Key Formulas | Class 10 — Maths

01

Real Numbers

Fundamental Theorem of Arithmetic: every composite number is a unique product of primes

HCF(a,b) × LCM(a,b) = a × b

If denominator of p/q is of the form 2^m × 5^n → decimal terminates

Otherwise → non-terminating recurring

02

Polynomials

For p(x) = ax^2 + bx + c with zeroes α and β:

Sum of zeroes: α + β = -b/a

Product of zeroes: α × β = c/a

Polynomial with zeroes α and β: k[x^2 - (α+β)x + αβ]

Zero of linear polynomial ax + b: x = -b/a

03

Pair of Linear Equations in Two Variables

Standard form: a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0

Unique solution: a1/a2 ≠ b1/b2 (intersecting lines)

Infinitely many solutions: a1/a2 = b1/b2 = c1/c2 (coincident lines)

No solution: a1/a2 = b1/b2 ≠ c1/c2 (parallel lines)

04

Quadratic Equations

Standard form: ax^2 + bx + c = 0, a ≠ 0

Quadratic formula: x = [-b ± √(b^2 - 4ac)] / (2a)

Discriminant: D = b^2 - 4ac

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

Sum of roots = -b/a | Product of roots = c/a

05

Arithmetic Progressions

AP: a, a+d, a+2d, ... | nth term: a_n = a + (n-1)d

Sum of first n terms: S_n = (n/2)[2a + (n-1)d]

or S_n = (n/2)(a + l), where l = last term

Common difference: d = a_n - a_(n-1)

06

Triangles

Basic Proportionality Theorem: a line parallel to one side divides the other two sides in the same ratio

For similar triangles: AB/PQ = BC/QR = CA/RP

Area ratio of similar triangles = (corresponding side ratio)^2

Pythagoras: hypotenuse^2 = perpendicular^2 + base^2

07

Coordinate Geometry

Distance formula: d = √[(x2-x1)^2 + (y2-y1)^2]

Section formula (internal division m:n): ((mx2+nx1)/(m+n), (my2+ny1)/(m+n))

Mid-point: ((x1+x2)/2, (y1+y2)/2)

Area of triangle: (1/2)|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|

08

Introduction to Trigonometry

sin A = opp/hyp | cos A = adj/hyp | tan A = opp/adj

cosec A = 1/sin A | sec A = 1/cos A | cot A = 1/tan A

Identity 1: sin^2 A + cos^2 A = 1

Identity 2: 1 + tan^2 A = sec^2 A

Identity 3: 1 + cot^2 A = cosec^2 A

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

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

Complementary: sin(90°-A)=cos A | cos(90°-A)=sin A | tan(90°-A)=cot A

09

Some Applications of Trigonometry

tan θ = height / horizontal distance

For tower of height h, angle θ, distance d: tan θ = h/d

Sun's altitude θ, shadow length s, object height h: tan θ = h/s

Angle of elevation: measured upward from horizontal

Angle of depression: measured downward from horizontal

10

Circles

Tangent length from external point: PT^2 = OP^2 - r^2

Radius at point of contact is perpendicular to the tangent

Two tangents from external point: PT1 = PT2 (equal lengths)

Equal tangents subtend equal angles at the centre

11

Areas Related to Circles

Area of circle = πr^2 | Circumference = 2πr

Area of sector (angle θ) = (θ/360°) × πr^2

Length of arc = (θ/360°) × 2πr

Area of minor segment = Area of sector - Area of triangle formed by two radii and chord

Area of triangle with sides a, b and included angle θ: (1/2)ab sin θ

12

Surface Areas and Volumes

Cube: SA = 6a^2 | V = a^3

Cuboid: SA = 2(lb+bh+hl) | V = l×b×h

Cylinder: CSA = 2πrh | TSA = 2πr(h+r) | V = πr^2h

Cone: l = √(h^2+r^2) | CSA = πrl | TSA = πr(l+r) | V = (1/3)πr^2h

Sphere: SA = 4πr^2 | V = (4/3)πr^3

Hemisphere: CSA = 2πr^2 | TSA = 3πr^2 | V = (2/3)πr^3

Frustum: V = (πh/3)(r1^2+r2^2+r1r2) | CSA = π(r1+r2)l where l = √(h^2+(r1-r2)^2)

13

Statistics

Mean (direct): x̄ = Σ(fi·xi) / Σfi

Mean (assumed mean): x̄ = a + Σ(fi·di)/Σfi, where di = xi - a

Mean (step deviation): x̄ = a + h·[Σ(fi·ui)/Σfi], where ui = (xi-a)/h

Median = l + [(n/2 - cf)/f] × h

Mode = l + [(f1-f0)/(2f1-f0-f2)] × h

Empirical relation: 3 Median = Mode + 2 Mean

14

Probability

P(E) = (Number of favourable outcomes) / (Total possible outcomes)

0 ≤ P(E) ≤ 1 | P(E) + P(not E) = 1

P(sure event) = 1 | P(impossible event) = 0

Standard deck: 52 cards, 4 suits of 13 each, 26 red and 26 black, 12 face cards (J, Q, K of each suit)

Class 10 — Maths — Formulas