Mathematics Formula Sheet
56 formulas across 11 chapters — with variables explained and exam tips where needed.
Ch 1Real Numbers(4 formulas)
Euclid's Division Lemma
a = bq + r, where 0 ≤ r < b
a = dividend, b = divisor, q = quotient, r = remainder
Fundamental Theorem of Arithmetic
Every integer > 1 is either prime or a unique product of primes
HCF × LCM relationship
HCF(a,b) × LCM(a,b) = a × b
Irrational number test (log form)
log₂ 3, √2, √3, √5 are all irrational
Ch 2Polynomials(5 formulas)
Sum of zeros (quadratic)
α + β = −b/a
ax² + bx + c = 0
Product of zeros (quadratic)
αβ = c/a
Sum of zeros (cubic)
α + β + γ = −b/a
ax³ + bx² + cx + d = 0
Product of zeros (cubic)
αβγ = −d/a
Sum of products of pairs (cubic)
αβ + βγ + γα = c/a
Ch 4Quadratic Equations(4 formulas)
Quadratic formula
x = [−b ± √(b²−4ac)] / 2a
b²−4ac is the discriminant D
D > 0: two distinct real roots; D = 0: equal roots; D < 0: no real roots
Discriminant
D = b² − 4ac
Sum of roots
x₁ + x₂ = −b/a
Product of roots
x₁ × x₂ = c/a
Ch 5Arithmetic Progressions(4 formulas)
nth term of AP
aₙ = a + (n−1)d
a = first term, d = common difference, n = term number
Sum of n terms
Sₙ = n/2 × [2a + (n−1)d]
Also written as Sₙ = n/2 × (a + l) where l is the last term
Common difference
d = aₙ₊₁ − aₙ (any consecutive pair)
Number of terms
n = [(l − a)/d] + 1
l = last term
Ch 6Triangles(4 formulas)
Basic Proportionality Theorem (BPT)
If DE ∥ BC, then AD/DB = AE/EC
Pythagoras Theorem
AC² = AB² + BC²
In right triangle, AC = hypotenuse
Converse: if AC² = AB² + BC², then angle B = 90°
Area ratio of similar triangles
Area(△ABC)/Area(△PQR) = (AB/PQ)² = (BC/QR)² = (CA/RP)²
Median length (Apollonius)
AB² + AC² = 2(AD² + BD²)
D = midpoint of BC, AD = median
Ch 7Coordinate Geometry(4 formulas)
Distance formula
d = √[(x₂−x₁)² + (y₂−y₁)²]
Section formula (internal)
P = [(mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)]
P divides AB in ratio m:n
Midpoint formula
M = [(x₁+x₂)/2, (y₁+y₂)/2]
Area of triangle
Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
If area = 0, points are collinear
Ch 8Introduction to Trigonometry(7 formulas)
Basic ratios
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
Reciprocal ratios
cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
Pythagorean identity 1
sin²θ + cos²θ = 1
Pythagorean identity 2
1 + tan²θ = sec²θ
Pythagorean identity 3
1 + cot²θ = cosec²θ
Standard angle values
sin 30°=½, sin 45°=1/√2, sin 60°=√3/2, sin 90°=1
cos values: reverse order. tan 0°=0, 30°=1/√3, 45°=1, 60°=√3, 90°=undefined
Complementary angles
sin(90°−θ) = cosθ, cos(90°−θ) = sinθ, tan(90°−θ) = cotθ
Ch 10Circles(3 formulas)
Tangent length
PT = √(d² − r²)
P = external point, T = point of tangency, d = distance from P to centre, r = radius
Two tangents from an external point are equal in length
Angle in semicircle
Angle in a semicircle = 90°
Tangent-radius angle
Angle between tangent and radius at point of contact = 90°
Ch 12Surface Areas and Volumes(11 formulas)
Sphere: Volume
cm³V = (4/3)πr³
Sphere: Surface area
cm²SA = 4πr²
Hemisphere: Volume
V = (2/3)πr³
Hemisphere: Curved SA
CSA = 2πr²
Cylinder: Volume
V = πr²h
Cylinder: Curved SA
CSA = 2πrh
Cone: Volume
V = (1/3)πr²h
Cone: Slant height
l = √(r² + h²)
Cone: Curved SA
CSA = πrl
Frustum: Volume
V = (πh/3)(r₁² + r₂² + r₁r₂)
r₁, r₂ = radii of two ends, h = height
Frustum: Curved SA
CSA = π(r₁+r₂)l
l = slant height = √[h² + (r₁−r₂)²]
Ch 13Statistics(5 formulas)
Mean (direct method)
x̄ = Σfᵢxᵢ / Σfᵢ
Mean (assumed mean)
x̄ = a + (Σfᵢdᵢ / Σfᵢ)
dᵢ = xᵢ − a, a = assumed mean
Median
M = l + [(n/2 − cf)/f] × h
l = lower limit of median class, cf = cumulative frequency before median class, f = frequency of median class, h = class size
Mode
Mode = l + [(f₁−f₀)/(2f₁−f₀−f₂)] × h
f₁ = modal class frequency, f₀ = preceding class frequency, f₂ = succeeding class frequency
Empirical relationship
Mode = 3(Median) − 2(Mean)
Ch 14Probability(5 formulas)
Probability of event
P(E) = Number of favourable outcomes / Total outcomes
Complement rule
P(Ē) = 1 − P(E)
Impossible event
P(impossible event) = 0
Sure event
P(sure event) = 1
Range
0 ≤ P(E) ≤ 1