Class 12 Maths formulas organised by unit, chapter and topic. Tap a card to see how and when to use each formula.
\[n(A)=m,\ n(B)=n \;\Rightarrow\; 2^{mn} \text{ relations from } A \text{ to } B\]
On a set with \(n\) elements there are \(2^{n^2}\) relations, \(2^{n^2-n}\) reflexive relations and \(2^{n(n+1)/2}\) symmetric relations.
\[a\equiv b \pmod n \iff n\mid(a-b)\]
An equivalence relation on \(\mathbb{Z}\) with exactly \(n\) classes: \([0],[1],\dots,[n-1]\).
\[\#(A\to B)=n^m,\qquad \#\text{one-one}=\frac{n!}{(n-m)!}\ (m\le n)\]
Bijections from an \(n\)-element set to itself: \(n!\). Onto functions from an \(m\)-set onto a 2-element set: \(2^m-2\).
| Function | Domain | Range (principal branch) |
|---|---|---|
| \(\sin^{-1}x\) | \([-1,1]\) | \([-\frac{\pi}{2},\frac{\pi}{2}]\) |
| \(\cos^{-1}x\) | \([-1,1]\) | \([0,\pi]\) |
| \(\tan^{-1}x\) | \(\mathbb{R}\) | \((-\frac{\pi}{2},\frac{\pi}{2})\) |
| \(\cot^{-1}x\) | \(\mathbb{R}\) | \((0,\pi)\) |
| \(\sec^{-1}x\) | \(\mathbb{R}-(-1,1)\) | \([0,\pi]-\{\frac{\pi}{2}\}\) |
| \(\csc^{-1}x\) | \(\mathbb{R}-(-1,1)\) | \([-\frac{\pi}{2},\frac{\pi}{2}]-\{0\}\) |
\[\sin^{-1}(-x)=-\sin^{-1}x,\quad \tan^{-1}(-x)=-\tan^{-1}x,\quad \csc^{-1}(-x)=-\csc^{-1}x\]\[\cos^{-1}(-x)=\pi-\cos^{-1}x,\quad \cot^{-1}(-x)=\pi-\cot^{-1}x,\quad \sec^{-1}(-x)=\pi-\sec^{-1}x\]
| x | 0 | 1/2 | 1/√2 | √3/2 | 1 |
|---|---|---|---|---|---|
| \(\sin^{-1}x\) | 0 | \(\frac\pi6\) | \(\frac\pi4\) | \(\frac\pi3\) | \(\frac\pi2\) |
| \(\cos^{-1}x\) | \(\frac\pi2\) | \(\frac\pi3\) | \(\frac\pi4\) | \(\frac\pi6\) | 0 |
\(\tan^{-1}1=\frac\pi4\), \(\tan^{-1}\sqrt3=\frac\pi3\), \(\tan^{-1}\frac{1}{\sqrt3}=\frac\pi6\).
\[\text{order } m\times n \;\Rightarrow\; mn \text{ entries}\]
Possible orders of a matrix with \(N\) entries correspond to factor pairs of \(N\). E.g. 12 entries: \(1\times12, 2\times6, 3\times4, 4\times3, 6\times2, 12\times1\).
\[(AB)_{ij} = \sum_{k=1}^{n} a_{ik}\,b_{kj}\]
Multiply row \(i\) of \(A\) by column \(j\) of \(B\) term by term and add. Requires columns of \(A\) = rows of \(B\).
\[(m\times n)(n\times p) = (m\times p)\]
The inner dimensions must match; the outer ones give the order of the product.
\[(AB)^{T} = B^{T}A^{T}\]
Also \((A^T)^T = A\), \((A+B)^T = A^T + B^T\), \((kA)^T = kA^T\).
\[(A^T)^T=A,\quad (kA)^T=kA^T,\quad (A+B)^T=A^T+B^T,\quad (AB)^T=B^TA^T\]
\[A = \tfrac12(A + A^{T}) + \tfrac12(A - A^{T})\]
The first part is symmetric, the second skew-symmetric. Works for every square matrix.
\[\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\]
Valid only when \(ad - bc \ne 0\). Swap the diagonal, negate the off-diagonal, divide by the determinant.
\[(AB)^{-1} = B^{-1}A^{-1}\]
Also \((A^{-1})^{-1} = A\) and \((A^T)^{-1} = (A^{-1})^T\).
\[\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc\]
Main diagonal product minus the other diagonal product.
\[|kA| = k^{n}|A| \quad (A \text{ of order } n)\]
Each of the \(n\) rows contributes a factor \(k\). Also \(|AB| = |A||B|\) and \(|A^T| = |A|\).
\[|A^T|=|A|,\quad |AB|=|A|\,|B|,\quad |kA|=k^n|A|,\quad |A^{-1}|=\frac{1}{|A|}\]
\[\Delta = \frac12\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right|\]
Zero determinant ⇔ the three points are collinear.
\[A_{ij} = (-1)^{i+j}M_{ij}\]
\(M_{ij}\): determinant left after deleting row \(i\) and column \(j\).
\[A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|\,I\]
Consequences: \(A^{-1} = \dfrac{\operatorname{adj}A}{|A|}\) and \(|\operatorname{adj}A| = |A|^{n-1}\).
\[A^{-1} = \frac{1}{|A|}\operatorname{adj}A, \quad |A| \ne 0\]
Also \(|A^{-1}| = 1/|A|\).
\[A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|I,\qquad |\operatorname{adj}A|=|A|^{n-1}\]
\[AX = B \;\Rightarrow\; X = A^{-1}B \quad (|A| \ne 0)\]
Note the order: \(A^{-1}\) multiplies \(B\) from the left.
\[(x^n)'=nx^{n-1},\ (\sin x)'=\cos x,\ (\cos x)'=-\sin x,\ (\tan x)'=\sec^2x\]\[(\sec x)'=\sec x\tan x,\ (\csc x)'=-\csc x\cot x,\ (\cot x)'=-\csc^2x\]
\[\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\]
\[(\sin^{-1}x)'=\frac{1}{\sqrt{1-x^2}},\ (\tan^{-1}x)'=\frac1{1+x^2},\ (\sec^{-1}x)'=\frac{1}{|x|\sqrt{x^2-1}}\]
\[(e^x)'=e^x,\ (a^x)'=a^x\log a,\ (\log x)'=\frac1x\]
\[\frac{d}{dx}\,u^v=u^v\left(v'\log u+\frac{v\,u'}{u}\right)\]
\[\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\]
\[\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}\]
Area of circle: dA/dt = 2πr dr/dt. Volume of sphere: dV/dt = 4πr² dr/dt. Cube: dV/dt = 3a² da/dt.
\[f'(c)=0:\quad f''(c)\lt 0\Rightarrow\text{local max},\qquad f''(c)\gt 0\Rightarrow\text{local min}\]
| \(f(x)\) | \(\int f(x)\,dx\) |
|---|---|
| \(x^n\ (n\neq-1)\) | \(\frac{x^{n+1}}{n+1}+C\) |
| \(\frac1x\) | \(\log|x|+C\) |
| \(e^x\) | \(e^x+C\) |
| \(a^x\) | \(\frac{a^x}{\log a}+C\) |
| \(\sin x\) | \(-\cos x+C\) |
| \(\cos x\) | \(\sin x+C\) |
| \(\sec^2x\) | \(\tan x+C\) |
| \(\csc^2x\) | \(-\cot x+C\) |
| \(\sec x\tan x\) | \(\sec x+C\) |
| \(\frac{1}{\sqrt{1-x^2}}\) | \(\sin^{-1}x+C\) |
| \(\frac1{1+x^2}\) | \(\tan^{-1}x+C\) |
\[\int\tan x=\log|\sec x|,\ \int\cot x=\log|\sin x|,\ \int\sec x=\log|\sec x+\tan x|,\ \int\csc x=\log|\csc x-\cot x|\]
| Integral | Result |
|---|---|
| \(\int\frac{dx}{x^2-a^2}\) | \(\frac1{2a}\log\left|\frac{x-a}{x+a}\right|+C\) |
| \(\int\frac{dx}{a^2-x^2}\) | \(\frac1{2a}\log\left|\frac{a+x}{a-x}\right|+C\) |
| \(\int\frac{dx}{x^2+a^2}\) | \(\frac1a\tan^{-1}\frac xa+C\) |
| \(\int\frac{dx}{\sqrt{x^2-a^2}}\) | \(\log\left|x+\sqrt{x^2-a^2}\right|+C\) |
| \(\int\frac{dx}{\sqrt{a^2-x^2}}\) | \(\sin^{-1}\frac xa+C\) |
| \(\int\frac{dx}{\sqrt{x^2+a^2}}\) | \(\log\left|x+\sqrt{x^2+a^2}\right|+C\) |
\[\int uv\,dx=u\int v\,dx-\int\left(u'\int v\,dx\right)dx\]
ILATE: Inverse trig, Log, Algebraic, Trig, Exponential.
\[\int e^x\left[f(x)+f\,'(x)\right]dx=e^xf(x)+C\]
\(\int_0^af(x)dx=\int_0^af(a-x)dx\); \(\int_{-a}^af=2\int_0^af\) (even) or 0 (odd).
\[A=\int_a^b y\,dx\quad\text{or}\quad A=\int_c^d x\,dy\]
\[\int_0^a\sqrt{a^2-x^2}\,dx=\frac{\pi a^2}{4},\qquad \text{ellipse area}=\pi ab\]
\[y^2=4ax,\ x=a:\quad A=\frac{8a^2}{3}\]
\[\frac{dP}{dt}=kP\;\Rightarrow\;P=P_0e^{kt}\]
\[y=vx,\quad \frac{dy}{dx}=v+x\frac{dv}{dx}\]
\[\frac{dy}{dx}+Py=Q:\quad \text{IF}=e^{\int P\,dx},\quad y\,\text{IF}=\int Q\,\text{IF}\,dx+C\]
\[|\vec a|=\sqrt{a_1^2+a_2^2+a_3^2},\qquad \hat a=\frac{\vec a}{|\vec a|}\]
\[\vec r=\frac{m\vec b+n\vec a}{m+n}\]
\[\vec a\cdot\vec b=a_1b_1+a_2b_2+a_3b_3,\quad \text{proj}_{\vec b}\vec a=\frac{\vec a\cdot\vec b}{|\vec b|}\]
\[|\vec a\times\vec b|=|\vec a||\vec b|\sin\theta,\quad \Delta=\tfrac12|\vec a\times\vec b|\]
\[l=\frac{a}{\sqrt{a^2+b^2+c^2}},\ m=\frac{b}{\sqrt{a^2+b^2+c^2}},\ n=\frac{c}{\sqrt{a^2+b^2+c^2}}\]
\[\vec r=\vec a+\lambda\vec b,\qquad \frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\]
\[\cos\theta=\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{\sum a_1^2}\sqrt{\sum a_2^2}}\]
\[d=\frac{|(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)|}{|\vec b_1\times\vec b_2|}\]
\[\text{Optimise } Z=px+qy\ \text{ subject to }\ a_ix+b_iy\ (\le,\ \ge)\ c_i,\quad x\ge0,\ y\ge0\]
If an LPP has an optimal value, it is attained at a corner point of the feasible region. A bounded feasible region always has both a maximum and a minimum value of \(Z\).
\[P(A\mid B)=\frac{P(A\cap B)}{P(B)}\]
\[P(A\cap B)=P(A)P(B\mid A);\quad \text{independent: }P(A\cap B)=P(A)P(B)\]
\[P(E_i\mid A)=\frac{P(E_i)P(A\mid E_i)}{\sum_jP(E_j)P(A\mid E_j)}\]
\[E(X)=\sum x_ip_i,\qquad \sum p_i=1\]
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