Formula Sheet

Ohm's Law

V = I × R

V in volts, I in amperes, R in ohms

Resistance of a conductor

R = ρ l / A

ρ = resistivity

Series resistance

Rs = R₁ + R₂ + R₃ …

Parallel resistance

1/Rp = 1/R₁ + 1/R₂ + 1/R₃ …

Electric power

P = VI = I²R = V²/R

Heating effect (Joule's law)

H = I²Rt

Charge & current

Q = I × t

1 A = 1 C/s

Commercial energy

1 kWh = 3.6 × 10⁶ J

Mirror formula

1/v + 1/u = 1/f

Magnification (mirror)

m = −v/u = h'/h

Focal length of mirror

f = R/2

Lens formula

1/v − 1/u = 1/f

Magnification (lens)

m = v/u = h'/h

Power of lens

P = 1/f (in metres)

unit: dioptre (D)

Refractive index

n = c/v = sin i / sin r

Lenses in contact

P = P₁ + P₂ + …

Least distance of distinct vision

D = 25 cm

Myopia correction

Concave lens, f = −(far point)

Hypermetropia correction

Convex lens, 1/f = 1/v − 1/u with v = −near point

Right-hand thumb rule

Thumb → current, curled fingers → magnetic field

Fleming's left-hand rule

Force on current-carrying conductor in a field (motor)

Fleming's right-hand rule

Induced current direction (generator)

pH scale

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

Neutralisation

Acid + Base → Salt + Water

Metal + Acid

Metal + dil. acid → Salt + H₂↑

Corrosion of iron

4Fe + 3O₂ + xH₂O → 2Fe₂O₃·xH₂O (rust)

Chlor-alkali process

2NaCl + 2H₂O → 2NaOH + Cl₂ + H₂

Respiration (aerobic)

C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + 38 ATP

Photosynthesis

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

Euclid's division lemma

a = bq + r, 0 ≤ r < b

HCF × LCM

HCF(a,b) × LCM(a,b) = a × b

Quadratic: sum of zeroes

α + β = −b/a

Quadratic: product of zeroes

αβ = c/a

Forming a quadratic

x² − (α+β)x + αβ = 0

Cubic relations

α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a

Unique solution

a₁/a₂ ≠ b₁/b₂

Infinitely many solutions

a₁/a₂ = b₁/b₂ = c₁/c₂

No solution

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Quadratic formula

x = [−b ± √(b² − 4ac)] / 2a

Discriminant

D = b² − 4ac

D>0 real distinct, D=0 equal, D<0 no real roots

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)

Distance formula

d = √[(x₂−x₁)² + (y₂−y₁)²]

Section formula

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

Midpoint

( (x₁+x₂)/2, (y₁+y₂)/2 )

Basic Proportionality (Thales)

AD/DB = AE/EC

Similar triangles area

ar(ΔABC)/ar(ΔPQR) = (AB/PQ)²

Pythagoras theorem

hyp² = base² + perpendicular²

Tangent property

Tangent ⊥ radius at point of contact

Tangents from external point

PA = PB

Basic ratios

sin θ = P/H, cos θ = B/H, tan θ = P/B

Identity 1

sin²θ + cos²θ = 1

Identity 2

1 + tan²θ = sec²θ

Identity 3

1 + cot²θ = cosec²θ

Complementary angles

sin(90° − θ) = cos θ, tan(90° − θ) = cot θ

Standard values

sin30 = 1/2, sin45 = 1/√2, sin60 = √3/2

Area of sector

(θ/360) × πr²

Length of arc

(θ/360) × 2πr

Cylinder

CSA = 2πrh, TSA = 2πr(r+h), V = πr²h

Cone

CSA = πrl, TSA = πr(l+r), V = (1/3)πr²h

Sphere

SA = 4πr², V = (4/3)πr³

Hemisphere

CSA = 2πr², TSA = 3πr², V = (2/3)πr³

Frustum

V = (1/3)πh(R² + r² + Rr)

Mean (direct)

x̄ = Σfᵢxᵢ / Σfᵢ

Mean (assumed mean)

x̄ = a + Σfᵢdᵢ / Σfᵢ

Mode

l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h

Median

l + [(n/2 − cf)/f] × h

Empirical relation

3 Median = Mode + 2 Mean

Probability

P(E) = favourable outcomes / total outcomes