ISC Class 12 Mathematics Formulas

A revision sheet of the ISC Class 12 Mathematics formulas that questions use, by chapter: 76 entries from relations and inverse trigonometric functions to probability. Learn them, then practise on the past questions of each chapter.

Relations and functions; inverse trigonometric functions

Number of relations from A to B: $2^{mn}$ where $n(A)=m$, $n(B)=n$
One-one and onto functions from A to B (finite sets, $n(A)=n(B)=n$): number of bijections $= n!$
Composite function: $(f\circ g)(x) = f(g(x))$
Principal value branches: $\sin^{-1}x \in [-\tfrac{\pi}{2},\tfrac{\pi}{2}]$, $\cos^{-1}x \in [0,\pi]$, $\tan^{-1}x \in (-\tfrac{\pi}{2},\tfrac{\pi}{2})$
Complementary pairs: $\sin^{-1}x + \cos^{-1}x = \tfrac{\pi}{2}$ and $\tan^{-1}x + \cot^{-1}x = \tfrac{\pi}{2}$
Sum of two inverse tangents: $\tan^{-1}x + \tan^{-1}y = \tan^{-1}\dfrac{x+y}{1-xy}$ for $xy \lt 1$
Double angle forms: $2\tan^{-1}x = \tan^{-1}\dfrac{2x}{1-x^2} = \sin^{-1}\dfrac{2x}{1+x^2}$

Matrices and determinants

Product of matrices: $(AB)_{ij} = \sum_k a_{ik}b_{kj}$; $AB \ne BA$ in general
Transpose rules: $(AB)^T = B^TA^T$, $(A+B)^T = A^T + B^T$
Symmetric and skew-symmetric: $A = A^T$ and $A = -A^T$; every square matrix is the sum of one of each
Determinant of a 2 × 2 matrix: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$
Area of a triangle: $\Delta = \tfrac12 \left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right|$
Adjoint and inverse: $A\,(\operatorname{adj}A) = (\operatorname{adj}A)\,A = |A|\,I$, so $A^{-1} = \dfrac{\operatorname{adj}A}{|A|}$ when $|A| \ne 0$
Orders of determinants: $|kA| = k^n|A|$ for order $n$; $|AB| = |A||B|$; $|\operatorname{adj}A| = |A|^{n-1}$
Solving $AX = B$: $X = A^{-1}B$ when $|A| \ne 0$

Continuity and differentiation

Continuity at $x=a$: $\lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) = f(a)$
Chain rule: $\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}$
Product and quotient: $(uv)' = u'v + uv'$ and $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$
Parametric form: $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$
Standard derivatives: $\dfrac{d}{dx}x^n = nx^{n-1}$, $\dfrac{d}{dx}e^x = e^x$, $\dfrac{d}{dx}a^x = a^x\log a$, $\dfrac{d}{dx}\log x = \dfrac1x$
Trigonometric derivatives: $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$, $(\tan x)' = \sec^2x$, $(\sec x)' = \sec x\tan x$, $(\cot x)' = -\operatorname{cosec}^2x$, $(\operatorname{cosec}x)' = -\operatorname{cosec}x\cot x$
Inverse trigonometric derivatives: $(\sin^{-1}x)' = \dfrac{1}{\sqrt{1-x^2}}$, $(\cos^{-1}x)' = -\dfrac{1}{\sqrt{1-x^2}}$, $(\tan^{-1}x)' = \dfrac{1}{1+x^2}$
Logarithmic differentiation: for $y = u^v$: $\log y = v\log u$, then differentiate both sides

Applications of derivatives

Rate of change: $\dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot\dfrac{dx}{dt}$
Tangent at $(x_1,y_1)$: $y - y_1 = m(x - x_1)$ with $m = \left.\dfrac{dy}{dx}\right|_{(x_1,y_1)}$
Normal at $(x_1,y_1)$: $y - y_1 = -\dfrac{1}{m}(x - x_1)$
Increasing / decreasing: $f'(x) \gt 0$ increasing; $f'(x) \lt 0$ decreasing
Maxima and minima (second derivative test): $f'(c)=0$ and $f''(c) \lt 0$ gives a maximum; $f''(c) \gt 0$ gives a minimum
Approximation: $f(x+\Delta x) \approx f(x) + f'(x)\,\Delta x$
Marginal cost and revenue: $MC = \dfrac{dC}{dx}$, $MR = \dfrac{dR}{dx}$, $AC = \dfrac{C}{x}$

Integrals

Power rule: $\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$, $n \ne -1$; $\int \dfrac1x\,dx = \log|x| + C$
Exponential: $\int e^x\,dx = e^x + C$, $\int a^x\,dx = \dfrac{a^x}{\log a} + C$
Trigonometric: $\int \sin x\,dx = -\cos x + C$, $\int \cos x\,dx = \sin x + C$, $\int \sec^2x\,dx = \tan x + C$, $\int \sec x\tan x\,dx = \sec x + C$
$\tan x$ and $\cot x$: $\int \tan x\,dx = \log|\sec x| + C$, $\int \cot x\,dx = \log|\sin x| + C$
$\sec x$ and $\operatorname{cosec}x$: $\int \sec x\,dx = \log|\sec x + \tan x| + C$, $\int \operatorname{cosec}x\,dx = \log|\operatorname{cosec}x - \cot x| + C$
Standard forms: $\int \dfrac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\dfrac xa + C$, $\int \dfrac{dx}{a^2+x^2} = \dfrac1a\tan^{-1}\dfrac xa + C$
More standard forms: $\int \dfrac{dx}{x^2-a^2} = \dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right| + C$, $\int \dfrac{dx}{\sqrt{x^2\pm a^2}} = \log\left|x + \sqrt{x^2\pm a^2}\right| + C$
Integration by parts: $\int u\,v\,dx = u\int v\,dx - \int\left(\dfrac{du}{dx}\int v\,dx\right)dx$ (choose $u$ by ILATE)
A useful form: $\int e^x\,[f(x) + f'(x)]\,dx = e^x f(x) + C$
Definite integral: $\int_a^b f(x)\,dx = F(b) - F(a)$
Properties: $\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$; $\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$; $\int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx$ if $f$ is even, $0$ if odd

Application of integrals and differential equations

Area under a curve: $A = \int_a^b y\,dx = \int_a^b f(x)\,dx$ (between the curve, the x-axis and $x=a$, $x=b$)
Area between two curves: $A = \int_a^b [f(x) - g(x)]\,dx$ where $f(x) \ge g(x)$
Separation of variables: $\dfrac{dy}{dx} = f(x)g(y) \Rightarrow \int \dfrac{dy}{g(y)} = \int f(x)\,dx$
Linear equation: $\dfrac{dy}{dx} + Py = Q$: integrating factor $e^{\int P\,dx}$, solution $y\,e^{\int P\,dx} = \int Q\,e^{\int P\,dx}\,dx + C$
Homogeneous equation: put $y = vx$ so that $\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}$
Order and degree: order is the highest derivative; degree is its power once the equation is a polynomial in the derivatives

Vector algebra

Magnitude and unit vector: $|\vec a| = \sqrt{a_1^2+a_2^2+a_3^2}$, $\hat a = \dfrac{\vec a}{|\vec a|}$
Direction cosines: $l = \cos\alpha = \dfrac{a_1}{|\vec a|}$, and $l^2+m^2+n^2 = 1$
Dot product: $\vec a\cdot\vec b = |\vec a||\vec b|\cos\theta = a_1b_1+a_2b_2+a_3b_3$
Projection of $\vec a$ on $\vec b$: $\dfrac{\vec a\cdot\vec b}{|\vec b|}$ (scalar); $\dfrac{\vec a\cdot\vec b}{|\vec b|^2}\vec b$ (vector)
Cross product: $\vec a\times\vec b = |\vec a||\vec b|\sin\theta\,\hat n = \begin{vmatrix} \hat i & \hat j & \hat k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$
Areas: parallelogram $= |\vec a\times\vec b|$; triangle $= \tfrac12|\vec a\times\vec b|$
Scalar triple product: $[\vec a\ \vec b\ \vec c] = \vec a\cdot(\vec b\times\vec c)$ = volume of the parallelepiped; coplanar if $0$
Section formula: internally in ratio $m:n$: $\dfrac{m\vec b + n\vec a}{m+n}$; midpoint $\dfrac{\vec a+\vec b}{2}$

Three-dimensional geometry

Distance between two points: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
Line through a point with direction ratios $a,b,c$: $\dfrac{x-x_1}{a} = \dfrac{y-y_1}{b} = \dfrac{z-z_1}{c}$; vector form $\vec r = \vec a + \lambda\vec b$
Angle between two lines: $\cos\theta = \dfrac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$
Shortest distance between skew lines: $d = \dfrac{|(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)|}{|\vec b_1\times\vec b_2|}$
Distance between parallel lines: $d = \dfrac{|\vec b\times(\vec a_2-\vec a_1)|}{|\vec b|}$
Plane: $ax+by+cz+d=0$; vector form $\vec r\cdot\vec n = d$; intercept form $\dfrac xa+\dfrac yb+\dfrac zc = 1$
Distance of a point from a plane: $\dfrac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}$
Angle between line and plane: $\sin\theta = \dfrac{|\vec b\cdot\vec n|}{|\vec b||\vec n|}$

Linear programming

Parts of a problem: objective function $Z = ax+by$ (maximise or minimise) under linear constraints and $x,y\ge 0$
Corner point method: evaluate $Z$ at every corner of the feasible region; the best value is the optimum
Bounded and unbounded regions: for an unbounded region the optimum exists only if the half-plane $Z \lt M$ (max) or $Z \gt m$ (min) has no point in common with it

Probability

Conditional probability: $P(A\mid B) = \dfrac{P(A\cap B)}{P(B)}$, $P(B) \ne 0$
Multiplication rule: $P(A\cap B) = P(A)\,P(B\mid A)$; independent events: $P(A\cap B) = P(A)P(B)$
Total probability: $P(A) = \sum_i P(E_i)\,P(A\mid E_i)$
Bayes’ theorem: $P(E_i\mid A) = \dfrac{P(E_i)\,P(A\mid E_i)}{\sum_j P(E_j)\,P(A\mid E_j)}$
Random variable: mean: $E(X) = \mu = \sum x_i\,p_i$
Random variable: variance: $\operatorname{Var}(X) = \sum x_i^2p_i - \mu^2$; standard deviation $= \sqrt{\operatorname{Var}(X)}$
Binomial distribution: $P(X=r) = {}^nC_r\,p^r q^{n-r}$ with $q = 1-p$; mean $= np$, variance $= npq$

Linear regression (for papers up to 2026)

Regression coefficients: $b_{yx} = r\dfrac{\sigma_y}{\sigma_x}$, $b_{xy} = r\dfrac{\sigma_x}{\sigma_y}$, and $r^2 = b_{yx}\,b_{xy}$
Lines of regression: $y - \bar y = b_{yx}(x - \bar x)$ and $x - \bar x = b_{xy}(y - \bar y)$
From the data: $b_{yx} = \dfrac{\sum xy - n\bar x\bar y}{\sum x^2 - n\bar x^2}$

Common questions

Which are the most important formulas in ISC Class 12 Mathematics?

Calculus carries the most marks (35 of 80 in the 2027 syllabus), so the derivative and integral tables, integration by parts and the properties of definite integrals are the ones to know cold, followed by the determinant and inverse formulas, the vector products and the probability rules.

Are formulas given in the ISC Maths paper?

The instructions state that mathematical tables and graph papers are provided, but formulas are not printed, so these have to be learned.

Is linear regression in the 2027 Maths syllabus?

No. Regression was in Section C of papers up to 2026; the 2027 syllabus has seven units without it. Its formulas are kept at the end of this page for those papers.

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Last checked 2026-10-16.