series

Rₛ = R₁ + R₂ + R₃ + …

Parallel (two)

1/Rₚ = 1/R₁ + 1/R₂

Parallel (two shortcut)

Rₚ = (R₁ R₂) / (R₁ + R₂)

Electric power

P = V I = I² R = V² / R

Electrical energy

E = P t = V I t

Heat (Joule)

H = I² R t

Commercial unit

1 kWh = 3.6 × 10⁶ J

From chapter list

V=IR; P=VI

Right-hand thumb rule

Thumb → current; curled fingers → field

Fleming’s left-hand rule

Force on a current in a magnetic field (motor)

Fleming’s right-hand rule

Induced current (generator)

Domestic fuse

Fuse is in the live wire, in series

10% law

About 10% of energy passes to the next trophic level