Class 9 — Maths — Formulas

Coordinates of a point: (x, y) — x = abscissa, y = ordinate

Point on x-axis: (x, 0) | Point on y-axis: (0, y) | Origin: (0, 0)

Quadrant I: x > 0, y > 0 | Quadrant II: x < 0, y > 0 | Quadrant III: x < 0, y < 0 | Quadrant IV: x > 0, y < 0

Distance of P(x, y) from x-axis = |y|

Distance of P(x, y) from y-axis = |x|

General polynomial: a_n x^n + ... + a_1 x + a_0

Linear polynomial: p(x) = ax + b, where a ≠ 0

Value at x = k: p(k) = ak + b

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

Graph of y = ax + b is a straight line | Degree of linear polynomial = 1

Rational number: p/q where p, q are integers and q ≠ 0

Laws of exponents: a^m · a^n = a^(m+n) | a^m / a^n = a^(m-n) | (a^m)^n = a^(mn)

(ab)^n = a^n · b^n | a^0 = 1 | a^(-n) = 1/a^n

√(ab) = √a · √b | √(a/b) = √a/√b

Rationalisation: 1/√a = √a/a | 1/(a + √b) = (a - √b)/(a^2 - b)

(a + b)^2 = a^2 + 2ab + b^2

(a - b)^2 = a^2 - 2ab + b^2

(a + b)(a - b) = a^2 - b^2

(x + a)(x + b) = x^2 + (a+b)x + ab

(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca

(a + b)^3 = a^3 + b^3 + 3ab(a+b) | (a - b)^3 = a^3 - b^3 - 3ab(a-b)

a^3 + b^3 = (a+b)(a^2 - ab + b^2) | a^3 - b^3 = (a-b)(a^2 + ab + b^2)

a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) | If a+b+c=0 then a^3+b^3+c^3 = 3abc

In a right triangle with angle θ:

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

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

Pythagoras: hypotenuse^2 = opposite^2 + adjacent^2

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

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

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

Perimeter of rectangle = 2(l + b) | Area of rectangle = l × b

Perimeter of square = 4a | Area of square = a^2

Area of triangle = (1/2) × base × height

Heron's formula: A = √(s(s-a)(s-b)(s-c)), s = (a+b+c)/2

Area of parallelogram = base × height

Area of rhombus = (1/2) × d1 × d2 | Area of trapezium = (1/2)(a+b) × h

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

Area of sector = (θ/360°) × πr^2 | Arc length = (θ/360°) × 2πr

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

0 ≤ P(E) ≤ 1

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

P(E) + P(not E) = 1, so P(not E) = 1 - P(E)

Empirical probability = (Number of times event happened) / (Total trials)

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

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

GP: a, ar, ar^2, ... | nth term: a_n = a · r^(n-1)

Sum of n terms of GP: S_n = a(r^n - 1)/(r - 1), for r ≠ 1