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CBSE Class 12 MathematicsAll chapters · one-page formula sheet · 2026–27 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 threeUse: Reflexive, symmetric or transitive: a counter-example when one fails
One-one: \(f(x_1)=f(x_2)\Rightarrow x_1=x_2\). Onto: range = codomain. Bijective: bothUse: Decide whether functions are one-one, onto, both or neitherWatch for: Prove one-one and onto algebraically, not by examples
Counting, \(|A|=m\), \(|B|=n\): \(\text{relations on }A:\ 2^{m^2};\) \(\text{functions }A\to B:\ n^m\)Use: Count relations and functions between finite sets
One-one \(A\to B\) (\(m\le n\)); bijections (\(m=n\)): \(\frac{n!}{(n-m)!};\) \(n!\)Use: Count one-one functions and bijections
Watch for: One-one: \(f(a) = f(b) \Rightarrow a = b\)
2 Inverse Trigonometric Functions
Principal ranges: \(\sin^{-1}:[-1,1]\to\left[-\tfrac\pi2,\tfrac\pi2\right],\) \(\cos^{-1}:[-1,1]\to[0,\pi],\) \(\tan^{-1}:\mathbb R\to\left(-\tfrac\pi2,\tfrac\pi2\right)\)Use: Principal values such as \(\sin^{-1}(-\tfrac{1}{2})\) and \(\cos^{-1}(-\frac{\sqrt{3}}{2})\)Watch for: Know the ranges: \(\sin^{-1}\) in \([-\frac{\pi}{2}, \frac{\pi}{2}]\), \(\cos^{-1}\) in \([0, \pi]\)
More ranges: \(\cot^{-1}:\mathbb R\to(0,\pi),\) \(\sec^{-1}:[0,\pi]\) \({}-\{\tfrac\pi2\},\) \(\cosec^{-1}:\left[-\tfrac\pi2,\tfrac\pi2\right]-\{0\}\)Use: Domain and range of each inverse trig function
\(\sec^{-1}\), \(\cosec^{-1}\) domain: \(\mathbb R-(-1,1)\)Use: Domain and range of each inverse trig function
\(\sin(\sin^{-1}x)=x\) for \(x\in[-1,1]\); \(\sin^{-1}(\sin x)=x\) only for \(x\in[-\tfrac\pi2,\tfrac\pi2]\) (similarly for the others)Use: Compositions such as \(\sin^{-1}(\sin x)\) outside the principal branch
Watch for: Know the principal value ranges
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 \(A,B\ne O\)Use: Multiply matrices when the product is defined
Transpose: \((A^T)^T=A,\) \((A+B)^T=A^T+B^T,\) \((kA)^T=kA^T,\) \((AB)^T=B^TA^T\)Use: Verify results such as \((AB)' = B'A'\)
Symmetric \(A^T=A\); skew-symmetric \(A^T=-A\) (diagonal entries 0)Use: Identify symmetric and skew-symmetric matrices
Any square \(A\): \(A=\tfrac12(A+A^T)\) \({}+\tfrac12(A-A^T)\)Use: Write a square matrix as symmetric plus skew-symmetric
Inverse: \(AB=BA=I\Rightarrow B\) \({}=A^{-1};\) \((AB)^{-1}=B^{-1}A^{-1}\)Use: Check whether two matrices are inverses of each other
4 Determinants
\(2\times2\): \(\begin{vmatrix}a&b\\c&d\end{vmatrix}\) \({}=ad-bc\)Use: Evaluate determinants of \(2 \times 2\) and \(3 \times 3\) matrices
Minor; cofactor: \(M_{ij};\) \(A_{ij}=(-1)^{i+j}M_{ij}\)Use: Minors and cofactors to expand a determinantWatch for: Cofactor signs follow the \(+ - +\) pattern
\(|A|=\sum_j a_{ij}A_{ij}\) along any row (or column); elements × cofactors of a different row sum to 0Use: Expand along the row or column with most zeroes
Area of triangle: \(\tfrac12\left|\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}\right|;\) \(\text{collinear}\iff\text{det}\) \({}=0\)Use: Area of a triangle from its vertices; zero area means collinear
Adjoint = transpose of the cofactor matrix: \(A(\operatorname{adj}A)=(\operatorname{adj}A)A\) \({}=|A|I\)Use: Find the adjoint of a square matrix
Inverse: \(A^{-1}=\frac{1}{|A|}\operatorname{adj}A\) \((|A|\ne0)\)Use: Inverse when the determinant is non-zeroWatch for: Check \(|A| \ne 0\), then divide adj \(A\) by |A|
Order \(n\): \(|AB|=|A||B|,\) \(|A^T|=|A|,\) \(|kA|=k^n|A|,\) \(|\operatorname{adj}A|=|A|^{n-1},\) \(|A^{-1}|=\tfrac1{|A|}\)Use: Simplify expressions and find unknown determinants
\(AX=B\): \(|A|\ne0:\ X=A^{-1}B\ \text{(unique)}\)Use: Solve three equations by the matrix method \(X = A^{-1}B\)
\(|A|=0\): \((\operatorname{adj}A)B\ne O\) ⇒ inconsistent; \((\operatorname{adj}A)B=O\) ⇒ infinitely many or noneUse: Decide whether a system is consistent or inconsistent
Watch for: Check \(|A| \ne 0\) and the cofactor signs
5 Continuity & Differentiability
Continuous at \(c\): \(\lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)\) \({}=f(c)\)Use: Test continuity by comparing left-hand and right-hand limits
Differentiable at \(c\): LHD = RHD; differentiable ⇒ continuous (not conversely, e.g. \(|x|\) at 0)Use: Test differentiability with left-hand and right-hand derivatives
Standard: \((x^n)'=nx^{n-1},\) \((\sin x)'=\cos x,\) \((\cos x)'=-\sin x,\) \((\tan x)'=\sec^2x\)Use: Differentiate composite functions with the chain rule
More trig: \((\cot x)'=-\cosec^2x,\) \((\sec x)'=\sec x\tan x,\) \((\cosec x)'=-\cosec x\cot x\)Use: Differentiate composite trigonometric functions
Exp, log: \((e^x)'=e^x,\) \((a^x)'=a^x\log a,\) \((\log x)'=\tfrac1x,\) \((\log_ax)'=\tfrac1{x\log a}\)Use: Differentiate \(e^{3x}\) and \(\log(\sin x)\)
Inverse trig: \((\sin^{-1}x)'=\tfrac1{\sqrt{1-x^2}},\) \((\cos^{-1}x)'=-\tfrac1{\sqrt{1-x^2}},\) \((\tan^{-1}x)'=\tfrac1{1+x^2}\)Use: Differentiate \(\sin^{-1}(2x)\) and \(\tan^{-1}(\sqrt{x})\)
Rules: \((uv)'=u'v+uv',\) \(\Big(\frac uv\Big)'=\frac{u'v-uv'}{v^2},\) \(\frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx}\)Use: Write each step of the chain rule clearly
Logarithmic, \(y=u^v\): \(\log y=v\log u\Rightarrow\frac1y\frac{dy}{dx}\) \({}=v'\log u+\frac{vu'}u\)Use: Differentiate \(x^{x}\) and products of several factors
Parametric: \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt},\) \(\frac{d^2y}{dx^2}=\frac{\frac d{dt}\left(\frac{dy}{dx}\right)}{dx/dt}\)Use: \(\frac{dy}{dx}\) for curves in parametric form
Implicit: differentiate term by term, \(\frac{d}{dx}(y^2)=2y\frac{dy}{dx}\)Use: \(\frac{dy}{dx}\) in terms of \(x\) and \(y\) for implicit functions
Log means natural log (base \(e\)) in NCERT.
6 Application of Derivatives
Rate of change: \(\frac{dA}{dt}=\frac{dA}{dr}\cdot\frac{dr}{dt}\)Use: Rate of change, such as how fast a circle's area grows
\(f'(x)>0\) on \((a,b)\): strictly increasing; \(f'(x)<0\): strictly decreasingUse: Intervals where a function is strictly increasing or decreasing
Critical point: \(f'(c)=0\) or \(f'(c)\) undefinedUse: Local maxima and minimaWatch for: Solve \(f'(x) = 0\) fully before testing
First derivative test: \(f'\) changes + → − (max), − → + (min), no change (neither)Use: Local maxima and minima by the first derivative test
Second derivative test: \(f'(c)=0:\) \(f''(c)<0\text{ max},\) \(f''(c)>0\text{ min},\) \(f''(c)=0\text{ test fails}\)Use: Optimisation: maximise a box's volume or minimise a costWatch for: Apply the test to each point; answer in context
Absolute max/min on \([a,b]\): compare \(f\) at critical points and at \(a\), \(b\)Use: Absolute maximum and minimum on a closed interval
Watch for: Solve \(f'(x) = 0\) fully, then test each point
7 Integrals
Basic: \(\int x^n\,dx=\tfrac{x^{n+1}}{n+1}\) \({}+C\ (n\ne-1),\) \(\int\tfrac1x\,dx=\log|x|+C\)Use: Indefinite integrals from the standard formulaeWatch for: Write \(+ C\) on every indefinite integral
Exp: \(\int e^x\,dx=e^x+C,\) \(\int a^x\,dx=\tfrac{a^x}{\log a}+C\)Use: Indefinite integrals from the standard formulae
Trig: \(\int\sin x\,dx=-\cos x+C,\) \(\int\cos x\,dx=\sin x+C,\) \(\int\sec^2x\,dx=\tan x+C\)Use: Integrate \(\sin^{2} x\) and \(\cos^{3} x\) with trigonometric identities
Trig: \(\int\cosec^2x\,dx=-\cot x+C,\) \(\int\sec x\tan x\,dx=\sec x+C,\) \(\int\cosec x\cot x\,dx=-\cosec x+C\)Use: Indefinite integrals from the standard formulae
Trig: \(\int\tan x\,dx=\log|\sec x|+C,\) \(\int\cot x\,dx=\log|\sin x|+C\)Use: Indefinite integrals from the standard formulae
sec, cosec: \(\int\sec x\,dx=\log|\sec x\) \({}+\tan x|+C,\) \(\int\cosec x\,dx\) \({}=\log|\cosec x-\cot x|+C\)Use: Indefinite integrals from the standard formulae
Inverse trig: \(\int\frac{dx}{\sqrt{1-x^2}}\) \({}=\sin^{-1}x+C,\) \(\int\frac{dx}{1+x^2}=\tan^{-1}x+C\)Use: Special forms such as \(\frac{1}{x^{2} + a^{2}}\) and \(\frac{1}{\sqrt{a^{2} - x^{2}}}\)
Special: \(\int\frac{dx}{x^2-a^2}=\frac1{2a}\log\left|\frac{x-a}{x+a}\right|+C,\) \(\int\frac{dx}{a^2-x^2}=\frac1{2a}\log\left|\frac{a+x}{a-x}\right|+C\)Use: Integrals that match the special standard forms
Special: \(\int\frac{dx}{x^2+a^2}=\frac1a\tan^{-1}\frac xa+C,\) \(\int\frac{dx}{\sqrt{a^2-x^2}}\) \({}=\sin^{-1}\frac xa+C\)Use: Integrals that match the special standard forms
Special: \(\int\frac{dx}{\sqrt{x^2\pm a^2}}\) \({}=\log\left|x+\sqrt{x^2\pm a^2}\right|+C\)Use: Complete the square to reach a special standard form
Root: \(\int\sqrt{a^2-x^2}\,dx=\frac x2\sqrt{a^2-x^2}\) \({}+\frac{a^2}2\sin^{-1}\frac xa+C\)Use: Complete the square under a root, then integrate
Root: \(\int\sqrt{x^2\pm a^2}\,dx\) \({}=\frac x2\sqrt{x^2\pm a^2}\) \({}\pm\frac{a^2}2\log\left|x+\sqrt{x^2\pm a^2}\right|+C\)Use: Complete the square under a root, then integrate
By parts (ILATE for the first function \(u\)): \(\int uv\,dx=u\int v\,dx\) \({}-\int\Big(u'\int v\,dx\Big)dx\)Use: Integrate \(x e^{x}\) and \(x \sin x\) by parts
Special form: \(\int e^x\big[f(x)+f'(x)\big]dx\) \({}=e^xf(x)+C\)Use: Integrals of \(e^{x}[f(x) + f'(x)]\)
\(ax^2+bx+c\): complete the square; for \(\frac{px+q}{ax^2+bx+c}\) write \(px+q=A\frac{d}{dx}(ax^2+bx+c)+B\)Use: Complete the square in a quadratic denominator
Partial fractions: \(\frac{px+q}{(x-a)(x-b)}=\frac A{x-a}\) \({}+\frac B{x-b},\) \(\frac{\dots}{(x-a)^2(x-b)}\) \({}=\frac A{x-a}+\frac B{(x-a)^2}\) \({}+\frac C{x-b}\)Use: Rational functions with linear and repeated linear factorsWatch for: Repeated \((x - a)^{2}\) needs \(\frac{A}{x - a} + \frac{B}{(x - a)^{2}}\)
Quadratic factor: \(\frac{\dots}{(x-a)(x^2+bx+c)}\) \({}=\frac A{x-a}+\frac{Bx+C}{x^2+bx+c}\)Use: Partial fractions with a quadratic factorWatch for: An irreducible quadratic needs \((Bx + C)\) on top
Fundamental theorem: \(\int_a^bf(x)\,dx=F(b)-F(a)\)Use: Evaluate definite integrals by substituting the limits
Properties: \(\int_a^bf=-\int_b^af,\) \(\int_a^bf=\int_a^cf+\int_c^bf,\) \(\int_a^bf(x)\,dx=\int_a^bf(a+b-x)\,dx\)Use: Split the interval and reverse the limits
Properties: \(\int_0^af(x)\,dx=\int_0^af(a-x)\,dx\)Use: Replace \(x\) by \(a + b - x\) to evaluate integrals
Properties: \(\int_0^{2a}f\,dx=2\int_0^af\,dx\text{ if }f(2a-x)\) \({}=f(x),\) \(0\text{ if }f(2a-x)=-f(x)\)Use: Simplify definite integrals with their properties
Even / odd: \(\int_{-a}^af\,dx=2\int_0^af\,dx\text{ (even)},\) \(0\text{ (odd)}\)Use: Definite integrals of odd and even functions
Watch for: Write \(+ C\) and show the substitution line
8 Application of Integrals
Area: \(\int_a^b|y|\,dx,\) \(\int_c^d|x|\,dy\)Use: Area bounded by a curve and the \(x\)-axis or \(y\)-axisWatch for: Sketch the region and find intersection points first
Circle \(x^2+y^2=a^2\): area \(\pi a^2\); ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\): area \(\pi ab\)Use: Areas enclosed by circles and ellipses using symmetry
Region between a curve and a line: integrate (upper − lower) between the intersection pointsUse: Area between a curve and a lineWatch for: Integrate (top \(-\) bottom) between the intersections
Watch for: Sketch the region and find 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 orderUse: Order and degree of a differential equation
Variables separable: \(\int\frac{dy}{g(y)}=\int f(x)\,dx+C\)Use: Separate the variables; use a given condition
Homogeneous \(\frac{dy}{dx}=F\big(\frac yx\big)\): \(y=vx,\) \(\frac{dy}{dx}=v+x\frac{dv}{dx}\)Use: Homogeneous equations with \(y = vx\)
Linear \(\frac{dy}{dx}+Py=Q\): \(\text{I.F.}=e^{\int P\,dx},\) \(y\cdot\text{I.F.}=\int Q\cdot\text{I.F.}\,dx+C\)Use: Linear equations \(\frac{dy}{dx} + Py = Q\) with an integrating factorWatch for: \(y \times\) IF \(= \int Q \times\) IF \(\mathrm{d}x + C\), then divide by IF
Linear \(\frac{dx}{dy}+P_1x=Q_1\): \(\text{I.F.}=e^{\int P_1\,dy},\) \(x\cdot\text{I.F.}=\int Q_1\cdot\text{I.F.}\,dy+C\)Use: Treat \(x\) as the dependent variable
Watch for: IF \(= e^{\int P}\,\mathrm{d}x\); then divide everything by IF
10 Vector Algebra
Magnitude; unit vector: \(|\vec a|=\sqrt{x^2+y^2+z^2},\) \(\hat a=\frac{\vec a}{|\vec a|}\)Use: Magnitude and direction of vectors
Direction cosines: \(l=\tfrac x{|\vec r|},\ m\) \({}=\tfrac y{|\vec r|},\ n\) \({}=\tfrac z{|\vec r|},\) \(l^2+m^2+n^2=1\)Use: Direction cosines and direction ratios of a vector
\(\overrightarrow{PQ}\): \((x_2-x_1)\hat i+(y_2-y_1)\hat j\) \({}+(z_2-z_1)\hat k\)Use: Write vectors in component form
Section, \(m:n\) internal; external: \(\frac{m\vec b+n\vec a}{m+n};\) \(\frac{m\vec b-n\vec a}{m-n}\)Use: Position vectors of dividing points
Dot product: \(\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta\) \({}=a_1b_1+a_2b_2+a_3b_3\)Use: Angle between vectors; projection of one on another
Perpendicular; projection of \(\vec a\) on \(\vec b\): \(\vec a\cdot\vec b=0;\) \(\frac{\vec a\cdot\vec b}{|\vec b|}\)Use: Projection of one vector on another
Cross product: \(\vec a\times\vec b=|\vec a||\vec b|\sin\theta\,\hat n\) \({}=\begin{vmatrix}\hat i&\hat j&\hat k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}\)Use: A vector perpendicular to both
Area: parallelogram; triangle: \(|\vec a\times\vec b|;\) \(\tfrac12|\vec a\times\vec b|\)Use: Areas of triangles and parallelograms
Parallelogram from its diagonals: \(\tfrac12|\vec d_1\times\vec d_2|\)Use: Areas of parallelograms with the vector product
Parallel \(\vec a\times\vec b=\vec 0\); \(\vec a\times\vec b=-\vec b\times\vec a\); \(\hat i\times\hat j=\hat k\), \(\hat j\times\hat k=\hat i\), \(\hat k\times\hat i=\hat j\)Use: Choose the right product for each part
11 Three Dimensional Geometry
DCs from direction ratios \(a,b,c\): \(l=\frac{a}{\sqrt{a^2+b^2+c^2}},\) \(\dots\)Use: Convert between direction cosines and direction ratios
Line through \(\vec a\) parallel to \(\vec b\): \(\vec r=\vec a+\lambda\vec b;\) \(\frac{x-x_1}a=\frac{y-y_1}b\) \({}=\frac{z-z_1}c\)Use: Line in space through a point parallel to a vector
Through two points: \(\vec r=\vec a+\lambda(\vec b-\vec a);\) \(\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}\) \({}=\frac{z-z_1}{z_2-z_1}\)Use: Line in space through two points
Angle between lines: \(\cos\theta=\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\)Use: Angle between two lines from direction ratios
Perpendicular; parallel: \(a_1a_2+b_1b_2+c_1c_2=0;\) \(\tfrac{a_1}{a_2}=\tfrac{b_1}{b_2}\) \({}=\tfrac{c_1}{c_2}\)Use: Decide whether lines are parallel or perpendicular
Shortest distance, skew lines: \(d=\frac{\big|(\vec b_1\times\vec b_2)\cdot(\vec a_2-\vec a_1)\big|}{|\vec b_1\times\vec b_2|}\)Use: Shortest distance between skew lines
Parallel lines: \(d=\frac{|\vec b\times(\vec a_2-\vec a_1)|}{|\vec b|}\)Use: Distance between two parallel lines in space
Lines intersect ⇔ shortest distance = 0. Foot of perpendicular from \(P\): general point \(Q\) on the line with \(\overrightarrow{PQ}\cdot\vec b=0\)Use: Foot of the perpendicular; image of a point
12 Linear Programming
Maximise/minimise \(Z=ax+by\) subject to linear constraints, \(x,y\ge0\)Use: Formulate problems: objective function and constraints
Corner point method: the optimum occurs at a corner of the feasible region. Bounded region: both max and min existUse: Evaluate the objective function at each corner point
Unbounded: \(M\) is the max only if \(ax+by>M\) has no point in the feasible region (use \(<m\) for the min)Use: Unbounded feasible regions and multiple optimal solutions
13 Probability
Conditional: \(P(E\mid F)=\frac{P(E\cap F)}{P(F)},\) \(P(F)\ne0\)Use: Conditional probabilities and their propertiesWatch for: The given event goes after the bar
Properties: \(P(S\mid F)=1,\) \(P(E'\mid F)=1-P(E\mid F)\)Use: Use the properties of conditional probability
Multiplication rule: \(P(E\cap F)=P(E)P(F\mid E)\) \({}=P(F)P(E\mid F)\)Use: Successive draws without replacement
Independent: \(P(E\cap F)=P(E)P(F)\)Use: Decide whether events are independent
Union: \(P(E\cup F)=P(E)+P(F)\) \({}-P(E\cap F)\)Use: Probabilities of combinations of events
Total probability (partition \(E_1,\dots,E_n\)): \(P(A)=\sum_{j=1}^nP(E_j)P(A\mid E_j)\)Use: An event that can happen in several ways
Bayes' theorem: \(P(E_i\mid A)=\frac{P(E_i)P(A\mid E_i)}{\sum_{j=1}^nP(E_j)P(A\mid E_j)}\)Use: Which machine produced a faulty itemWatch for: Define events in words; show the total-probability line
Random variable: \(p_i\ge0,\) \(\sum p_i=1,\) \(E(X)=\mu=\sum x_ip_i\)Use: Mean of a random variable from its distribution
Watch for: Write \(P(A \mid B)\) with the given event after the bar
