Math Quick Cheatsheet | Last Minute Hacks

Relation Counting Total Relations = $2^{n^2}$.
Reflexive Relations = $2^{n^2-n}$.

Principal Range $\sin^{-1}, \tan^{-1} \to [-\pi/2, \pi/2]$
$\cos^{-1}, \cot^{-1} \to [0, \pi]$

Symmetry $A = A^T$ (Symmetric).
$A = -A^T$ (Skew-Symmetric).

Properties $|adj A| = |A|^{n-1}$
$|kA| = k^n|A|$

Formula If $y = \sqrt{f(x) + \dots}$, $\frac{dy}{dx} = \frac{f'(x)}{2y-1}$

Tangent/Normal Slope of Normal = $-1 / (dy/dx)$.

Special Case $\int_0^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} dx = \pi/4$

Area (Parabola) Between $y^2=4ax, x^2=4by \implies 16ab/3$

Linear Solution $I.F. = e^{\int P dx}$. Sol: $y \cdot I.F. = \int Q \cdot I.F. dx + C$

Projection Proj of $\vec{a}$ on $\vec{b} = (\vec{a} \cdot \vec{b})/|\vec{b}|$

Shortest Distance $d = \frac{|(\vec{a}_2-\vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2)|}{|\vec{b}_1\times\vec{b}_2|}$

Feasible Region Check Corner Points for Max/Min value.

Bayes' Theorem $P(Ei|A) = \frac{P(Ei)P(A|Ei)}{\sum P(Ej)P(A|Ej)}$