Mathematics Formula Reference

01 Relations and Functions 3 marks

Empty Relation

$R = \phi \subset A \times A$

No element is related to any element

Universal Relation

$R = A \times A$

Every element is related to every element

One-one test

$f(x_1) = f(x_2) \Rightarrow x_1 = x_2, \; \forall \; x_1, x_2 \in X$

Equivalent to: x1 ≠ x2 ⇒ f(x1) ≠ f(x2)

Onto condition

$\forall \; y \in Y, \; \exists \; x \in X \text{ such that } f(x) = y$

Equivalently, f is onto if and only if Range of f = Y (codomain)

Composition of functions

$g \circ f(x) = g(f(x)), \; \forall \; x \in A$

If f: A → B and g: B → C, then gof: A → C

Inverse function condition

$g \circ f = I_X \text{ and } f \circ g = I_Y$

f is invertible ⟺ f is one-one and onto (bijective)

Inverse verification

$f^{-1}(y) = x \iff f(x) = y$

f⁻¹ o f = I_X and f o f⁻¹ = I_Y

02 Inverse Trigonometric Functions 3 marks

Domain and Range of sin⁻¹

$\sin^{-1} : [-1, 1] \to \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

Principal value branch

Domain and Range of cos⁻¹

$\cos^{-1} : [-1, 1] \to [0, \pi]$

Principal value branch

Domain and Range of cosec⁻¹

$\csc^{-1} : \mathbb{R} - (-1, 1) \to \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}$

Principal value branch. Domain is |x| ≥ 1, i.e., x ≤ −1 or x ≥ 1

Domain and Range of sec⁻¹

$\sec^{-1} : \mathbb{R} - (-1, 1) \to [0, \pi] - \left\{\frac{\pi}{2}\right\}$

Principal value branch. Domain is |x| ≥ 1, i.e., x ≤ −1 or x ≥ 1

Domain and Range of tan⁻¹

$\tan^{-1} : \mathbb{R} \to \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

Principal value branch

Domain and Range of cot⁻¹

$\cot^{-1} : \mathbb{R} \to (0, \pi)$

Principal value branch

Sine inverse-forward composition

$\sin(\sin^{-1} x) = x, \; x \in [-1, 1]$

Composition of function with its inverse

Sine forward-inverse composition

$\sin^{-1}(\sin x) = x, \; x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

Composition of inverse with function

Cancellation property (sin)

$\sin(\sin^{-1} x) = x, \; x \in [-1,1] \text{ and } \sin^{-1}(\sin x) = x, \; x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

Similar results hold for other trigonometric functions for suitable values of domain

Double angle formula for sin⁻¹

$\sin^{-1}(2x\sqrt{1 - x^2}) = 2\sin^{-1} x, \; -\frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}}$

Derived by substituting x = sin θ

Double angle formula for cos⁻¹

$\sin^{-1}(2x\sqrt{1 - x^2}) = 2\cos^{-1} x, \; \frac{1}{\sqrt{2}} \leq x \leq 1$

Derived by substituting x = cos θ

Simplification of cot⁻¹(1/√(x² − 1))

$\cot^{-1}\!\left(\frac{1}{\sqrt{x^2 - 1}}\right) = \sec^{-1} x, \; x > 1$

Derived by substituting x = sec θ

03 Matrices 5 marks

General m x n matrix

$A = [a_{ij}]_{m \times n}, \; 1 \leq i \leq m, \; 1 \leq j \leq n$

The i-th row consists of elements aᵢ₁, aᵢ₂, ..., aᵢₙ and the j-th column consists of elements a₁ⱼ, a₂ⱼ, ..., aₘⱼ

Matrix addition

$A + B = [a_{ij} + b_{ij}]_{m \times n}$

Both matrices must be of the same order

Scalar multiplication

$kA = [k \cdot a_{ij}]_{m \times n}$

The (i,j)-th element of kA is k · aᵢⱼ

Matrix multiplication element

$c_{ik} = a_{i1}b_{1k} + a_{i2}b_{2k} + \cdots + a_{in}b_{nk} = \sum_{j=1}^{n} a_{ij} b_{jk}$

Number of columns of A must equal number of rows of B

Commutative law of addition

$A + B = B + A$

For matrices of the same order

Associative law of addition

$(A + B) + C = A + (B + C)$

For matrices of the same order

Additive identity

$A + O = O + A = A$

O is the zero matrix of the same order as A

Additive inverse

$A + (-A) = (-A) + A = O$

-A = [-aᵢⱼ]ₘ ₓ ₙ

Scalar distributive over matrix addition

$k(A + B) = kA + kB$

A, B are matrices of same order, k is a scalar

Scalar sum distributive

$(k + l)A = kA + lA$

k and l are scalars

Associative law of multiplication

$(AB)C = A(BC)$

Whenever both sides of the equality are defined

Distributive law (left)

$A(B + C) = AB + AC$

Whenever both sides are defined

Distributive law (right)

$(A + B)C = AC + BC$

Whenever both sides are defined

Multiplicative identity

$IA = AI = A$

I is the identity matrix of appropriate order

Transpose of transpose

$(A^{T})^{T} = A$

Taking transpose twice gives back the original matrix

Transpose of scalar multiple

$(kA)^{T} = kA^{T}$

Where k is any constant

Transpose of sum

$(A + B)^{T} = A^{T} + B^{T}$

For matrices A and B of suitable orders

Transpose of product

$(AB)^{T} = B^{T}A^{T}$

The order reverses when taking transpose of a ∏

Symmetric part of a matrix

$\frac{1}{2}(A + A') \text{ is symmetric}$

For any square matrix A with real number entries

Skew symmetric part of a matrix

$\frac{1}{2}(A - A') \text{ is skew-symmetric}$

For any square matrix A with real number entries

Decomposition of a square matrix

$A = \frac{1}{2}(A + A') + \frac{1}{2}(A - A')$

Any square matrix can be expressed as the ∑ of a symmetric and a skew symmetric matrix

Inverse of a product

$(AB)^{-1} = B^{-1} A^{-1}$

If A and B are invertible matrices of the same order

04 Determinants 5 marks

$\Delta = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$

$\Delta = \frac{1}{2} |\text{determinant value}|$

$\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0 \; (\text{collinear})$

$M_{ij} = \text{minor of } a_{ij}$

$A_{ij} = (-1)^{i+j} \cdot M_{ij}$

$Minor of an element of a determinant of order n (n >= 2) is a determinant of order n-1$

$\Delta = a_{i1}A_{i1} + a_{i2}A_{i2} + a_{i3}A_{i3}$

$\Delta = a_{1j}A_{1j} + a_{2j}A_{2j} + a_{3j}A_{3j}$

$a_{i1}A_{j1} + a_{i2}A_{j2} + a_{i3}A_{j3} = 0, \; i \neq j$

$For A = [a11 a12 a13; a21 a22 a23; a31 a32 a33]$

adj A = Transpose of [A11 A12 A13 ┃ A21 A22 A23 ┃ A31 A32 A33] = [A11 A21 A31 ┃ A12 A22 A32 ┃ A13 A23 A33]

$For 2x2 matrix A = [a11 a12; a21 a22]$

adj A = [a22 -a12 ┃ -a21 a11] (interchange diagonal elements, change sign of off-diagonal elements)

$A(\text{adj } A) = (\text{adj } A)A = |A| \cdot I$

$|\text{adj}(A)| = |A|^{n-1}$

$A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A), \; |A| \neq 0$

$|AB| = |A| \cdot |B|$

$If AB = BA = I, then B is called the inverse of A, i.e., B = A^(-1)$

$A^{-1} = B, \; B^{-1} = A, \; (A^{-1})^{-1} = A$

For system

$a1*x + b1*y + c1*z = d1, a2*x + b2*y + c2*z = d2, a3*x + b3*y + c3*z = d3: Matrix form AX = B where A = [a1 b1 c1; a2 b2 c2; a3 b3 c3], X = [x; y; z], B = [d1; d2; d3]$

$|A| \neq 0 \Rightarrow X = A^{-1}B$

$If |A| = 0 and (adj A)B != O, system is inconsistent (no solution)$

$If |A| = 0 and (adj A)B = O, system may be consistent (infinitely many solutions) or inconsistent$

05 Continuity and Differentiability 8 marks

Sum/Difference Rule

$(u \pm v)' = u' \pm v'$

Product Rule (Leibnitz Rule)

$(uv)' = u'v + uv'$

Derivative of a ∏ of two functions

Quotient Rule

$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}, \; v \neq 0$

Chain Rule (two functions)

$\frac{df}{dx} = \frac{dv}{dt} \cdot \frac{dt}{dx}$

Chain Rule (three functions)

$\frac{df}{dx} = \frac{dw}{ds} \cdot \frac{ds}{dt} \cdot \frac{dt}{dx}$

Derivative of sin^(-1) x

$\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}}$

Derivative of cos^(-1) x

$\frac{d}{dx}(\cos^{-1} x) = \frac{-1}{\sqrt{1 - x^2}}$

Derivative of tan^(-1) x

$\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2}$

Derivative of e^x

$\frac{d}{dx}(e^x) = e^x$

Derivative of log x (natural log)

$\frac{d}{dx}(\log x) = \frac{1}{x}$

Derivative of a^x

$\frac{d}{dx}(a^x) = a^x \log a$

Change of base formula

$\log_a p = \frac{\log_b p}{\log_b a}$

Log of product

$\log_b(pq) = \log_b p + \log_b q$

Log of power

$\log_b(p^n) = n \log_b p$

Log of quotient

$\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y$

Logarithmic differentiation formula

$\frac{dy}{dx} = y\left[v(x) \cdot \frac{u'(x)}{u(x)} + v'(x) \cdot \log u(x)\right]$

Parametric differentiation

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}, \; f'(t) \neq 0$

06 Application of Derivatives 8 marks

Rate of change using Chain Rule

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \; \frac{dx}{dt} \neq 0$

Rate of change of area of circle

$\frac{dA}{dr} = 2\pi r$

07 Integrals 8 marks

Power Rule

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \; n \neq -1$

Particularly, ∫ dx = x + C

Cosine integral

$\int \cos x\,dx = \sin x + C$

Sine integral

$\int \sin x\,dx = -\cos x + C$

Secant squared integral

$\int \sec^2 x\,dx = \tan x + C$

Cosecant squared integral

$\int \csc^2 x\,dx = -\cot x + C$

Secant-tangent integral

$\int \sec x \tan x\,dx = \sec x + C$

Cosecant-cotangent integral

$\int \csc x \cot x\,dx = -\csc x + C$

Inverse sine integral

$\int \frac{dx}{\sqrt{1 - x^2}} = \sin^{-1} x + C$

Negative inverse cosine integral

$\int \frac{dx}{\sqrt{1 - x^2}} = -\cos^{-1} x + C$

Inverse tangent integral

$\int \frac{dx}{1 + x^2} = \tan^{-1} x + C$

Exponential integral

$\int e^x\,dx = e^x + C$

Logarithmic integral

$\int \frac{1}{x}\,dx = \log |x| + C$

General exponential integral

$\int a^x\,dx = \frac{a^x}{\log a} + C$

$\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log \left|\frac{x-a}{x+a}\right| + C$

$\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \log \left|\frac{a+x}{a-x}\right| + C$

$\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}\!\left(\frac{x}{a}\right) + C$

$\int \frac{dx}{\sqrt{x^2 - a^2}} = \log \left|x + \sqrt{x^2 - a^2}\right| + C$

$\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\!\left(\frac{x}{a}\right) + C$

$\int \frac{dx}{\sqrt{x^2 + a^2}} = \log \left|x + \sqrt{x^2 + a^2}\right| + C$

Distinct linear factors

$\frac{px+q}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}, \; a \neq b$

Two distinct linear factors ∈ denominator

Repeated linear factor

$\frac{px+q}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}$

Same linear factor repeated twice

Three distinct linear factors

$\frac{px^2+qx+r}{(x-a)(x-b)(x-c)} = \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c}$

Three distinct linear factors ∈ denominator

Repeated and distinct linear factors

$\frac{px^2+qx+r}{(x-a)^2(x-b)} = \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}$

One repeated and one distinct linear factor

Linear and irreducible quadratic factors

$\frac{px^2+qx+r}{(x-a)(x^2+bx+c)} = \frac{A}{x-a} + \frac{Bx+C}{x^2+bx+c}$

Where x² + bx + c cannot be factorised further

Integration by Parts formula

$\int f(x)g(x)\,dx = f(x)\!\int g(x)\,dx - \int \left[f'(x)\!\int g(x)\,dx\right]dx$

Special exponential formula

$\int e^x [f(x) + f'(x)]\,dx = e^x f(x) + C$

Property P0

$\int_a^b f(x)\,dx = \int_a^b f(t)\,dt$

The variable of integration is a dummy variable.

Property P1

$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$

Interchanging limits changes the sign.

Property P2

$\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx$

Splitting the interval at an intermediate point c.

Property P3

$\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$

Substitution x → a + b - x.

Property P4

$\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx$

Particular case of P3 with lower limit 0.

Property P5

$\int_0^{2a} f(x)\,dx = \int_0^a f(x)\,dx + \int_0^a f(2a - x)\,dx$

Splitting [0, 2a] using substitution.

Property P6

$\int_0^{2a} f(x)\,dx = \begin{cases} 2\int_0^a f(x)\,dx & \text{if } f(2a-x) = f(x) \\ 0 & \text{if } f(2a-x) = -f(x) \end{cases}$

Useful for symmetric/antisymmetric functions about x = a.

Property P7

$\int_{-a}^{a} f(x)\,dx = \begin{cases} 2\int_0^a f(x)\,dx & \text{if } f(-x) = f(x) \\ 0 & \text{if } f(-x) = -f(x) \end{cases}$

Even and odd function properties for symmetric intervals.

08 Application of Integrals 5 marks

Area using vertical strips

$A = \int_a^b y\,dx = \int_a^b f(x)\,dx$

Area bounded by curve y = f(x), x-axis, and lines x = a, x = b

Area using horizontal strips

$A = \int_c^d x\,dy = \int_c^d g(y)\,dy$

Area bounded by curve x = g(y), y-axis, and lines y = c, y = d

Area when curve is below x-axis

$A = \left|\int_a^b f(x)\,dx\right|$

If f(x) < 0 from x = a to x = b, the area is the absolute value of the ∫.

Area when curve crosses x-axis

$A = |A_{1}| + A_{2}$

When part of the curve is above and part below x-axis, take absolute value of negative area and add to positive area.

09 Differential Equations 5 marks

Variable Separable Form

$\frac{dy}{dx} = h(y) \cdot g(x)$

Standard form for variable separable equations

Separated Form

$\frac{1}{h(y)}\,dy = g(x)\,dx$

After separating the variables

General Solution

$\int \frac{1}{h(y)}\,dy = \int g(x)\,dx + C$

Integrate both sides to get the solution; H(y) = G(x) + C

Substitution for dy/dx form

$y = vx, \; \frac{dy}{dx} = v + x\frac{dv}{dx}$

Used when dy/dx = g(y/x)

Substitution for dx/dy form

$x = vy, \; \frac{dx}{dy} = v + y\frac{dv}{dy}$

Used when dx/dy = h(x/y)

Reduced form

$x\frac{dv}{dx} = g(v) - v, \; \frac{dv}{g(v) - v} = \frac{dx}{x}$

After substitution, separate variables ∈ v and x

General Solution

$\int \frac{dv}{g(v) - v} = \int \frac{1}{x}\,dx + C$

Integrate and replace v by y/x to get the solution

Standard Form (Type 1)

$\frac{dy}{dx} + Py = Q$

P, Q are constants or functions of x only

Integrating Factor (Type 1)

$\text{I.F.} = e^{\int P\,dx}$

Integrating factor for dy/dx + Py = Q

General Solution (Type 1)

$y \cdot (\text{I.F.}) = \int (Q \times \text{I.F.})\,dx + C$

y · e^{∫P dx} = ∫(Q · e^{∫P dx}) dx + C

Standard Form (Type 2)

$\frac{dx}{dy} + P_1 x = Q_1$

P₁, Q₁ are constants or functions of y only

Integrating Factor (Type 2)

$\text{I.F.} = e^{\int P_1\,dy}$

Integrating factor for dx/dy + P₁x = Q₁

General Solution (Type 2)

$x \cdot (\text{I.F.}) = \int (Q_1 \times \text{I.F.})\,dy + C$

x · e^{∫P₁ dy} = ∫(Q₁ · e^{∫P₁ dy}) dy + C

10 Vector Algebra 5 marks

Magnitude of position vector

$|\vec{OP}| = \sqrt{x^2 + y^2 + z^2}$

Direction cosines

$\cos\alpha = \frac{x}{r}, \; \cos\beta = \frac{y}{r}, \; \cos\gamma = \frac{z}{r}$

Direction cosine identity

$l^{2} + m^{2} + n^{2} = 1$

Triangle law

$\vec{AC} = \vec{AB} + \vec{BC}$

Vector difference

$\vec{a} - \vec{b} = \vec{AB} + \vec{BC'} \text{ where } \vec{BC'} = -\vec{BC}$

Sides of triangle sum to zero

$\vec{AB} + \vec{BC} + \vec{CA} = \vec{AA} = \vec{0}$

Scalar multiplication magnitude

$|\lambda \vec{a}| = |\lambda| \cdot |\vec{a}|$

Negative of a vector

$\vec{a} + (-\vec{a}) = (-\vec{a}) + \vec{a} = \vec{0}$

Unit vector

$\hat{a} = \frac{1}{|\vec{a}|} \cdot \vec{a}, \; \vec{a} \neq \vec{0}$

For any scalar k

$k \cdot \vec{0} = \vec{0}$

Position vector in component form

$\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$

Magnitude from components

$|\vec{r}| = |x\hat{i} + y\hat{j} + z\hat{k}| = \sqrt{x^2 + y^2 + z^2}$

Sum of vectors in component form

$\vec{a} + \vec{b} = (a_1+b_1)\hat{i} + (a_2+b_2)\hat{j} + (a_3+b_3)\hat{k}$

Difference of vectors in component form

$\vec{a} - \vec{b} = (a_1-b_1)\hat{i} + (a_2-b_2)\hat{j} + (a_3-b_3)\hat{k}$

Equality of vectors

$\vec{a} = \vec{b} \iff a_1 = b_1, \; a_2 = b_2, \; a_3 = b_3$

Scalar multiplication in component form

$\lambda\vec{a} = (\lambda a_1)\hat{i} + (\lambda a_2)\hat{j} + (\lambda a_3)\hat{k}$

Distributive law 1

$k\vec{a} + m\vec{a} = (k + m)\vec{a}$

Distributive law 2

$k(m\vec{a}) = (km)\vec{a}$

Distributive law 3

$k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$

Collinearity condition

$\vec{b} = \lambda\vec{a} \iff \frac{b_1}{a_1} = \frac{b_2}{a_2} = \frac{b_3}{a_3} = \lambda$

Vector joining two points

$\vec{P_1P_2} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}$

Distance between two points

$|\vec{P_1P_2}| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$

Section formula (internal)

$\vec{OR} = \frac{m\vec{b} + n\vec{a}}{m + n}$

Section formula (external)

$\vec{OR} = \frac{m\vec{b} - n\vec{a}}{m - n}$

Midpoint formula

$\vec{OR} = \frac{\vec{a} + \vec{b}}{2}$

Scalar (dot) product definition

$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$

Dot product of perpendicular vectors

$\vec{a} \cdot \vec{b} = 0 \iff \vec{a} \perp \vec{b}$

Dot product when theta = 0

$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|; \; \vec{a} \cdot \vec{a} = |\vec{a}|^2$

Dot product when theta = pi

$\vec{a} \cdot \vec{b} = -|\vec{a}||\vec{b}|$

Dot products of unit vectors

$\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1; \; \hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$

Angle between vectors using dot product

$\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}, \; \theta = \cos^{-1}\!\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\right)$

Commutative property of dot product

$\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$

Distributive property of dot product

$\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$

Scalar factor in dot product

$(\lambda\vec{a}) \cdot \vec{b} = \lambda(\vec{a} \cdot \vec{b}) = \vec{a} \cdot (\lambda\vec{b})$

Dot product in component form

$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$

Projection of a on b

$\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$

Projection of a on b (vector form)

$\vec{a} \cdot \frac{\vec{b}}{|\vec{b}|} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$

Direction cosines from dot product

$\cos\alpha = \frac{a_1}{|\vec{a}|}, \; \cos\beta = \frac{a_2}{|\vec{a}|}, \; \cos\gamma = \frac{a_3}{|\vec{a}|}$

Unit vector in terms of direction cosines

$\hat{a} = \cos\alpha\,\hat{i} + \cos\beta\,\hat{j} + \cos\gamma\,\hat{k}$

Vector (cross) product definition

$\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \; \hat{n}$

Cross product of parallel/collinear vectors

$\vec{a} \times \vec{b} = \vec{0} \iff \vec{a} \parallel \vec{b}$

Cross product of self

$\vec{a} \times \vec{a} = \vec{0}$

Cross product when theta = pi/2

$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|$

Cross products of unit vectors

$\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = \vec{0}; \; \hat{i} \times \hat{j} = \hat{k}, \; \hat{j} \times \hat{k} = \hat{i}, \; \hat{k} \times \hat{i} = \hat{j}$

Reverse cross products of unit vectors

$\hat{j} \times \hat{i} = -\hat{k}, \; \hat{k} \times \hat{j} = -\hat{i}, \; \hat{i} \times \hat{k} = -\hat{j}$

Angle from cross product

$\sin\theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}||\vec{b}|}$

Anti-commutative property

$\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$

Area of triangle

$\text{Area}_{\triangle} = \frac{1}{2}|\vec{a} \times \vec{b}|$

Area of parallelogram

$\text{Area}_{\square} = |\vec{a} \times \vec{b}|$

Distributive property of cross product

$\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$

Scalar factor in cross product

$\lambda(\vec{a} \times \vec{b}) = (\lambda\vec{a}) \times \vec{b} = \vec{a} \times (\lambda\vec{b})$

Cross product in component form (determinant)

$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

Cauchy-Schwarz Inequality

$|\vec{a} \cdot \vec{b}| \leq |\vec{a}||\vec{b}|$

Triangle Inequality

$|\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}|$

11 Three Dimensional Geometry 5 marks

Direction cosines identity

$l^{2} + m^{2} + n^{2} = 1$

Sum of squares of direction cosines equals 1

Relation between direction ratios and direction cosines

$\frac{l}{a} = \frac{m}{b} = \frac{n}{c} = \pm\frac{1}{\sqrt{a^2 + b^2 + c^2}}$

Connecting direction ratios (a,b,c) to direction cosines (l,m,n)

Direction cosines from direction ratios

$l = \pm\frac{a}{\sqrt{a^2+b^2+c^2}}, \; m = \pm\frac{b}{\sqrt{a^2+b^2+c^2}}, \; n = \pm\frac{c}{\sqrt{a^2+b^2+c^2}}$

Direction cosines computed from direction ratios

Direction cosines of line joining two points

$l = \frac{x_2-x_1}{PQ}, \; m = \frac{y_2-y_1}{PQ}, \; n = \frac{z_2-z_1}{PQ}$

Direction cosines of line segment joining P(x1,y1,z1) and Q(x2,y2,z2)

Direction ratios of line joining two points

$x_{2}-x_{1}, \; y_{2}-y_{1}, \; z_{2}-z_{1}$

Direction ratios of line segment from P(x1,y1,z1) to Q(x2,y2,z2)

Vector equation of a line (point + direction)

$\vec{r} = \vec{a} + \lambda\vec{b}$

Line through point with position vector a, parallel to vector b. λ is a real parameter.

Cartesian equation of a line (point + direction ratios)

$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$

Line through (x1,y1,z1) with direction ratios a, b, c

Cartesian equation using direction cosines

$\frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n}$

Line through (x1,y1,z1) with direction cosines l, m, n

Parametric equations of a line

$x = x_1 + \lambda a, \; y = y_1 + \lambda b, \; z = z_1 + \lambda c$

Parametric form of line through (x1,y1,z1) with direction ratios a, b, c

Vector equation of a line through two points

$\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$

Line through two points with position vectors a and b

Angle between two lines (direction ratios)

$\cos\theta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2+b_1^2+c_1^2} \cdot \sqrt{a_2^2+b_2^2+c_2^2}}$

Angle between lines with direction ratios (a1,b1,c1) and (a2,b2,c2)

Angle between two lines (direction cosines)

$\cos\theta = |l_1 l_2 + m_1 m_2 + n_1 n_2|$

Angle between lines with direction cosines (l1,m1,n1) and (l2,m2,n2), since l²+m²+n²=1

sin(theta) using direction ratios

$\sin\theta = \frac{\sqrt{(a_1 b_2-a_2 b_1)^2 + (b_1 c_2-b_2 c_1)^2 + (c_1 a_2-c_2 a_1)^2}}{\sqrt{a_1^2+b_1^2+c_1^2} \cdot \sqrt{a_2^2+b_2^2+c_2^2}}$

Sine of angle between two lines

sin(theta) using direction cosines

$\sin\theta = \sqrt{(l_1 m_2-l_2 m_1)^2 + (m_1 n_2-m_2 n_1)^2 + (n_1 l_2-n_2 l_1)^2}$

Sine of angle between two lines using direction cosines

Angle between lines in vector form

$\cos\theta = \frac{|\vec{b_1} \cdot \vec{b_2}|}{|\vec{b_1}| \cdot |\vec{b_2}|}$

For lines r = a1 + λ*b1 and r = a2 + μ*b2

Condition for perpendicular lines (direction ratios)

$a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0$

Two lines are perpendicular when θ = 90 deg

Condition for parallel lines (direction ratios)

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Two lines are parallel when θ = 0

Shortest distance between skew lines (vector form)

$d = \frac{|(\vec{b_1} \times \vec{b_2}) \cdot (\vec{a_2} - \vec{a_1})|}{|\vec{b_1} \times \vec{b_2}|}$

For lines r = a1 + λ*b1 and r = a2 + μ*b2

Shortest distance between skew lines (Cartesian form)

$d = \frac{\begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix}}{\sqrt{(b_1 c_2-b_2 c_1)^2 + (c_1 a_2-c_2 a_1)^2 + (a_1 b_2-a_2 b_1)^2}}$

For lines (x-x1)/a1 = (y-y1)/b1 = (z-z1)/c1 and (x-x2)/a2 = (y-y2)/b2 = (z-z2)/c2

Distance between parallel lines

$d = \frac{|\vec{b} \times (\vec{a_2} - \vec{a_1})|}{|\vec{b}|}$

For parallel lines r = a1 + λ*b and r = a2 + μ*b

12 Linear Programming 5 marks

General Objective Function

$Z = ax + by$

a, b are constants; x, y are decision variables; Z is to be maximised or minimised

13 Probability 8 marks

Conditional Probability

$P(E|F) = \frac{P(E \cap F)}{P(F)}, \; P(F) \neq 0$

Also written as P(E|F) = n(E ∩ F) / n(F) for equally likely outcomes

Complement Rule

$P(A') = 1 - P(A)$

Probability of event not occurring

Addition Theorem

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

For any two events A and B

Mutually Exclusive Events

$P(A \cup B) = P(A) + P(B)$

When A ∩ B = ϕ (no common outcomes)

Property 1: P(S|F)

$P(S|F) = P(F|F) = 1$

The conditional probability of the sample space S given F is 1

Property 2: Addition rule for conditional probability

$P((A \cup B)|F) = P(A|F) + P(B|F) - P((A \cap B)|F)$

For disjoint events A and B: P((A ∪ B)|F) = P(A|F) + P(B|F)

Property 3: Complement rule

$P(E'|F) = 1 - P(E|F)$

Follows from P(S|F) = 1 and E, E' being disjoint with E ∪ E' = S

Multiplication Rule (two events)

$P(E \cap F) = P(E) \cdot P(F|E) = P(F) \cdot P(E|F)$

Provided P(E) ≠ 0 and P(F) ≠ 0

Multiplication Rule (three events)

$P(E \cap F \cap G) = P(E) \cdot P(F|E) \cdot P(G|E \cap F)$

Can be extended to four or more events similarly

Test for Independence

$P(E \cap F) = P(E) \cdot P(F)$

If this holds, E and F are independent events

Probability of at least one of two independent events

$P(A \cup B) = 1 - P(A') \cdot P(B')$

For independent events A and B

Three Independent Events

$P(A \cap B \cap C) = P(A) \cdot P(B) \cdot P(C)$

Extends to n mutually independent events

Theorem of Total Probability

$P(A) = \sum_{j=1}^{n} P(E_j) P(A|E_j)$

Where {E₁, E₂, ..., Eₙ} is a partition of S and each Eᵢ has nonzero probability

Bayes' Theorem (Simple Form)

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$

Gives posterior probability of A given B has occurred

Bayes' Theorem (General Form)

$P(E_i|A) = \frac{P(E_i) P(A|E_i)}{\sum_{j=1}^{n} P(E_j) P(A|E_j)}$

For partition {E₁, E₂, …, Eₙ} of S. Also called the formula for the probability of 'causes'.

Mean (Expected Value)

$E(X) = \mu = \sum_{i=1}^{n} x_i p_i$

xᵢ are values of X and pᵢ are corresponding probabilities

Variance

$\text{Var}(X) = E(X^2) - [E(X)]^2 = \sum x_i^2 p_i - \left(\sum x_i p_i\right)^2$

Also written as σ²

Variance (Alternative)

$\text{Var}(X) = \sum_{i=1}^{n} (x_i - \mu)^2 \cdot p_i$

Direct formula using deviations from the mean

Standard Deviation

$\sigma = \sqrt{\text{Var}(X)}$

Non-negative square root of the variance

Binomial Probability

$P(X = r) = \binom{n}{r} p^r q^{n-r}, \; q = 1 - p$

r = 0, 1, 2, ..., n. Here n = number of trials, p = probability of success, q = probability of failure

Mean of Binomial Distribution

$E(X) = np$

n = number of trials, p = probability of success

Variance of Binomial Distribution

$\text{Var}(X) = npq = np(1 - p)$

Always less than or equal to the mean np since q ≤ 1

Standard Deviation of Binomial Distribution

$\sigma = \sqrt{npq}$

Where q = 1 − p

Mathematics Formula Reference

⚡ Critical Formulas — Memorise These First