128 formulas across 13 chapters, in syllabus order.

Chapter 1. Relations and Functions

11 formulas

Relation as a subset
R \subseteq A \times B
A relation from A to B is any set of ordered pairs (a,b) with a \in A and b \in B.
Reflexive relation
(a,a) \in R \quad \forall\, a \in A
Every element of the set is related to itself.
Valid when
Must hold for every element without exception.
Symmetric relation
(a,b) \in R \;\Rightarrow\; (b,a) \in R
Whenever one element is related to another, the reverse also holds.
Transitive relation
(a,b) \in R \text{ and } (b,c) \in R \;\Rightarrow\; (a,c) \in R
Relations chain: related through an intermediate means related directly.
Equivalence class
[a] = \{\, x \in A : (x,a) \in R \,\}
The set of all elements related to a. Classes are disjoint and together cover A.
Valid when
Defined when R is an equivalence relation on A.
Number of relations on a finite set
2^{\,n^2}
A set with n elements has n^2 ordered pairs, and a relation is any subset of them.
Valid when
|A| = n; counts relations from A to A.
Number of relations from $A$ to $B$
2^{\,mn}
Where |A| = m and |B| = n.
One-one (injective) function
f(x_1) = f(x_2) \;\Rightarrow\; x_1 = x_2
Distinct inputs are never sent to the same output.
Onto (surjective) function
\text{Range}(f) = B
Every element of the codomain is the image of at least one element of the domain.
Number of one-one functions
{}^{n}P_{m} = \dfrac{n!}{(n-m)!}
One-one functions from a set of size m to a set of size n.
Valid when
Zero when m > n.
Number of functions from $A$ to $B$
n^{m}
Each of the m elements of A may be sent to any of the n elements of B.

Chapter 2. Inverse Trigonometric Functions

9 formulas

Principal value ranges
\sin^{-1}x \in \left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right], \quad \cos^{-1}x \in [0, \pi], \quad \tan^{-1}x \in \left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)
The three ranges every other result in the chapter depends on.
Remaining principal value ranges
\csc^{-1}x \in \left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]\setminus\{0\}, \quad \sec^{-1}x \in [0,\pi]\setminus\left\{\tfrac{\pi}{2}\right\}, \quad \cot^{-1}x \in (0, \pi)
Note the excluded points: \csc and \sec are undefined where their reciprocals vanish.
Negative arguments — symmetric branches
\sin^{-1}(-x) = -\sin^{-1}x, \quad \tan^{-1}(-x) = -\tan^{-1}x, \quad \csc^{-1}(-x) = -\csc^{-1}x
Valid because these branches are symmetric about 0, so a negative answer is available.
Valid when
x within the appropriate domain.
Negative arguments — non-negative branches
\cos^{-1}(-x) = \pi - \cos^{-1}x, \quad \cot^{-1}(-x) = \pi - \cot^{-1}x, \quad \sec^{-1}(-x) = \pi - \sec^{-1}x
These ranges lie inside [0,\pi], so the answer cannot be negative; it reflects about \frac{\pi}{2} instead.
Inverse composed with the function
\sin\!\left(\sin^{-1}x\right) = x
Holds for every x in the domain of the inverse function.
Valid when
x \in [-1,1] for sine and cosine; x \in \mathbb{R} for tangent.
Function composed with the inverse
\sin^{-1}(\sin x) = x
The conditional direction: true only when x already lies in the principal branch.
Valid when
x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] for sine; x \in [0,\pi] for cosine; x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) for tangent. Otherwise reduce first.
Reciprocal relationships
\csc^{-1}x = \sin^{-1}\!\left(\tfrac{1}{x}\right), \quad \sec^{-1}x = \cos^{-1}\!\left(\tfrac{1}{x}\right), \quad \cot^{-1}x = \tan^{-1}\!\left(\tfrac{1}{x}\right)
Useful for converting to the three functions whose values you know best.
Valid when
|x| \geq 1 for the first two. The cotangent relation holds as written for x > 0; for x < 0, \cot^{-1}x = \pi + \tan^{-1}\frac{1}{x}.
Standard principal values
\sin^{-1}\!\left(\tfrac{1}{2}\right) = \tfrac{\pi}{6}, \quad \sin^{-1}\!\left(\tfrac{1}{\sqrt2}\right) = \tfrac{\pi}{4}, \quad \sin^{-1}\!\left(\tfrac{\sqrt3}{2}\right) = \tfrac{\pi}{3}
The values that appear most often in examination questions.
Values at the boundaries
\sin^{-1}(0) = 0,\quad \sin^{-1}(1) = \tfrac{\pi}{2},\quad \cos^{-1}(0) = \tfrac{\pi}{2},\quad \cos^{-1}(1) = 0,\quad \cos^{-1}(-1) = \pi

Chapter 3. Matrices

10 formulas

Order of a matrix product
A_{m \times n} \, B_{n \times p} = (AB)_{m \times p}
The product exists only when the columns of A match the rows of B; the outer dimensions give the result's order.
Entry of a product
(AB)_{ij} = \sum_{k=1}^{n} a_{ik}\,b_{kj}
Row i of A paired term by term with column j of B.
Transpose of a product
(AB)^{T} = B^{T} A^{T}
The order reverses — this is forced by the dimension requirement.
Transpose properties
(A^{T})^{T} = A, \quad (A + B)^{T} = A^{T} + B^{T}, \quad (kA)^{T} = k\,A^{T}
Symmetric matrix
A^{T} = A \quad \text{i.e.} \quad a_{ij} = a_{ji}
Entries mirror across the main diagonal.
Valid when
Defined only for square matrices.
Skew-symmetric matrix
A^{T} = -A \quad \text{i.e.} \quad a_{ij} = -a_{ji}
Forces a_{ii} = 0: every diagonal entry of a skew-symmetric matrix is zero.
Symmetric–skew decomposition
A = \tfrac{1}{2}\left(A + A^{T}\right) + \tfrac{1}{2}\left(A - A^{T}\right)
The unique way to write a square matrix as symmetric plus skew-symmetric.
Valid when
A square.
Inverse of a product
(AB)^{-1} = B^{-1} A^{-1}
Like the transpose, the order reverses.
Valid when
A and B both invertible and of the same order.
Definition of the inverse
AB = BA = I \;\Rightarrow\; B = A^{-1}
The inverse must work on both sides; when it exists it is unique.
Valid when
A square.
Number of entries
\text{An } m \times n \text{ matrix has } mn \text{ entries.}
Used in counting questions: the possible orders of a matrix with mn entries correspond to the factor pairs of mn.

Chapter 4. Determinants

13 formulas

Determinant of order 2
\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc
Determinant of order 3 (expansion along the first row)
|A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}
Any row or column may be used. Choose the one with the most zeros — it removes whole terms.
Cofactor
C_{ij} = (-1)^{\,i+j}\, M_{ij}
M_{ij} is the minor: the determinant remaining after deleting row i and column j.
Adjoint
\operatorname{adj}(A) = \left[C_{ij}\right]^{T}
The transpose of the cofactor matrix. Forgetting the transpose is the classic error.
Inverse of a matrix
A^{-1} = \dfrac{1}{|A|}\operatorname{adj}(A)
Valid when
Exists if and only if |A| \neq 0.
Fundamental adjoint identity
A\,(\operatorname{adj} A) = (\operatorname{adj} A)\,A = |A|\,I
A quick way to check an adjoint before using it.
Determinant of the adjoint
|\operatorname{adj} A| = |A|^{\,n-1}
Valid when
A of order n. For n = 3: |\operatorname{adj} A| = |A|^{2}.
Determinant of a product
|AB| = |A|\,|B|
Determinants do commute, even though the matrices do not.
Determinant of a scalar multiple
|kA| = k^{\,n}\,|A|
Valid when
A of order n. The scalar is taken from every one of the n rows.
Determinant of an inverse
\left|A^{-1}\right| = \dfrac{1}{|A|}
Valid when
|A| \neq 0.
Area of a triangle
\Delta = \dfrac{1}{2}\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right|
The outer absolute value matters: area is never negative.
Condition for collinear points
\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0
Three points are collinear exactly when the triangle they form has zero area.
Matrix method for linear systems
AX = B \;\Rightarrow\; X = A^{-1}B
Valid when
Valid when |A| \neq 0; then the solution is unique.

Chapter 5. Continuity and Differentiability

10 formulas

Continuity at a point
\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) = f(a)
All three quantities must exist and be equal.
Derivative from first principles
f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}
Chain rule
\dfrac{d}{dx}\,f(g(x)) = f'(g(x)) \cdot g'(x)
Differentiate the outer function, then multiply by the derivative of the inner.
Product rule
(uv)' = u'v + uv'
Quotient rule
\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^{2}}
Valid when
v \neq 0. Note the order in the numerator — it is not symmetric.
Derivatives of inverse trigonometric functions
\dfrac{d}{dx}\sin^{-1}x = \dfrac{1}{\sqrt{1-x^{2}}}, \quad \dfrac{d}{dx}\cos^{-1}x = \dfrac{-1}{\sqrt{1-x^{2}}}, \quad \dfrac{d}{dx}\tan^{-1}x = \dfrac{1}{1+x^{2}}
Valid when
|x| < 1 for the first two.
Exponential and logarithmic derivatives
\dfrac{d}{dx}e^{x} = e^{x}, \quad \dfrac{d}{dx}a^{x} = a^{x}\ln a, \quad \dfrac{d}{dx}\ln x = \dfrac{1}{x}
Valid when
a > 0, a \neq 1; x > 0 for the logarithm.
Logarithmic differentiation
y = f(x)^{g(x)} \;\Rightarrow\; \dfrac{1}{y}\dfrac{dy}{dx} = \dfrac{d}{dx}\left[g(x)\ln f(x)\right]
For a variable base with a variable exponent.
Valid when
f(x) > 0.
Parametric differentiation
\dfrac{dy}{dx} = \dfrac{\;\dfrac{dy}{dt}\;}{\dfrac{dx}{dt}}
Valid when
\dfrac{dx}{dt} \neq 0.
Second derivative of a parametric function
\dfrac{d^{2}y}{dx^{2}} = \dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right) \cdot \dfrac{1}{\;\dfrac{dx}{dt}\;}
The extra factor \dfrac{dt}{dx} is essential — differentiating \frac{dy}{dx} with respect to t alone is not the second derivative.

Chapter 6. Application of Derivatives

9 formulas

Rate of change
\dfrac{dy}{dx} \text{ is the rate of change of } y \text{ with respect to } x
For related rates, use the chain rule: \dfrac{dA}{dt} = \dfrac{dA}{dr}\cdot\dfrac{dr}{dt}.
Strictly increasing function
f'(x) > 0 \quad \text{for all } x \text{ in the interval}
Strictly decreasing function
f'(x) < 0 \quad \text{for all } x \text{ in the interval}
Critical points
f'(c) = 0 \quad \text{or} \quad f'(c) \text{ does not exist}
The only candidates for a local extremum in the interior.
Second derivative test
f'(c) = 0 \text{ and } \begin{cases} f''(c) < 0 & \Rightarrow \text{local maximum} \\ f''(c) > 0 & \Rightarrow \text{local minimum} \end{cases}
Valid when
Inconclusive when f''(c) = 0 — use the first derivative test instead.
Absolute extrema on a closed interval
\max/\min \text{ of } \{\, f(a),\; f(b),\; f(c_i) \,\}
Compare the values at the endpoints and at every critical point c_i in [a,b].
Volume and surface area of a sphere
V = \dfrac{4}{3}\pi r^{3}, \qquad S = 4\pi r^{2}
The most frequently used solid in rate-of-change questions.
Volume and curved surface area of a cylinder
V = \pi r^{2} h, \qquad \text{CSA} = 2\pi r h, \qquad \text{TSA} = 2\pi r(r + h)
Volume and curved surface area of a cone
V = \dfrac{1}{3}\pi r^{2} h, \qquad \text{CSA} = \pi r l, \qquad l = \sqrt{r^{2} + h^{2}}

Chapter 7. Integrals

15 formulas

Power rule
\int x^{n}\,dx = \dfrac{x^{n+1}}{n+1} + C
Valid when
n \neq -1. For n = -1 the integral is \ln|x| + C.
Reciprocal
\int \dfrac{1}{x}\,dx = \ln|x| + C
The modulus matters — the integrand is defined for negative x too.
Standard trigonometric integrals
\int \sin x\,dx = -\cos x + C, \quad \int \cos x\,dx = \sin x + C, \quad \int \sec^{2}x\,dx = \tan x + C
Integrals giving logarithms of trigonometric functions
\int \tan x\,dx = \ln|\sec x| + C, \qquad \int \cot x\,dx = \ln|\sin x| + C
Integration by parts
\int u\,v'\,dx = uv - \int u'\,v\,dx
Choose u by ILATE: Inverse, Logarithmic, Algebraic, Trigonometric, Exponential.
The exponential shortcut
\int e^{x}\left[f(x) + f'(x)\right]dx = e^{x}f(x) + C
Look for an e^x multiplying a bracket where one term is the derivative of the other.
Standard form: difference of squares in the denominator
\int \dfrac{dx}{x^{2} - a^{2}} = \dfrac{1}{2a}\ln\left|\dfrac{x-a}{x+a}\right| + C
Standard form: sum of squares in the denominator
\int \dfrac{dx}{x^{2} + a^{2}} = \dfrac{1}{a}\tan^{-1}\!\left(\dfrac{x}{a}\right) + C
Standard form: square root of a difference
\int \dfrac{dx}{\sqrt{a^{2} - x^{2}}} = \sin^{-1}\!\left(\dfrac{x}{a}\right) + C
Valid when
|x| < a.
Standard form: square root of a sum
\int \dfrac{dx}{\sqrt{x^{2} + a^{2}}} = \ln\left|x + \sqrt{x^{2}+a^{2}}\right| + C
Integral of a square root
\int \sqrt{a^{2} - x^{2}}\,dx = \dfrac{x}{2}\sqrt{a^{2}-x^{2}} + \dfrac{a^{2}}{2}\sin^{-1}\!\left(\dfrac{x}{a}\right) + C
Fundamental Theorem of Calculus
\int_{a}^{b} f(x)\,dx = F(b) - F(a) \quad \text{where } F' = f
Reversal property
\int_{a}^{b} f(x)\,dx = -\int_{b}^{a} f(x)\,dx
The reflection property
\int_{0}^{a} f(x)\,dx = \int_{0}^{a} f(a-x)\,dx
The single most useful property in examinations — it often turns an intractable integrand into a tractable sum.
Even and odd functions
\int_{-a}^{a} f(x)\,dx = \begin{cases} 2\displaystyle\int_{0}^{a} f(x)\,dx, & f \text{ even} \\[6pt] 0, & f \text{ odd} \end{cases}
Check the parity before integrating — an odd integrand vanishes immediately.

Chapter 8. Application of Integrals

7 formulas

Area under a curve (with respect to $x$)
A = \int_{a}^{b} y\,dx = \int_{a}^{b} f(x)\,dx
Valid when
Valid where f(x) \geq 0; otherwise take the absolute value piecewise.
Area under a curve (with respect to $y$)
A = \int_{c}^{d} x\,dy
Use horizontal strips when they cross the region without splitting.
Area below the axis
A = \left|\int_{a}^{b} f(x)\,dx\right|
Valid when
When f(x) \leq 0 throughout [a,b].
Area between two curves
A = \int_{a}^{b}\left(y_{\text{upper}} - y_{\text{lower}}\right)dx
Valid when
a and b are the x-coordinates of the intersection points.
Area of a circle by integration
A = 4\int_{0}^{a}\sqrt{a^{2} - x^{2}}\,dx = \pi a^{2}
One quadrant computed and multiplied by four, using symmetry.
Area of an ellipse by integration
A = 4\int_{0}^{a}\frac{b}{a}\sqrt{a^{2}-x^{2}}\,dx = \pi a b
Valid when
For \dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1.
Standard integral needed for circles
\int \sqrt{a^{2}-x^{2}}\,dx = \frac{x}{2}\sqrt{a^{2}-x^{2}} + \frac{a^{2}}{2}\sin^{-1}\!\left(\frac{x}{a}\right) + C

Chapter 9. Differential Equations

8 formulas

Order of a differential equation
\text{the order of the highest derivative present}
Degree of a differential equation
\text{the power of the highest-order derivative, once the equation is a polynomial in its derivatives}
Valid when
Undefined if the equation cannot be written as a polynomial in the derivatives — for example if a derivative appears inside a sine, a logarithm or a radical.
Variable separable form
\int \frac{dy}{g(y)} = \int f(x)\,dx + C
Valid when
Applies when \dfrac{dy}{dx} = f(x)g(y).
Homogeneous substitution
y = vx \;\Longrightarrow\; \frac{dy}{dx} = v + x\,\frac{dv}{dx}
Note the product rule — both terms are needed.
Linear differential equation in $y$
\frac{dy}{dx} + P(x)\,y = Q(x)
Integrating factor
\text{I.F.} = e^{\int P\,dx}
No constant of integration is included when forming the integrating factor.
Solution of a linear differential equation
y \cdot (\text{I.F.}) = \int Q \cdot (\text{I.F.})\,dx + C
Linear in $x$ instead
\frac{dx}{dy} + P(y)\,x = Q(y) \;\Longrightarrow\; \text{I.F.} = e^{\int P\,dy}
Use when the equation is linear in x but not in y.

Chapter 10. Vector Algebra

13 formulas

Magnitude of a vector
\left|\vec{a}\right| = \sqrt{a_1^{2} + a_2^{2} + a_3^{2}} \quad \text{for } \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}
Unit vector
\hat{a} = \dfrac{\vec{a}}{\left|\vec{a}\right|}
Valid when
\vec{a} \neq \vec{0}.
Scalar (dot) product
\vec{a}\cdot\vec{b} = \left|\vec{a}\right|\left|\vec{b}\right|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3
Angle between two vectors
\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{\left|\vec{a}\right|\left|\vec{b}\right|}
Perpendicularity test
\vec{a} \perp \vec{b} \iff \vec{a}\cdot\vec{b} = 0
Valid when
\vec{a}, \vec{b} non-zero.
Projection of one vector on another
\text{Projection of } \vec{a} \text{ on } \vec{b} = \dfrac{\vec{a}\cdot\vec{b}}{\left|\vec{b}\right|}
A signed scalar — negative when the angle is obtuse.
Vector (cross) product
\vec{a}\times\vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}
Magnitude of the cross product
\left|\vec{a}\times\vec{b}\right| = \left|\vec{a}\right|\left|\vec{b}\right|\sin\theta
Anti-commutativity
\vec{a}\times\vec{b} = -\left(\vec{b}\times\vec{a}\right)
Order reverses direction — unlike the dot product, which commutes.
Area of a triangle from two sides
\text{Area} = \dfrac{1}{2}\left|\vec{a}\times\vec{b}\right|
Area of a parallelogram
\text{Area} = \left|\vec{a}\times\vec{b}\right|
With diagonals \vec{d_1}, \vec{d_2}, the area is \frac{1}{2}\left|\vec{d_1}\times\vec{d_2}\right|.
Direction cosines
l = \dfrac{a_1}{|\vec{a}|}, \quad m = \dfrac{a_2}{|\vec{a}|}, \quad n = \dfrac{a_3}{|\vec{a}|}, \qquad l^{2}+m^{2}+n^{2} = 1
Section formula (internal division)
\vec{r} = \dfrac{m\vec{b} + n\vec{a}}{m + n}
Position vector of the point dividing AB internally in the ratio m:n.

Chapter 11. Three Dimensional Geometry

10 formulas

Direction cosines of a line
l^{2} + m^{2} + n^{2} = 1
Direction cosines are the components of a unit vector along the line.
Direction ratios from two points
\left(x_2 - x_1,\; y_2 - y_1,\; z_2 - z_1\right)
Direction ratios of the line joining (x_1,y_1,z_1) and (x_2,y_2,z_2).
Direction cosines from direction ratios
l = \dfrac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}, \quad m = \dfrac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}, \quad n = \dfrac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}
Vector equation of a line
\vec{r} = \vec{a} + \lambda\vec{b}
Through the point with position vector \vec{a}, in the direction \vec{b}.
Cartesian equation of a line
\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}
Through (x_1,y_1,z_1) with direction ratios a, b, c.
Line through two points
\vec{r} = \vec{a} + \lambda\left(\vec{b} - \vec{a}\right)
Angle between two lines
\cos\theta = \left|\dfrac{\vec{b_1}\cdot\vec{b_2}}{\left|\vec{b_1}\right|\left|\vec{b_2}\right|}\right|
The modulus gives the acute angle, which is the convention for the angle between lines.
Perpendicular and parallel conditions
\text{Perpendicular: } a_1a_2 + b_1b_2 + c_1c_2 = 0; \qquad \text{Parallel: } \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}
Shortest distance between two skew lines
d = \left|\dfrac{\left(\vec{a_2}-\vec{a_1}\right)\cdot\left(\vec{b_1}\times\vec{b_2}\right)}{\left|\vec{b_1}\times\vec{b_2}\right|}\right|
Valid when
Requires \vec{b_1} \not\parallel \vec{b_2}.
Distance between two parallel lines
d = \dfrac{\left|\vec{b}\times\left(\vec{a_2}-\vec{a_1}\right)\right|}{\left|\vec{b}\right|}
Valid when
Both lines share the direction \vec{b}.

Chapter 12. Linear Programming

5 formulas

Objective function
Z = ax + by
The quantity to be maximised or minimised.
Non-negativity constraints
x \geq 0, \qquad y \geq 0
Almost always present, since the decision variables represent physical quantities.
Corner point theorem
\text{The optimum of } Z \text{ over a feasible region occurs at a corner point.}
Valid when
The region must be bounded for the optimum to be guaranteed to exist.
Unbounded region — maximum test
M \text{ is the maximum} \iff ax + by > M \text{ has no point in the feasible region}
Unbounded region — minimum test
m \text{ is the minimum} \iff ax + by < m \text{ has no point in the feasible region}

Chapter 13. Probability

8 formulas

Conditional probability
P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}
Valid when
P(B) \neq 0.
Multiplication theorem
P(A \cap B) = P(B)\,P(A \mid B) = P(A)\,P(B \mid A)
Independent events
P(A \cap B) = P(A)\,P(B)
Equivalently P(A\mid B) = P(A) — knowing B changes nothing about A.
Theorem of total probability
P(A) = \sum_{i=1}^{n} P(E_i)\,P(A \mid E_i)
Valid when
E_1, \dots, E_n must be mutually exclusive and exhaustive.
Bayes' theorem
P(E_i \mid A) = \dfrac{P(E_i)\,P(A \mid E_i)}{\displaystyle\sum_{j=1}^{n} P(E_j)\,P(A \mid E_j)}
Complement
P(A') = 1 - P(A)
Addition rule
P(A \cup B) = P(A) + P(B) - P(A \cap B)
The subtraction avoids counting the overlap twice; it vanishes for mutually exclusive events.
At least one of two independent events
P(\text{at least one}) = 1 - P(A')\,P(B')
Valid when
A and B independent.

Chapter 1. Relations and Functions

Relation as a subset

Reflexive relation

Symmetric relation

Transitive relation

Equivalence class

Number of relations on a finite set

Number of relations from $A$ to $B$

One-one (injective) function

Onto (surjective) function

Number of one-one functions

Number of functions from $A$ to $B$

Chapter 2. Inverse Trigonometric Functions

Principal value ranges

Remaining principal value ranges

Negative arguments — symmetric branches

Negative arguments — non-negative branches

Inverse composed with the function

Function composed with the inverse

Reciprocal relationships

Standard principal values

Values at the boundaries

Chapter 3. Matrices

Order of a matrix product

Entry of a product

Transpose of a product

Transpose properties

Symmetric matrix

Skew-symmetric matrix

Symmetric–skew decomposition

Inverse of a product

Definition of the inverse

Number of entries

Chapter 4. Determinants

Determinant of order 2

Determinant of order 3 (expansion along the first row)

Cofactor

Adjoint

Inverse of a matrix

Fundamental adjoint identity

Determinant of the adjoint

Determinant of a product

Determinant of a scalar multiple

Determinant of an inverse

Area of a triangle

Condition for collinear points

Matrix method for linear systems

Chapter 5. Continuity and Differentiability

Continuity at a point

Derivative from first principles

Chain rule

Product rule

Quotient rule

Derivatives of inverse trigonometric functions

Exponential and logarithmic derivatives

Logarithmic differentiation

Parametric differentiation

Second derivative of a parametric function

Chapter 6. Application of Derivatives

Rate of change

Strictly increasing function

Strictly decreasing function

Critical points

Second derivative test

Absolute extrema on a closed interval

Volume and surface area of a sphere

Volume and curved surface area of a cylinder

Volume and curved surface area of a cone

Chapter 7. Integrals

Power rule

Reciprocal

Standard trigonometric integrals

Integrals giving logarithms of trigonometric functions

Integration by parts

The exponential shortcut

Standard form: difference of squares in the denominator

Standard form: sum of squares in the denominator

Standard form: square root of a difference

Standard form: square root of a sum

Integral of a square root