The 65 formulas CDS Maths actually tests

The three identities

sin²θ + cos²θ = 1 sec²θ − tan²θ = 1 cosec²θ − cot²θ = 1

θ = any angle

The reciprocal pairs

(sec θ + tan θ)(sec θ − tan θ) = 1 (cosec θ + cot θ)(cosec θ − cot θ) = 1

from the identities above

Note:If sec θ + tan θ = p, then sec θ − tan θ = 1/p.

Complementary angles

sin(90° − θ) = cos θ tan(90° − θ) = cot θ sec(90° − θ) = cosec θ

θ = acute angle

Note:tan 1° · tan 2° · … · tan 89° = 1, because each tan θ pairs with tan(90° − θ) = cot θ.

Greatest and least values

−√(a² + b²) ≤ a sin θ + b cos θ ≤ √(a² + b²) x + 1/x ≥ 2 for x > 0

a, b = constants

Note:sin θ and cos θ never exceed 1 in size, so values like sin θ = 1.2 are impossible.

HCF and LCM of two numbers

HCF × LCM = a × b

a, b = two positive integers

Note:True for TWO numbers only.

Number of divisors

N = pᵃ qᵇ rᶜ ⇒ divisors = (a + 1)(b + 1)(c + 1)

p, q, r = distinct primes

Trailing zeros of n!

⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …

⌊ ⌋ = whole-number part

Unit digits of powers

unit digit of aⁿ repeats every 4 powers; use n mod 4 (0 → 4)

a = base, n = exponent

Heron's formula

Area = √(s(s − a)(s − b)(s − c)), s = (a + b + c)/2

a, b, c = sides

Equilateral triangle

Area = (√3/4) a² height = (√3/2) a inradius = a/(2√3) circumradius = a/√3

a = side

Sector and arc

Arc = (θ/360) × 2πr Sector area = (θ/360) × πr²

θ = angle at centre in degrees

Note:A sector's perimeter adds two radii to the arc.

Inradius of any triangle

r = Area / s

s = semi-perimeter

Cuboid

V = lbh TSA = 2(lb + bh + hl) diagonal = √(l² + b² + h²)

l, b, h = length, breadth, height

Cylinder and cone

Cylinder: V = πr²h, CSA = 2πrh Cone: V = ⅓πr²h, CSA = πrl, l = √(r² + h²)

r = radius, h = height, l = slant height

Sphere and hemisphere

Sphere: V = ⁴⁄₃πr³, SA = 4πr² Solid hemisphere: V = ⅔πr³, TSA = 3πr²

r = radius

Frustum

V = ⅓πh(R² + r² + Rr) CSA = πl(R + r)

R, r = end radii, h = height, l = slant height

Scaling

lengths × k ⇒ areas × k², volumes × k³

k = scale factor

Classify by the largest side

c² = a² + b²: right c² < a² + b²: acute c² > a² + b²: obtuse

c = largest side

Altitude to the hypotenuse

h² = pq h = ab/c

p, q = parts of the hypotenuse; a, b = legs; c = hypotenuse

Apollonius (median)

AB² + AC² = 2(AD² + BD²)

D = midpoint of BC

Note:The centroid divides each median 2 : 1 from the vertex.

Similar triangles

area ratio = (side ratio)²

Mean under a change

x → x + c: mean + c x → kx: mean × k Σ(x − x̄) = 0

c, k = constants

Median of grouped data

Median = l + ((n/2 − cf) / f) × h

l = lower limit of median class

cf = cumulative frequency before it

f = its frequency, h = class width

Mean, median and mode

Mode ≈ 3 Median − 2 Mean

for a moderately skewed distribution

Componendo-dividendo

a/b = c/d ⇒ (a + b)/(a − b) = (c + d)/(c − d)

Variation

x ∝ y: x = ky x ∝ 1/y: xy = k

k = constant

Alligation

cheaper : dearer = (d − m) : (m − c)

c, d = prices of the two; m = mean price

Average speed over equal distances

2uv / (u + v)

u, v = the two speeds

Note:Not (u + v)/2.

Relative speed

towards each other: u + v same direction: u − v

Boats and streams

boat = (down + up)/2 stream = (down − up)/2

Clock hands

angle = |30H − 5.5M|

H = hour, M = minutes

Two successive changes

net % = a + b + ab/100

a, b = the changes (negative for a fall)

Reversing a percentage

x is r% more than y ⇒ y is (r/(100 + r)) × 100% less than x

Profit and discount

SP = CP(1 + p/100) SP = MP(1 − d/100)

p = profit %, d = discount %

Pie-chart share

share = angle / 360° value = share × total

Percentage change

(new − old) / old × 100

Mean and total

total = mean × count

Combined mean

(n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)

n = group sizes, x̄ = group means

Two workers together

time = ab / (a + b)

a, b = days each takes alone

Man-days

M₁D₁H₁ / W₁ = M₂D₂H₂ / W₂

M = men, D = days, H = hours, W = work

Simple interest

SI = PRT / 100

P = principal, R = rate %, T = years

Compound amount

A = P(1 + R/100)ⁿ

n = years (half-yearly: R/2 and 2n)

CI − SI for 2 years

P(R/100)²

Reciprocal sums

x + 1/x = k ⇒ x² + 1/x² = k² − 2, x³ + 1/x³ = k³ − 3k

The cube identity

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

Note:If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Square of a trinomial

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Sum and product of roots

α + β = −b/a αβ = c/a

ax² + bx + c = 0

Nature of roots

D = b² − 4ac: D > 0 real and distinct, D = 0 equal, D < 0 not real

Forming an equation

x² − (α + β)x + αβ = 0

Laws of indices

aᵐ · aⁿ = aᵐ⁺ⁿ (aᵐ)ⁿ = aᵐⁿ a⁻ⁿ = 1/aⁿ a⁰ = 1

Rationalising

1/(√a + √b) = (√a − √b)/(a − b)

Square root of a surd

√(a + 2√b) = √x + √y where x + y = a, xy = b

Remainder and factor theorems

remainder of p(x) ÷ (x − a) = p(a); (x − a) is a factor ⇔ p(a) = 0

Sum and difference of cubes

a³ ± b³ = (a ± b)(a² ∓ ab + b²)

Chord and distance

r² = d² + (c/2)²

d = distance of chord from centre, c = chord

Tangent length

PT = √(OP² − r²)

O = centre, P = external point

Power of a point

PA × PB = PC × PD = PT²

chords or secants through P; PT = tangent

Common tangents

direct: √(d² − (r₁ − r₂)²) transverse: √(d² − (r₁ + r₂)²)

d = distance between centres

Rhombus

area = ½ d₁d₂ side² = (d₁/2)² + (d₂/2)²

d₁, d₂ = diagonals

Trapezium

area = ½(a + b)h

a, b = parallel sides

One line of sight

height = distance × tan θ

θ = angle of elevation

Complementary elevations

height = √(ab)

a, b = distances with elevations α and 90° − α

Laws of logarithms

log ab = log a + log b log aⁿ = n log a log_b a = log a / log b

Number of digits

digits of N = ⌊log₁₀ N⌋ + 1

Two equations in two unknowns

a₁/a₂ ≠ b₁/b₂: one solution a₁/a₂ = b₁/b₂ ≠ c₁/c₂: none all equal: infinitely many