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CBSE Class 12 MathematicsAll chapters · one-page formula sheet · 2025–26 rationalised syllabus
CBSE gives no formula booklet: learn every result on this sheet. Organised by NCERT chapter.
1 Relations & Functions
Reflexive \((a,a)\in R\ \forall a\); symmetric \((a,b)\in R\Rightarrow(b,a)\in R\); transitive \((a,b),(b,c)\in R\Rightarrow(a,c)\in R\); equivalence = all three
One-one: \(f(x_1)=f(x_2)\Rightarrow x_1=x_2\). Onto: range = codomain. Bijective: both
Counting, \(|A|=m\), \(|B|=n\): \(\text{relations on }A:\ 2^{m^2};\) \(\text{functions }A\to B:\ n^m\)
One-one \(A\to B\) (\(m\le n\)); bijections (\(m=n\)): \(\frac{n!}{(n-m)!};\) \(n!\)
2 Inverse Trigonometric Functions
Principal ranges: \(\sin^{-1}:[-1,1]\to\left[-\tfrac\pi2,\tfrac\pi2\right],\) \(\cos^{-1}:[-1,1]\to[0,\pi],\) \(\tan^{-1}:\mathbb R\to\left(-\tfrac\pi2,\tfrac\pi2\right)\)
More ranges: \(\cot^{-1}:\mathbb R\to(0,\pi),\) \(\sec^{-1}:[0,\pi]-\{\tfrac\pi2\},\) \(\cosec^{-1}:\left[-\tfrac\pi2,\tfrac\pi2\right]-\{0\}\)
\(\sec^{-1}\), \(\cosec^{-1}\) domain: \(\mathbb R-(-1,1)\)
\(\sin(\sin^{-1}x)=x\) for \(x\in[-1,1]\); \(\sin^{-1}(\sin x)=x\) only for \(x\in[-\tfrac\pi2,\tfrac\pi2]\) (similarly for the others)
3 Matrices
Product \(A_{m\times n}B_{n\times p}\) is \(m\times p\); in general \(AB\ne BA\); \(AB=O\) is possible with \(A,B\ne O\)
Transpose: \((A^T)^T=A,\) \((A+B)^T=A^T+B^T,\) \((kA)^T=kA^T,\) \((AB)^T=B^TA^T\)
Symmetric \(A^T=A\); skew-symmetric \(A^T=-A\) (diagonal entries 0)
Any square \(A\): \(A=\tfrac12(A+A^T)+\tfrac12(A-A^T)\)
Inverse: \(AB=BA=I\Rightarrow B\) \({}=A^{-1};\) \((AB)^{-1}=B^{-1}A^{-1}\)
4 Determinants
\(2\times2\): \(\begin{vmatrix}a&b\\c&d\end{vmatrix}\) \({}=ad-bc\)
Minor; cofactor: \(M_{ij};\) \(A_{ij}=(-1)^{i+j}M_{ij}\)
\(|A|=\sum_j a_{ij}A_{ij}\) along any row (or column); elements × cofactors of a different row sum to 0
Area of triangle: \(\tfrac12\left|\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}\right|;\) \(\text{collinear}\iff\text{det}\) \({}=0\)
Adjoint = transpose of the cofactor matrix: \(A(\operatorname{adj}A)=(\operatorname{adj}A)A\) \({}=|A|I\)
Inverse: \(A^{-1}=\frac{1}{|A|}\operatorname{adj}A\) \((|A|\ne0)\)
Order \(n\): \(|AB|=|A||B|,\) \(|A^T|=|A|,\) \(|kA|=k^n|A|,\) \(|\operatorname{adj}A|=|A|^{n-1},\) \(|A^{-1}|=\tfrac1{|A|}\)
\(AX=B\): \(|A|\ne0:\ X=A^{-1}B\ \text{(unique)}\)
\(|A|=0\): \((\operatorname{adj}A)B\ne O\) ⇒ inconsistent; \((\operatorname{adj}A)B=O\) ⇒ infinitely many or none
5 Continuity & Differentiability
Continuous at \(c\): \(\lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)\) \({}=f(c)\)
Differentiable at \(c\): LHD = RHD; differentiable ⇒ continuous (not conversely, e.g. \(|x|\) at 0)
Standard: \((x^n)'=nx^{n-1},\) \((\sin x)'=\cos x,\) \((\cos x)'=-\sin x,\) \((\tan x)'=\sec^2x\)
More trig: \((\cot x)'=-\cosec^2x,\) \((\sec x)'=\sec x\tan x,\) \((\cosec x)'=-\cosec x\cot x\)
Exp, log: \((e^x)'=e^x,\) \((a^x)'=a^x\log a,\) \((\log x)'=\tfrac1x,\) \((\log_ax)'=\tfrac1{x\log a}\)
Inverse trig: \((\sin^{-1}x)'=\tfrac1{\sqrt{1-x^2}},\) \((\cos^{-1}x)'=-\tfrac1{\sqrt{1-x^2}},\) \((\tan^{-1}x)'=\tfrac1{1+x^2}\)
Rules: \((uv)'=u'v+uv',\) \(\Big(\frac uv\Big)'=\frac{u'v-uv'}{v^2},\) \(\frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx}\)
Logarithmic, \(y=u^v\): \(\log y=v\log u\Rightarrow\frac1y\frac{dy}{dx}\) \({}=v'\log u+\frac{vu'}u\)
Parametric: \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt},\) \(\frac{d^2y}{dx^2}=\frac{\frac d{dt}\left(\frac{dy}{dx}\right)}{dx/dt}\)
Implicit: differentiate term by term, \(\frac{d}{dx}(y^2)=2y\frac{dy}{dx}\)
Log means natural log (base \(e\)) in NCERT.
6 Application of Derivatives
Rate of change: \(\frac{dA}{dt}=\frac{dA}{dr}\cdot\frac{dr}{dt}\)
\(f'(x)>0\) on \((a,b)\): strictly increasing; \(f'(x)<0\): strictly decreasing
Critical point: \(f'(c)=0\) or \(f'(c)\) undefined
First derivative test: \(f'\) changes + → − (max), − → + (min), no change (neither)
Second derivative test: \(f'(c)=0:\) \(f''(c)<0\text{ max},\) \(f''(c)>0\text{ min},\) \(f''(c)=0\text{ test fails}\)
Absolute max/min on \([a,b]\): compare \(f\) at critical points and at \(a\), \(b\)
7 Integrals
Basic: \(\int x^n\,dx=\tfrac{x^{n+1}}{n+1}\) \({}+C\ (n\ne-1),\) \(\int\tfrac1x\,dx=\log|x|+C\)
Exp: \(\int e^x\,dx=e^x+C,\) \(\int a^x\,dx=\tfrac{a^x}{\log a}+C\)
Trig: \(\int\sin x\,dx=-\cos x+C,\) \(\int\cos x\,dx=\sin x+C,\) \(\int\sec^2x\,dx=\tan x+C\)
Trig: \(\int\cosec^2x\,dx=-\cot x+C,\) \(\int\sec x\tan x\,dx=\sec x+C,\) \(\int\cosec x\cot x\,dx=-\cosec x+C\)
Trig: \(\int\tan x\,dx=\log|\sec x|+C,\) \(\int\cot x\,dx=\log|\sin x|+C\)
sec, cosec: \(\int\sec x\,dx=\log|\sec x+\tan x|+C,\) \(\int\cosec x\,dx\) \({}=\log|\cosec x-\cot x|+C\)
Inverse trig: \(\int\frac{dx}{\sqrt{1-x^2}}\) \({}=\sin^{-1}x+C,\) \(\int\frac{dx}{1+x^2}=\tan^{-1}x+C\)
Special: \(\int\frac{dx}{x^2-a^2}=\frac1{2a}\log\left|\frac{x-a}{x+a}\right|+C,\) \(\int\frac{dx}{a^2-x^2}=\frac1{2a}\log\left|\frac{a+x}{a-x}\right|+C\)
Special: \(\int\frac{dx}{x^2+a^2}=\frac1a\tan^{-1}\frac xa+C,\) \(\int\frac{dx}{\sqrt{a^2-x^2}}\) \({}=\sin^{-1}\frac xa+C\)
Special: \(\int\frac{dx}{\sqrt{x^2\pm a^2}}\) \({}=\log\left|x+\sqrt{x^2\pm a^2}\right|+C\)
Root: \(\int\sqrt{a^2-x^2}\,dx=\frac x2\sqrt{a^2-x^2}\) \({}+\frac{a^2}2\sin^{-1}\frac xa+C\)
Root: \(\int\sqrt{x^2\pm a^2}\,dx\) \({}=\frac x2\sqrt{x^2\pm a^2}\) \({}\pm\frac{a^2}2\log\left|x+\sqrt{x^2\pm a^2}\right|+C\)
By parts (ILATE for the first function \(u\)): \(\int uv\,dx=u\int v\,dx\) \({}-\int\Big(u'\int v\,dx\Big)dx\)
Special form: \(\int e^x\big[f(x)+f'(x)\big]dx\) \({}=e^xf(x)+C\)
\(ax^2+bx+c\): complete the square; for \(\frac{px+q}{ax^2+bx+c}\) write \(px+q=A\frac{d}{dx}(ax^2+bx+c)+B\)
Partial fractions: \(\frac{px+q}{(x-a)(x-b)}=\frac A{x-a}\) \({}+\frac B{x-b},\) \(\frac{\dots}{(x-a)^2(x-b)}\) \({}=\frac A{x-a}+\frac B{(x-a)^2}\) \({}+\frac C{x-b}\)
Quadratic factor: \(\frac{\dots}{(x-a)(x^2+bx+c)}\) \({}=\frac A{x-a}+\frac{Bx+C}{x^2+bx+c}\)
Fundamental theorem: \(\int_a^bf(x)\,dx=F(b)-F(a)\)
Properties: \(\int_a^bf=-\int_b^af,\) \(\int_a^bf=\int_a^cf+\int_c^bf,\) \(\int_a^bf(x)\,dx=\int_a^bf(a+b-x)\,dx\)
Properties: \(\int_0^af(x)\,dx=\int_0^af(a-x)\,dx\)
Properties: \(\int_0^{2a}f\,dx=2\int_0^af\,dx\text{ if }f(2a-x)\) \({}=f(x),\) \(0\text{ if }f(2a-x)=-f(x)\)
Even / odd: \(\int_{-a}^af\,dx=2\int_0^af\,dx\text{ (even)},\) \(0\text{ (odd)}\)
8 Application of Integrals
Area: \(\int_a^b|y|\,dx,\) \(\int_c^d|x|\,dy\)
Circle \(x^2+y^2=a^2\): area \(\pi a^2\); ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\): area \(\pi ab\)
Region between a curve and a line: integrate (upper − lower) between the intersection points
9 Differential Equations
Order = highest derivative; degree = power of it (when polynomial in derivatives). General solution has as many constants as the order
Variables separable: \(\int\frac{dy}{g(y)}=\int f(x)\,dx+C\)
Homogeneous \(\frac{dy}{dx}=F\big(\frac yx\big)\): \(y=vx,\) \(\frac{dy}{dx}=v+x\frac{dv}{dx}\)
Linear \(\frac{dy}{dx}+Py=Q\): \(\text{I.F.}=e^{\int P\,dx},\) \(y\cdot\text{I.F.}=\int Q\cdot\text{I.F.}\,dx+C\)
Linear \(\frac{dx}{dy}+P_1x=Q_1\): \(\text{I.F.}=e^{\int P_1\,dy},\) \(x\cdot\text{I.F.}=\int Q_1\cdot\text{I.F.}\,dy+C\)
10 Vector Algebra
Magnitude; unit vector: \(|\vec a|=\sqrt{x^2+y^2+z^2},\) \(\hat a=\frac{\vec a}{|\vec a|}\)
Direction cosines: \(l=\tfrac x{|\vec r|},\ m\) \({}=\tfrac y{|\vec r|},\ n\) \({}=\tfrac z{|\vec r|},\) \(l^2+m^2+n^2=1\)
\(\overrightarrow{PQ}\): \((x_2-x_1)\hat i+(y_2-y_1)\hat j\) \({}+(z_2-z_1)\hat k\)
Section, \(m:n\) internal; external: \(\frac{m\vec b+n\vec a}{m+n};\) \(\frac{m\vec b-n\vec a}{m-n}\)
Dot product: \(\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta\) \({}=a_1b_1+a_2b_2+a_3b_3\)
Perpendicular; projection of \(\vec a\) on \(\vec b\): \(\vec a\cdot\vec b=0;\) \(\frac{\vec a\cdot\vec b}{|\vec b|}\)
Cross product: \(\vec a\times\vec b=|\vec a||\vec b|\sin\theta\,\hat n\) \({}=\begin{vmatrix}\hat i&\hat j&\hat k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}\)
Area: parallelogram; triangle: \(|\vec a\times\vec b|;\) \(\tfrac12|\vec a\times\vec b|\)
Parallelogram from its diagonals: \(\tfrac12|\vec d_1\times\vec d_2|\)
Parallel \(\vec a\times\vec b=\vec 0\); \(\vec a\times\vec b=-\vec b\times\vec a\); \(\hat i\times\hat j=\hat k\), \(\hat j\times\hat k=\hat i\), \(\hat k\times\hat i=\hat j\)
11 Three Dimensional Geometry
DCs from direction ratios \(a,b,c\): \(l=\frac{a}{\sqrt{a^2+b^2+c^2}},\) \(\dots\)
Line through \(\vec a\) parallel to \(\vec b\): \(\vec r=\vec a+\lambda\vec b;\) \(\frac{x-x_1}a=\frac{y-y_1}b\) \({}=\frac{z-z_1}c\)
Through two points: \(\vec r=\vec a+\lambda(\vec b-\vec a);\) \(\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}\) \({}=\frac{z-z_1}{z_2-z_1}\)
Angle between lines: \(\cos\theta=\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\)
Perpendicular; parallel: \(a_1a_2+b_1b_2+c_1c_2=0;\) \(\tfrac{a_1}{a_2}=\tfrac{b_1}{b_2}\) \({}=\tfrac{c_1}{c_2}\)
Shortest distance, skew lines: \(d=\frac{\big|(\vec b_1\times\vec b_2)\cdot(\vec a_2-\vec a_1)\big|}{|\vec b_1\times\vec b_2|}\)
Parallel lines: \(d=\frac{|\vec b\times(\vec a_2-\vec a_1)|}{|\vec b|}\)
Lines intersect ⇔ shortest distance = 0. Foot of perpendicular from \(P\): general point \(Q\) on the line with \(\overrightarrow{PQ}\cdot\vec b=0\)
12 Linear Programming
Maximise/minimise \(Z=ax+by\) subject to linear constraints, \(x,y\ge0\)
Corner point method: the optimum occurs at a corner of the feasible region. Bounded region: both max and min exist
Unbounded: \(M\) is the max only if \(ax+by>M\) has no point in the feasible region (use \(<m\) for the min)
13 Probability
Conditional: \(P(E\mid F)=\frac{P(E\cap F)}{P(F)},\) \(P(F)\ne0\)
Properties: \(P(S\mid F)=1,\) \(P(E'\mid F)=1-P(E\mid F)\)
Multiplication rule: \(P(E\cap F)=P(E)P(F\mid E)\) \({}=P(F)P(E\mid F)\)
Independent: \(P(E\cap F)=P(E)P(F)\)
Union: \(P(E\cup F)=P(E)+P(F)-P(E\cap F)\)
Total probability (partition \(E_1,\dots,E_n\)): \(P(A)=\sum_{j=1}^nP(E_j)P(A\mid E_j)\)
Bayes' theorem: \(P(E_i\mid A)=\frac{P(E_i)P(A\mid E_i)}{\sum_{j=1}^nP(E_j)P(A\mid E_j)}\)
Random variable: \(p_i\ge0,\) \(\sum p_i=1,\) \(E(X)=\mu=\sum x_ip_i\)