The 191 formulas JEE Mains Maths actually tests

Circle in general form

x² + y² + 2gx + 2fy + c = 0: centre (−g, −f), r = √(g² + f² − c)

g, f, c = coefficients of the equation

Note:x-intercept = 2√(g² − c), y-intercept = 2√(f² − c).

Chords and tangent length

chord = 2√(r² − d²) tangent length from P = √S₁ chord with midpoint P: T = S₁

d = distance from the centre to the chord

S₁ = the circle's expression evaluated at P(x₁, y₁)

T = xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c

Two circles

|r₁ − r₂| < C₁C₂ < r₁ + r₂ ⇔ they cut in two points common chord: S₁ − S₂ = 0

C₁C₂ = distance between the centres

S₁, S₂ = the two circle equations with RHS 0

Note:Circles through the meet of a circle S and a line L: S + λL = 0.

Standard parabola

y² = 4ax: focus (a, 0), directrix x = −a, LR = 4a, point (at², 2at)

a = focal length

t = parameter

Note:Focal chord: t₁t₂ = −1, and its length at angle θ to the axis is 4a/sin²θ.

Parabola tangents and normals

tangent y = mx + a/m touches at (a/m², 2a/m) tangents at t₁, t₂ meet at (at₁t₂, a(t₁ + t₂)) normal y = mx − 2am − am³

m = slope

t₁, t₂ = parameters of the points of contact

Ellipse

x²/a² + y²/b² = 1 (a > b): b² = a²(1 − e²), foci (±ae, 0), LR = 2b²/a, SP + S′P = 2a

e = eccentricity (0 < e < 1)

S, S′ = foci

P = a point on the ellipse

Note:SP = a − ex₁, S′P = a + ex₁. Point (a cos θ, b sin θ); area πab. If b > a, swap the roles: a² = b²(1 − e²).

Hyperbola

x²/a² − y²/b² = 1: b² = a²(e² − 1), foci (±ae, 0), LR = 2b²/a, |S′P − SP| = 2a

e = eccentricity (e > 1)

Note:Rectangular hyperbola xy = c²: point (ct, c/t), e = √2.

When y = mx + c is a tangent

circle x² + y² = a²: c² = a²(1 + m²) ellipse: c² = a²m² + b² hyperbola: c² = a²m² − b²

m = slope, c = y-intercept of the line

Note:Perpendicular tangents to the ellipse meet on the director circle x² + y² = a² + b².

Direction cosines

(l, m, n) = (a, b, c) / √(a² + b² + c²) l² + m² + n² = 1

a, b, c = direction ratios

Angle between two lines

cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / (√(a₁² + b₁² + c₁²) · √(a₂² + b₂² + c₂²))

(a₁, b₁, c₁), (a₂, b₂, c₂) = direction ratios

Note:Perpendicular lines: a₁a₂ + b₁b₂ + c₁c₂ = 0.

Plane through a point

a(x − x₁) + b(y − y₁) + c(z − z₁) = 0 normal from two in-plane directions: n = u × v

(a, b, c) = normal

u, v = directions lying in the plane

Note:Planes through the line of intersection of P₁ = 0 and P₂ = 0: P₁ + λP₂ = 0.

Distance from a plane

distance = |ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²)

plane: ax + by + cz + d = 0

(x₁, y₁, z₁) = the point

Foot and image in a plane

k = (ax₁ + by₁ + cz₁ + d) / (a² + b² + c²): foot = P − k(a, b, c), image = P − 2k(a, b, c)

P = (x₁, y₁, z₁)

plane: ax + by + cz + d = 0

Angles with planes

line and plane: sin θ = |d · n| / (|d| |n|) two planes: cos θ = |n₁ · n₂| / (|n₁| |n₂|)

d = direction of the line

n, n₁, n₂ = normals

Foot of a perpendicular on a line

t = (P − A) · d / |d|², foot M = A + t d, image = 2M − P distance = |AP × d| / |d|

line through A with direction d

P = the external point

Shortest distance

skew: SD = |(a₂ − a₁) · (d₁ × d₂)| / |d₁ × d₂| parallel: d = |(a₂ − a₁) × d| / |d|

a₁, a₂ = points on the two lines

d₁, d₂ = their directions

Note:SD = 0 means the lines are coplanar: (a₂ − a₁) · (d₁ × d₂) = 0.

Counting functions

all maps: nᵐ one-one: n!/(n − m)! (m ≤ n) onto: nᵐ − C(n, 1)(n − 1)ᵐ + C(n, 2)(n − 2)ᵐ − …

m = size of the domain

n = size of the codomain

Counting relations on an n-element set

all: 2^(n²) reflexive: 2^(n² − n) symmetric: 2^(n(n + 1)/2)

n = number of elements

Note:The smallest equivalence relation with classes of sizes k₁, k₂, … has Σ kᵢ² pairs.

Domain rules

sin⁻¹u, cos⁻¹u: −1 ≤ u ≤ 1 log u: u > 0 √u: u ≥ 0 logₐ(log_b u) defined ⇔ u > 1 (a, b > 1)

u = the inner expression

Note:Domain of f∘g: the x in the domain of g with g(x) in the domain of f.

Composition and cycles

(f∘g)(x) = f(g(x)) fᵏ = identity ⇒ fⁿ = f^(n mod k)

fᵏ = f composed with itself k times

Note:f(x) = (ax + b)/(cx + d) with a + d = 0 is its own inverse: f∘f = identity.

The two tests

one-one: f(a) = f(b) ⇒ a = b onto: range of f = codomain

f : A → B

Note:For y = p(x)/q(x), solve for x as a quadratic and require its discriminant ≥ 0 to get the range.

Functional equations

f(x + y) = f(x) + f(y) ⇒ f(n) = n f(1) f(x) = bˣ/(bˣ + √b) ⇒ f(x) + f(1 − x) = 1

b > 0

n = positive integer

Note:af(x) + bf(g(x)) = h(x): write the equation again with x replaced by g(x) and solve the pair.

Three sets

n(A ∪ B ∪ C) = Σ n(A) − Σ n(A ∩ B) + n(A ∩ B ∩ C)

sums run over the three sets and the three pairs

Note:A − B = A ∩ B′.

Arithmetic progression

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

a = first term, d = common difference

l = last term

Note:Terms equidistant from the ends add equally: aₖ + aₙ₊₁₋ₖ = a₁ + aₙ.

Geometric progression

aₙ = arⁿ⁻¹ Sₙ = a(rⁿ − 1)/(r − 1) S∞ = a/(1 − r), |r| < 1

a = first term, r = common ratio

The three means

a, b, c in AP: 2b = a + c GP: b² = ac HP: b = 2ac/(a + c) A ≥ G ≥ H, G² = AH

A, G, H = arithmetic, geometric, harmonic means of two positive numbers

AM ≥ GM

(x₁ + x₂ + … + xₙ)/n ≥ (x₁x₂…xₙ)^(1/n)

xᵢ > 0

Note:Equality exactly when all the xᵢ are equal.

Sums of powers

Σk = n(n + 1)/2 Σk² = n(n + 1)(2n + 1)/6 Σk³ = [n(n + 1)/2]²

k runs from 1 to n

Arithmetico-geometric series

S∞ = a/(1 − r) + dr/(1 − r)², |r| < 1

terms a, (a + d)r, (a + 2d)r², …

Note:Standard route for a finite sum: write S, subtract rS, sum the GP that appears.

Telescoping

Σ [f(k) − f(k + 1)] = f(1) − f(n + 1) k · k! = (k + 1)! − k!

k runs from 1 to n

Note:k/(k⁴ + k² + 1) = ½[1/(k² − k + 1) − 1/(k² + k + 1)].

Terms common to two APs

common difference of the shared terms = lcm(d₁, d₂)

d₁, d₂ = the two common differences

Reflection property

∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx

a, b = limits

Note:Add the two forms; the troublesome part usually cancels to a constant.

Symmetric limits

∫₋ₐᵃ f(x) dx = 0 if f is odd, 2∫₀ᵃ f(x) dx if f is even

odd: f(−x) = −f(x); even: f(−x) = f(x)

Note:Split any integrand into odd and even parts first.

The 1 + bᵍ halving

∫₋ₐᵃ h(x)/(1 + b^g(x)) dx = ∫₀ᵃ h(x) dx

h = even function

g = odd function, b > 0

Many periods

∫₀^(nT) f(x) dx = n ∫₀ᵀ f(x) dx

T = period of f

n = positive integer

Note:The same idea handles {x}: ∫₀ⁿ g({x}) dx = n ∫₀¹ g(x) dx.

Leibniz rule

d/dx ∫ from u(x) to v(x) of f(t) dt = f(v(x)) v′(x) − f(u(x)) u′(x)

u, v = variable limits

Note:f(x) = g(x) + ∫ₐᵇ k(t) f(t) dt forces f(x) = g(x) + A with A a constant to be found.

Reduction for sine powers

∫₀^(π/2) sinⁿx dx = ((n − 1)/n) ∫₀^(π/2) sinⁿ⁻²x dx

n ≥ 2

Note:The same holds for cosⁿx on [0, π/2].

Limit of a sum

lim (n → ∞) (1/n) Σ f(k/n) = ∫₀¹ f(x) dx

k runs from 1 to n

Note:Replace k/n by x and 1/n by dx.

Bounds on an integral

m(b − a) ≤ ∫ₐᵇ f(x) dx ≤ M(b − a)

m, M = least and greatest values of f on [a, b]

Arrangements and selections

ⁿPᵣ = n!/(n − r)! C(n, r) = n!/(r!(n − r)!)

n = objects, r = chosen

Note:One-one maps from an m-set to an n-set: ⁿPₘ; strictly increasing ones: C(n, m).

Arrangements with repeats

n!/(p! q! r! …)

p, q, r = counts of identical items

No two together

n! × ⁿ⁺¹Pₖ

n = other items arranged first

k = items placed in the n + 1 gaps

Stars and bars

x₁ + … + xₖ = n, xᵢ ≥ 0: C(n + k − 1, k − 1) xᵢ ≥ 1: C(n − 1, k − 1)

n = identical items, k = distinct boxes

Power of a prime in n!

⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + …

p = prime

⌊ ⌋ = greatest integer

Points and polygons

triangles: C(n, 3) − C(k, 3) diagonals: n(n − 3)/2 triangles with no side of the polygon: n(n − 4)(n − 5)/6

n = points or polygon vertices

k = points lying on one line

Sum of all numbers formed

(number of arrangements ÷ n) × (digit sum) × 11…1 (n ones)

n = number of digit places, all digits used

Note:If 0 is among the digits, subtract the numbers that start with 0.

Rank in the dictionary

rank = 1 + Σ cᵢ (n − i)!

cᵢ = unused letters smaller than the i-th letter

n = length of the word (distinct letters)

Dot product and projection

a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃ projection of a on b = (a · b)/|b|

θ = angle between a and b

Note:a ⊥ b ⇔ a · b = 0.

Square of a sum

|a + b|² = |a|² + 2 a · b + |b|²

a, b = vectors

Note:Square first, then use a · a = |a|².

Cross product and area

|a × b| = |a||b| sin θ triangle area = ½ |AB × AC|

a × b is perpendicular to both a and b

Note:A vector perpendicular to a and b is λ(a × b).

Lagrange's identity

|a × b|² + (a · b)² = |a|² |b|²

a, b = vectors

Scalar triple product

[a b c] = a · (b × c) [a b c] = 0 ⇔ a, b, c coplanar

a, b, c = vectors

Note:Tetrahedron volume = ⅙ |[AB AC AD]|.

Vector triple product

a × (b × c) = (a · c) b − (a · b) c

a, b, c = vectors

Section formula and bisector

p = (n a + m b)/(m + n) bisector direction = a/|a| + b/|b|

p divides AB in the ratio m : n internally

a, b = position vectors

Solving r × a = b × a

r × a = b × a ⇒ r = b + λa

λ = scalar

Note:a × c = b with a · c known: c = ((a · c) a − a × b)/|a|².

Order and degree

order = number of arbitrary constants eliminated degree = power of the highest derivative after clearing radicals

degree is defined only when the equation is a polynomial in the derivatives

Variables separable

dy/dx = f(x) g(y) ⇒ ∫ dy/g(y) = ∫ f(x) dx + C

C = constant

Note:For dy/dx = F(ax + by + c), put t = ax + by + c.

Homogeneous equation

y = vx: v + x dv/dx = F(v) ⇒ ∫ dv/(F(v) − v) = ln|x| + C

dy/dx = F(y/x)

Note:With constant terms, shift the origin: x = X + h, y = Y + k.

Linear equation

dy/dx + Py = Q ⇒ y · e^(∫P dx) = ∫ Q e^(∫P dx) dx + C

P, Q = functions of x

e^(∫P dx) = integrating factor

Note:If x is the easier dependent variable, use dx/dy + P(y) x = Q(y).

Quick integrating factors

P = k tan x ⇒ IF = secᵏx P = g′(x)/g(x) ⇒ IF = g(x) P = 1/(1 + x²) ⇒ IF = e^(tan⁻¹x)

IF = integrating factor

Bernoulli equation

y′ + Py = Qyⁿ, z = y^(1 − n) ⇒ z′ + (1 − n)P z = (1 − n)Q

n ≠ 0, 1

Newton's law of cooling

dT/dt = −k(T − A) ⇒ T − A = (T₀ − A) e^(−kt)

A = surrounding temperature

T₀ = temperature at t = 0

Intercepts of the tangent

x-intercept = x − y/y′ y-intercept = y − x y′

(x, y) = point of contact, y′ = dy/dx there

Classical probability

P(E) = n(E)/n(S) P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

n(S) = equally likely outcomes

Note:k items fall in one given relative order with probability 1/k!.

Conditional probability and independence

P(A | B) = P(A ∩ B)/P(B) independent: P(A ∩ B) = P(A) P(B)

P(B) > 0

Total probability

P(E) = Σ P(Hᵢ) P(E | Hᵢ)

Hᵢ = mutually exclusive cases covering every outcome

Bayes' theorem

P(Hₖ | E) = P(Hₖ) P(E | Hₖ) / Σ P(Hᵢ) P(E | Hᵢ)

Hₖ = the case asked about

Binomial distribution

P(X = r) = C(n, r) pʳ qⁿ⁻ʳ mean = np variance = npq

n = trials, p = success chance, q = 1 − p

Mean and variance of X

E(X) = Σ xᵢ pᵢ Var(X) = E(X²) − (E(X))²

pᵢ = P(X = xᵢ), Σ pᵢ = 1

Alternate turns

P(A wins) = p_A / (1 − q_A q_B)

A goes first

p = chance of success on a turn, q = 1 − p

Sum of two dice

P(sum = k) = (6 − |k − 7|)/36, k = 2, …, 12

two fair dice

General term

Tᵣ₊₁ = C(n, r) aⁿ⁻ʳ bʳ

expansion of (a + b)ⁿ

r = 0, 1, …, n

Note:C(n, r) = C(n, n − r).

Ratio of neighbouring coefficients

C(n, r) / C(n, r − 1) = (n − r + 1)/r

r ≥ 1

Note:Use it for consecutive coefficients in a given ratio and for the greatest term.

Sums of coefficients

sum of coefficients = f(1) even-index sum = (f(1) + f(−1))/2 odd-index sum = (f(1) − f(−1))/2

f(x) = the expanded polynomial

Note:Σ C(n, r) = 2ⁿ.

Weighted sums

Σ r C(n, r) = n 2^(n − 1) Σ r² C(n, r) = n(n + 1) 2^(n − 2)

r runs from 0 to n

Note:Σ C(n, r)/(r + 1) = (2^(n + 1) − 1)/(n + 1).

Vandermonde's identity

Σ C(m, r) C(n, k − r) = C(m + n, k)

r runs over all valid values

Note:Special case: Σ C(n, r)² = C(2n, n).

Hockey stick

Σ C(k, r) = C(n + 1, r + 1)

k runs from r to n

Rational terms

C(n, r) p^((n − r)/a) q^(r/b) is rational ⇔ a divides n − r and b divides r

p, q = integers that are not perfect a-th, b-th powers

Remainders

(mq ± 1)ⁿ = (a multiple of m) + (±1)ⁿ

write the base as a multiple of the divisor plus or minus 1

Note:(1 + m)ⁿ − mn − 1 is divisible by m².

Area between curves

A = ∫ₐᵇ (upper − lower) dx

a, b = x-coordinates where the curves meet

Note:If the upper curve changes, split at the switch point.

Horizontal strips

A = ∫ from c to d of (x_right − x_left) dy

c, d = y-limits

Note:Use when the curves are x = g(y), as with sideways parabolas.

Parabola against a chord

∫ from α to β of a(x − α)(β − x) dx = a(β − α)³/6

α, β = x-coordinates where the parabola meets the line

Note:y = ax² and y = mx enclose m³/(6a²).

Under a circular arc

∫ √(r² − x²) dx = (x/2)√(r² − x²) + (r²/2) sin⁻¹(x/r) + C

r = radius

Segment and ellipse

minor segment = ½ r²(θ − sin θ) ellipse area = πab

θ = angle at the centre in radians

a, b = semi-axes

Under a square-root curve

∫₀ᶜ √(kx) dx = (2/3) c √(kc)

k > 0

Logs and exponentials

∫ ln x dx = x ln x − x + C ∫ aˣ dx = aˣ/ln a + C ∫ₐᵇ (k/x) dx = k ln(b/a)

a, b > 0

Modulus and min/max curves

∫ |f(x)| dx = Σ |∫ f(x) dx| over the pieces between the zeros of f

split at each zero of f

Note:For min{f, g} or max{f, g}, integrate whichever curve applies on each piece.

Conjugate identities

z z̄ = |z|² z + z̄ = 2 Re(z) z − z̄ = 2i Im(z) 1/z = z̄/|z|²

z̄ = conjugate of z

Note:|a ± b|² = |a|² + |b|² ± 2 Re(a b̄).

Polar form

z = r(cos θ + i sin θ) = r e^(iθ) |z₁z₂| = |z₁||z₂| arg(z₁z₂) = arg z₁ + arg z₂

r = |z|, θ = arg z

De Moivre's theorem

(r e^(iθ))ⁿ = rⁿ e^(inθ)

n = integer

Cube roots of unity

ω³ = 1 1 + ω + ω² = 0 ω = (−1 + i√3)/2

ω = a non-real cube root of 1

Note:Reduce powers of ω by the exponent mod 3.

n-th roots of unity

zⁿ = 1 ⇒ z = e^(2πik/n), k = 0, 1, …, n − 1

n ≥ 2

Note:Their sum is 0; they are vertices of a regular n-gon on the unit circle.

Rotation

(z₃ − z₁)/(z₂ − z₁) = (|z₃ − z₁| / |z₂ − z₁|) e^(iθ)

θ = angle at z₁ from z₂ to z₃

Circle and distance range

|z − a| = r ⇒ ||c − a| − r| ≤ |z − c| ≤ |c − a| + r

a = centre, r = radius

c = a fixed point

Note:z z̄ + ᾱz + αz̄ + d = 0 is a circle with centre −α and radius √(|α|² − d).

Arc from an argument

arg((z − a)/(z − b)) = θ ⇒ arc of a circle with radius |a − b| / (2 sin θ)

a, b = end points of the chord

Note:|z − z₁| + |z − z₂| = 2a > |z₁ − z₂| is an ellipse with foci z₁, z₂.

Roots and discriminant

x = (−b ± √D)/(2a), D = b² − 4ac

ax² + bx + c = 0, a ≠ 0

Note:D > 0: real, distinct. D = 0: equal. D < 0: complex. Rational coefficients and D a perfect square: rational roots.

Sum and product

α + β = −b/a αβ = c/a (α − β)² = (b² − 4ac)/a²

α, β = roots

Note:The equation with roots α, β: x² − (α + β)x + αβ = 0.

Cubic

α + β + γ = −b/a αβ + βγ + γα = c/a αβγ = −d/a

ax³ + bx² + cx + d = 0

Power sums of the roots

α² + β² = (α + β)² − 2αβ Pₙ = s Pₙ₋₁ − p Pₙ₋₂

Pₙ = αⁿ + βⁿ

s = α + β, p = αβ

A number between the roots

α < k < β ⇔ a f(k) < 0

f(x) = ax² + bx + c

One common root

(c₁a₂ − c₂a₁)² = (a₁b₂ − a₂b₁)(b₁c₂ − b₂c₁)

a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0

Reciprocal substitution

x² + 1/x² = (x + 1/x)² − 2

put t = x + 1/x

Note:logₐ b = 1/log_b a turns swapped-base equations into t + 1/t = k.

Modulus and greatest integer

|A + B| = |A| + |B| ⇔ AB ≥ 0 [x] = n ⇔ n ≤ x < n + 1

[x] = greatest integer ≤ x

n = integer

Standard limits (x → 0)

sin x/x → 1 tan x/x → 1 (1 − cos x)/x² → ½ (eˣ − 1)/x → 1 ln(1 + x)/x → 1

x in radians

Note:(aˣ − 1)/x → ln a.

1 to the power ∞

lim f(x)^g(x) = e^(lim g(x)(f(x) − 1))

f → 1, g → ∞

Note:lim (n → ∞) (1 + a/n)^(bn) = e^(ab).

Series expansions

sin x = x − x³/6 + … cos x = 1 − x²/2 + x⁴/24 − … eˣ = 1 + x + x²/2 + x³/6 + … ln(1 + x) = x − x²/2 + x³/3 − …

valid near x = 0

Note:For f(x)/xⁿ to have a finite limit, every coefficient below xⁿ must vanish.

Limits at infinity

lim (x → ∞) (aₙxⁿ + …)/(bₙxⁿ + …) = aₙ/bₙ

same degree n top and bottom

Note:Higher degree on top: ±∞. Higher degree below: 0.

Rationalising

√p − √q = (p − q)/(√p + √q)

p, q ≥ 0

Continuity at a point

lim (x → a⁻) f(x) = lim (x → a⁺) f(x) = f(a)

a = the point tested

Note:Intermediate value theorem: f continuous and f(a) f(b) < 0 ⇒ a root in (a, b).

Greatest integer bounds

t − 1 < [t] ≤ t

[t] = greatest integer ≤ t

Note:[g(x)] can break only where g(x) is an integer.

Powers in the limit

lim (n → ∞) x^(2n) = 0 if |x| < 1, 1 if |x| = 1, ∞ if |x| > 1

n = positive integer

Angle between two lines

tan θ = |(m₁ − m₂)/(1 + m₁m₂)|

m₁, m₂ = slopes

Note:Parallel: m₁ = m₂. Perpendicular: m₁m₂ = −1.

Forms of a line

intercept: x/a + y/b = 1 normal: x cos α + y sin α = p parametric: (x − x₁)/cos θ = (y − y₁)/sin θ = r

p = perpendicular distance from the origin

r = distance from (x₁, y₁)

Note:Lines through the meet of L₁ = 0 and L₂ = 0: L₁ + λL₂ = 0.

Distances

point to line: |ax₁ + by₁ + c|/√(a² + b²) parallel lines: |c₁ − c₂|/√(a² + b²)

line: ax + by + c = 0

Note:Two points are on the same side when ax + by + c has the same sign at both.

Area of a triangle

Δ = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

(xᵢ, yᵢ) = vertices

Note:Δ = 0 means the points are collinear.

Foot and image in a line

foot: (x − x₁)/a = (y − y₁)/b = −(ax₁ + by₁ + c)/(a² + b²) image: same with −2(ax₁ + by₁ + c)/(a² + b²)

line: ax + by + c = 0

(x₁, y₁) = the point

Angle bisectors

(a₁x + b₁y + c₁)/√(a₁² + b₁²) = ±(a₂x + b₂y + c₂)/√(a₂² + b₂²)

the two given lines

Centres of a triangle

centroid: ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3) incentre: (aA + bB + cC)/(a + b + c)

a, b, c = side lengths opposite A, B, C

Note:Circumcentre: equidistant from the vertices. Orthocentre: where two altitudes meet.

Pair of lines through the origin

ax² + 2hxy + by² = 0: tan θ = 2√(h² − ab)/|a + b|

θ = angle between the two lines

Note:Perpendicular when a + b = 0.

Product and transpose rules

(AB)ᵀ = BᵀAᵀ (AB)⁻¹ = B⁻¹A⁻¹ (A + B)² = A² + AB + BA + B²

A, B = square matrices of the same order

Note:AB ≠ BA in general, so (A + B)² ≠ A² + 2AB + B².

Determinant rules

|AB| = |A||B| |Aᵀ| = |A| |kA| = kⁿ|A|

n = order of A

Adjoint identities

A(adj A) = |A| I |adj A| = |A|^(n − 1) adj(adj A) = |A|^(n − 2) A adj(kA) = k^(n − 1) adj A

n = order of A

Note:|adj(adj A)| = |A|^((n − 1)²).

Inverse

A⁻¹ = adj A / |A| (|A| ≠ 0) |A⁻¹| = 1/|A|

A = square matrix

Note:PQ = kI ⇒ Q = kP⁻¹ and |Q| = kⁿ/|P|.

Cayley–Hamilton for 2 × 2

A² − (tr A) A + (det A) I = O A⁻¹ = ((tr A) I − A)/det A

tr A = sum of the diagonal

Note:A² = pA + qI ⇒ A³ = (p² + q)A + pqI.

Powers that collapse

A² = A ⇒ (I + A)ⁿ = I + (2ⁿ − 1)A A = I + N, N² = O ⇒ Aⁿ = I + nN

N = nilpotent part

Note:A^d = I ⇒ Aⁿ = A^(n mod d).

Symmetric and skew-symmetric

A = ½(A + Aᵀ) + ½(A − Aᵀ) symmetric count: k^(n(n + 1)/2) skew count: k^(n(n − 1)/2)

first part symmetric, second skew-symmetric

k = allowed entry values (skew needs 0 among them)

Note:A skew-symmetric matrix of odd order has determinant 0.

Orthogonal matrix

AAᵀ = I ⇒ A⁻¹ = Aᵀ, |A| = ±1

A = square matrix

Row operations

Rᵢ → Rᵢ − kRⱼ: Δ unchanged Rᵢ ↔ Rⱼ: sign changes common factor of a row comes out

the same rules hold for columns

Note:Two equal or proportional rows give Δ = 0.

Vandermonde determinant

|1 a a²; 1 b b²; 1 c c²| = (a − b)(b − c)(c − a)

rows separated by semicolons

A symmetric determinant

|k 1 1; 1 k 1; 1 1 k| = (k + 2)(k − 1)²

rows separated by semicolons

Cramer's rule

x = Δₓ/Δ y = Δ_y/Δ z = Δ_z/Δ (Δ ≠ 0)

Δₓ = Δ with the x-column replaced by the constants

Classifying a system

Δ ≠ 0: unique Δ = 0 and some Δₓ, Δ_y, Δ_z ≠ 0: no solution Δ = Δₓ = Δ_y = Δ_z = 0: infinitely many or none

three equations in three unknowns

Note:In the last case, test whether one equation is a combination of the other two.

Homogeneous system

AX = O has a non-zero solution ⇔ |A| = 0

A = coefficient matrix

Product and adjoint

|AB| = |A||B| |kA| = kⁿ|A| |adj A| = |A|^(n − 1)

n = order

Derivative of a determinant

d/dx Δ = sum of the determinants formed by differentiating one row at a time

other rows stay unchanged in each term

Variance from sums

σ² = Σx²/n − (Σx/n)²

n = number of observations

Frequency table

x̄ = Σfx/Σf σ² = Σfx²/Σf − x̄²

f = frequency of each value x

Linear change of data

y = ax + b ⇒ ȳ = a x̄ + b, σ_y² = a² σₓ²

a, b = constants

Note:Adding a constant leaves the variance unchanged.

Totals from mean and variance

Σx = n x̄ Σx² = n(σ² + x̄²)

x̄ = mean, σ² = variance

Correcting a wrong entry

Σx → Σx − w + c Σx² → Σx² − w² + c²

w = wrong value, c = correct value

Note:Fix both totals, then recompute the mean and variance.

Combined variance

σ² = (n₁σ₁² + n₂σ₂²)/(n₁ + n₂) + n₁n₂(x̄₁ − x̄₂)²/(n₁ + n₂)²

nᵢ, x̄ᵢ, σᵢ² = size, mean, variance of each group

Mean deviation

MD about a = (1/n) Σ |xᵢ − a|

a = mean or median

Note:Mean deviation is least about the median.

Two unknown values

(a − b)² = 2(a² + b²) − (a + b)²

a + b and a² + b² come from the totals

Tangent and normal

tangent: y − y₀ = f′(x₀)(x − x₀) normal slope = −1/f′(x₀)

(x₀, y₀) = point on the curve

Related rates

dV/dt = (dV/dr) · (dr/dt)

V, r = linked quantities

Increasing and decreasing

f′(x) > 0 on I ⇒ f increasing on I f′(x) < 0 on I ⇒ f decreasing on I

I = interval

Note:ax² + bx + c ≥ 0 for all x when a > 0 and b² − 4ac ≤ 0.

Local maxima and minima

f′(c) = 0 and f″(c) < 0: maximum f′(c) = 0 and f″(c) > 0: minimum

c = critical point: f′(c) = 0 or f′(c) does not exist

Note:For a cubic with positive leading coefficient, the maximum comes before the minimum.

Greatest and least on [a, b]

max f = greatest of f(a), f(b), f(c) over the critical points c

f continuous on [a, b]

Mean value and Rolle's theorems

f′(c) = (f(b) − f(a))/(b − a), c ∈ (a, b) Rolle: f(a) = f(b) ⇒ f′(c) = 0

f continuous on [a, b], differentiable on (a, b)

Note:n zeros of f ⇒ at least n − 1 zeros of f′.

Roots of a cubic

f(x) = k has three distinct real roots when local min < k < local max

f = cubic with two turning points

Note:f′ > 0 everywhere ⇒ f(x) = 0 has at most one root.

AM ≥ GM shortcut

a + b ≥ 2√(ab), equality when a = b

a, b > 0

Pythagorean identities

sin²x + cos²x = 1 1 + tan²x = sec²x 1 + cot²x = cosec²x

x = any angle where defined

Compound angles

sin(A ± B) = sin A cos B ± cos A sin B cos(A ± B) = cos A cos B ∓ sin A sin B

A, B = angles

Tangent of a sum

tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)

A, B = angles

Double and triple angles

sin 2θ = 2 sin θ cos θ cos 2θ = 1 − 2sin²θ = 2cos²θ − 1 sin 3θ = 3 sin θ − 4sin³θ cos 3θ = 4cos³θ − 3 cos θ

θ = angle

Note:sin 18° = (√5 − 1)/4, cos 36° = (√5 + 1)/4.

Sum to product

sin C + sin D = 2 sin((C + D)/2) cos((C − D)/2) cos C + cos D = 2 cos((C + D)/2) cos((C − D)/2)

C, D = angles

Note:cos C − cos D = −2 sin((C + D)/2) sin((C − D)/2).

Product chains

cos θ cos 2θ cos 4θ … cos 2^(n − 1)θ = sin 2ⁿθ / (2ⁿ sin θ) sin θ sin(60° − θ) sin(60° + θ) = ¼ sin 3θ

n = number of cosine factors

Note:Likewise cos θ cos(60° − θ) cos(60° + θ) = ¼ cos 3θ.

Fourth and sixth powers

sin⁴θ + cos⁴θ = 1 − 2p sin⁶θ + cos⁶θ = 1 − 3p

p = sin²θ cos²θ = ¼ sin²2θ, so 0 ≤ p ≤ ¼

Range of a sin θ + b cos θ

−√(a² + b²) ≤ a sin θ + b cos θ ≤ √(a² + b²)

a, b = constants

Domains and principal ranges

sin⁻¹: [−1, 1] → [−π/2, π/2] cos⁻¹: [−1, 1] → [0, π] tan⁻¹: all reals → (−π/2, π/2) cot⁻¹: all reals → (0, π)

sec⁻¹ and cosec⁻¹ need |x| ≥ 1

Complementary pairs

sin⁻¹x + cos⁻¹x = π/2 tan⁻¹x + cot⁻¹x = π/2 sec⁻¹x + cosec⁻¹x = π/2

x in the common domain of each pair

Negative arguments

sin⁻¹(−x) = −sin⁻¹x tan⁻¹(−x) = −tan⁻¹x cos⁻¹(−x) = π − cos⁻¹x cot⁻¹(−x) = π − cot⁻¹x

x in the domain

Adding inverse tangents

tan⁻¹a + tan⁻¹b = tan⁻¹((a + b)/(1 − ab)) (ab < 1)

a, b = reals

Note:If ab > 1 with a, b > 0, add π.

Telescoping term

tan⁻¹((b − a)/(1 + ab)) = tan⁻¹b − tan⁻¹a (ab > −1)

write each term of the series in this shape

2 tan⁻¹x

2 tan⁻¹x = sin⁻¹(2x/(1 + x²)) (|x| ≤ 1) = cos⁻¹((1 − x²)/(1 + x²)) (x ≥ 0) = tan⁻¹(2x/(1 − x²)) (|x| < 1)

valid only in the stated ranges

Back into the principal range

sin⁻¹(sin x) = π − x (π/2 ≤ x ≤ 3π/2) cos⁻¹(cos x) = 2π − x (π ≤ x ≤ 2π)

sin⁻¹(sin x) = x only on [−π/2, π/2]

Note:tan⁻¹(tan x) = x − π for π/2 < x < 3π/2.

Compositions to algebra

sin(tan⁻¹x) = x/√(1 + x²) cos(tan⁻¹x) = 1/√(1 + x²) cos(sin⁻¹x) = √(1 − x²)

draw the right triangle for the inner angle

Standard forms

∫ dx/(x² + a²) = (1/a) tan⁻¹(x/a) ∫ dx/(x² − a²) = (1/(2a)) ln|(x − a)/(x + a)| ∫ dx/√(a² − x²) = sin⁻¹(x/a) ∫ dx/√(x² ± a²) = ln|x + √(x² ± a²)|

a > 0; add + C to each

Note:Complete the square to reach one of these.

Function and its derivative

∫ f′(x) [f(x)]ᵏ dx = [f(x)]^(k + 1)/(k + 1) + C (k ≠ −1) ∫ f′(x)/f(x) dx = ln|f(x)| + C

f = inner function

Integration by parts

∫ u dv = uv − ∫ v du

u = the part that simplifies on differentiating

The eˣ pattern

∫ eˣ (f(x) + f′(x)) dx = eˣ f(x) + C

spot f and f′ side by side

Note:Reverse quotient rule: ∫ (u′v − uv′)/v² dx = u/v + C.

Partial fractions (cover-up)

(px + q)/((x − a)(x − b)) = A/(x − a) + B/(x − b), A = (pa + q)/(a − b)

B follows by swapping a and b

Split the numerator

∫ (A g(x) + B g′(x))/g(x) dx = Ax + B ln|g(x)| + C

write the numerator as A · denominator + B · its derivative

a² sin²x + b² cos²x

∫ dx/(a² sin²x + b² cos²x) = (1/(ab)) tan⁻¹((a tan x)/b) + C

divide by cos²x and put t = tan x

Ratio substitution

∫ dx/((x − a)ᵖ (x + b)^(2 − p)) = (1/((a + b)(1 − p))) ((x − a)/(x + b))^(1 − p) + C

p ≠ 1

put t = (x − a)/(x + b)

Chain rule

(g∘f)′(a) = g′(f(a)) · f′(a)

both derivatives must exist

Note:f′(a) = lim (h → 0) (f(a + h) − f(a))/h.

Derivative of an inverse

g′(k) = 1/f′(a), where f(a) = k

g = inverse of f

Parametric derivatives

dy/dx = (dy/dt)/(dx/dt) d²y/dx² = [d/dt (dy/dx)] ÷ (dx/dt)

t = parameter

Note:The second derivative is NOT (d²y/dt²)/(d²x/dt²).

Variable power

d/dx (f^g) = f^g (g′ ln f + g f′/f)

f > 0

Note:Take logs first: ln y = g ln f.

A standard substitution

x = tan θ: tan⁻¹(2x/(1 − x²)) = 2 tan⁻¹x (|x| < 1)

derivative is then 2/(1 + x²)

Derivative from a functional equation

f(x + y) = f(x) f(y) ⇒ f′(x) = f′(0) f(x)

found from the limit definition

Smooth join

f₁(a) = f₂(a) and f₁′(a) = f₂′(a)

f₁, f₂ = the pieces meeting at x = a

Note:The first condition gives continuity, the second differentiability.

Corners from a modulus

k|x − a|: f′(a⁺) − f′(a⁻) = 2k

k ≠ 0

Note:f is not differentiable at a unless the corner terms cancel.

General solutions

sin θ = sin α ⇒ θ = nπ + (−1)ⁿα cos θ = cos α ⇒ θ = 2nπ ± α tan θ = tan α ⇒ θ = nπ + α

n = any integer

Squared equations

sin²θ = sin²α, cos²θ = cos²α or tan²θ = tan²α ⇒ θ = nπ ± α

n = any integer

Roots of cos θ = c in order

α, 2π − α, 2π + α, 4π − α, …

α = cos⁻¹c, 0 < α < π

Note:Count roots in an interval by listing them this way.

Auxiliary angle

a cos x + b sin x = √(a² + b²) cos(x − φ), tan φ = b/a

a, b = constants

Note:a cos x + b sin x = c has a solution exactly when |c| ≤ √(a² + b²).

Existence of a solution

f(x) = k has a solution ⇔ min f ≤ k ≤ max f

f continuous

Forced equality

f(x) ≤ m ≤ g(x) for all x: f(x) = g(x) ⇔ f(x) = g(x) = m

m = a common bound

Note:Typical case: sin x + cos y = 2 forces sin x = cos y = 1.

Triple angles

cos 3x = 4cos³x − 3 cos x tan 3x = (3 tan x − tan³x)/(1 − 3tan²x)

x = angle

Exponent substitution

t = a^(sin²x): a^(sin²x) + a^(cos²x) = t + a/t

a > 0

Note:Then 1 ≤ t ≤ a (for a > 1) and the equation is a quadratic in t.