Class 10 — Maths Formula Sheet

a = bq + r,  0 ≤ r < b

For any two positive integers a and b.

📌 a = dividend, b = divisor, q = quotient, r = remainder

Every integer > 1 has a unique prime factorisation

Used to find HCF and LCM.

HCF(a,b) × LCM(a,b) = a × b

Product rule for HCF and LCM.

💡 HCF(12,18)=6, LCM=36, 6×36=12×18=216

x = [−b ± √(b²−4ac)] / 2a

Roots of ax²+bx+c=0.

📌 a, b, c = coefficients, a ≠ 0

💡 x²−5x+6=0: x=(5±1)/2 → x=3 or 2

D = b² − 4ac

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

α + β = −b/a
α × β = c/a

Relationship between roots and coefficients of ax²+bx+c=0.

📌 α, β = roots

x² − (α+β)x + αβ = 0

Build a quadratic from its roots α and β.

aₙ = a + (n−1)d

General term of an arithmetic progression.

📌 a = first term, d = common difference, n = term number

💡 AP 2,5,8: a₅=2+(4)3=14

Sₙ = n/2 × [2a + (n−1)d]
or  Sₙ = n/2 × (a + l)

Sum of first n terms. Use second form when last term l is known.

📌 l = last term

💡 S₅ for 2,5,8: 5/2×[4+12]=40

d = a₂ − a₁ = aₙ − aₙ₋₁

Constant difference between consecutive terms.

sin θ = Opposite / Hypotenuse

Basic trig ratio for a right-angled triangle.

cos θ = Adjacent / Hypotenuse

Basic trig ratio.

tan θ = Opposite / Adjacent = sin θ / cos θ

Basic trig ratio.

cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ

Reciprocal trigonometric ratios.

sin²θ + cos²θ = 1

Most fundamental trig identity.

💡 sin²30°+cos²30° = ¼+¾ = 1 ✓

1 + tan²θ = sec²θ

Derived from dividing the first identity by cos²θ.

1 + cot²θ = cosec²θ

Derived from dividing the first identity by sin²θ.

θ   | 0°  | 30° | 45°  | 60°  | 90°
sin | 0   | ½   | 1/√2 | √3/2 | 1
cos | 1   | √3/2| 1/√2 | ½    | 0
tan | 0   | 1/√3| 1    | √3   | ∞

Exact values for standard angles — must memorise for exams.

tan θ = height / horizontal distance

Used in height and distance problems.

💡 Tower 100m, distance 100m: tan θ=1, θ=45°

PT² = PA × PB  (secant–tangent)
PT = √(d² − r²)

Length of tangent from external point P to circle.

📌 d = distance from centre, r = radius

OT ⊥ PT  (∠OTP = 90°)

Tangent at point of contact is perpendicular to the radius.

PA = PB

Two tangents drawn from an external point are equal in length.

V = (πh/3)(r₁² + r₂² + r₁r₂)

Volume of a cone with top cut off.

📌 h = height, r₁, r₂ = top and bottom radii

CSA = π(r₁+r₂)l
l = √[h²+(r₁−r₂)²]

Curved surface area of a frustum.

📌 l = slant height of frustum

TSA = π[r₁²+r₂²+(r₁+r₂)l]

Total surface area including both circular faces.

x̄ = Σfᵢxᵢ / Σfᵢ

Mean for grouped data using class marks.

📌 fᵢ = frequency, xᵢ = class mark

x̄ = a + (Σfᵢdᵢ / Σfᵢ)
where dᵢ = xᵢ − a

Assumed mean method to simplify calculation.

📌 a = assumed mean

Median = l + [(n/2 − cf) / f] × h

Median for grouped frequency distribution.

📌 l = lower boundary, cf = cumulative freq before, f = freq of median class, h = class width

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

Mode for grouped data.

📌 f₁ = modal class freq, f₀ = preceding, f₂ = following, l = lower boundary, h = width

P(E) = (Favourable outcomes) / (Total outcomes)
0 ≤ P(E) ≤ 1

Classical definition of probability.

P(E') = 1 − P(E)

Probability of event NOT occurring.