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Class 12 Maths Formula Sheet

The Class 12 Mathematics curriculum is intensive and demanding, serving as a gateway to higher education in engineering, sciences, and commerce. Success at this level requires absolute fluency in a vast number of complex formulas. To aid your preparation, we have compiled this exhaustive Class 12 Maths formula sheet. It encompasses all the critical equations across advanced topics like algebra, calculus, vectors, probability, trigonometry, and logarithms. From complex integrals to 3D geometry formulas, everything you need is organised here for quick reference. Mastering these formulas is non-negotiable if you want to excel in your board exams and competitive entrance tests. Make this sheet an integral part of your daily study routine, use it to verify your solutions, and ensure you are completely exam-ready.

algebra

Quadratic Formula

Class 10algebra
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Roots of a quadratic equation ax^2 + bx + c = 0

#roots#equations#quadratic

Difference of Squares

Class 10algebra
a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)

Factoring difference of two squares

#factoring#polynomials

Sum of Cubes

Class 10algebra
a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Factoring sum of two cubes

#factoring#cubes

Difference of Cubes

Class 10algebra
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Factoring difference of two cubes

#factoring#cubes

Binomial Theorem

Class 12algebra
(x+y)n=∑k=0n(nk)xn−kyk(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

Expansion of a binomial power

#expansion#series#binomial

Arithmetic Progression (nth Term)

Class 10algebra
an=a1+(n−1)da_n = a_1 + (n-1)d

nth term of an AP with common difference d

#sequences#ap

Arithmetic Progression (Sum)

Class 10algebra
Sn=n2(2a1+(n−1)d)=n2(a1+an)S_n = \frac{n}{2}(2a_1 + (n-1)d) = \frac{n}{2}(a_1 + a_n)

Sum of first n terms of an AP

#series#ap

Geometric Progression (nth Term)

Class 12algebra
an=a1rn−1a_n = a_1 r^{n-1}

nth term of a GP with common ratio r

#sequences#gp

Geometric Progression (Sum)

Class 12algebra
Sn=a1(1−rn)1−r(r≠1)S_n = \frac{a_1(1-r^n)}{1-r} \quad (r \neq 1)

Sum of first n terms of a GP

#series#gp

Infinite Geometric Series

Class 12algebra
S∞=a11−r(∣r∣<1)S_\infty = \frac{a_1}{1-r} \quad (|r| < 1)

Sum of infinite GP when ratio is strictly between -1 and 1

#series#infinite

Vieta's Formulas (Quadratic)

Class 10algebra
x1+x2=−ba,x1x2=cax_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

Sum and product of roots of ax^2 + bx + c = 0

#roots#vieta#polynomials

Completing the Square

Class 10algebra
ax2+bx+c=a(x+b2a)2+(c−b24a)ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)

Rewriting quadratic into vertex form

#quadratic#factoring

calculus

Power Rule (Derivative)

Class 12calculus
ddx[xn]=nxn−1\frac{d}{dx}[x^n] = n x^{n-1}

Derivative of a power function

#derivative#power

Product Rule

Class 12calculus
ddx[u⋅v]=u′v+uv′\frac{d}{dx}[u \cdot v] = u'v + uv'

Derivative of product of two functions

#derivative#product

Quotient Rule

Class 12calculus
ddx[uv]=u′v−uv′v2\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}

Derivative of quotient of two functions

#derivative#quotient

Chain Rule

Class 12calculus
ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Derivative of composite functions

#derivative#chain

Derivative of Exponential

Class 12calculus
ddx[ex]=ex,ddx[ax]=axln⁡a\frac{d}{dx}[e^x] = e^x, \quad \frac{d}{dx}[a^x] = a^x \ln a

Derivative of natural and base-a exponential functions

#derivative#exponential

Derivative of Logarithm

Class 12calculus
ddx[ln⁡x]=1x,ddx[log⁡ax]=1xln⁡a\frac{d}{dx}[\ln x] = \frac{1}{x}, \quad \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

Derivative of natural and base-a logarithms

#derivative#logarithm

Standard Trig Derivatives

Class 12calculus
ddx[sin⁡x]=cos⁡x,ddx[cos⁡x]=−sin⁡x,ddx[tan⁡x]=sec⁡2x\frac{d}{dx}[\sin x] = \cos x, \quad \frac{d}{dx}[\cos x] = -\sin x, \quad \frac{d}{dx}[\tan x] = \sec^2 x

Derivatives of basic trigonometric functions

#derivative#trigonometry

Power Rule (Integral)

Class 12calculus
∫xn dx=xn+1n+1+C(n≠−1)\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

Indefinite integral of power function

#integral#power

Reciprocal Integral

Class 12calculus
∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln|x| + C

Integral of 1/x yielding natural logarithm

#integral#logarithm

Integration by Parts

Higher Edcalculus
∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du

Standard integration technique for products

#integral#parts

Fundamental Theorem of Calculus

Higher Edcalculus
∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a)

Evaluating definite integral from antiderivative F(x)

#integral#theorem

L'Hôpital's Rule

Higher Edcalculus
lim⁡x→cf(x)g(x)=lim⁡x→cf′(x)g′(x)\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

Evaluating indeterminate forms (0/0 or ∞/∞)

#limit#lhopital

vectors

Vector Dot Product

Class 12vectors
a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ=axbx+ayby+azbz\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_x b_x + a_y b_y + a_z b_z

Scalar product of two vectors

#vectors#dot-product

Vector Cross Product

Class 12vectors
a⃗×b⃗=∣a⃗∣∣b⃗∣sin⁡θ n^=∣i^j^k^axayazbxbybz∣\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \, \hat{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}

Vector product yielding orthogonal vector

#vectors#cross-product

Vector Magnitude

Class 12vectors
∣a⃗∣=ax2+ay2+az2|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}

Euclidean norm or length of a vector

#vectors#magnitude

Unit Vector

Class 12vectors
a^=a⃗∣a⃗∣\hat{a} = \frac{\vec{a}}{|\vec{a}|}

Vector of length 1 pointing in same direction

#vectors#unit-vector

Scalar Projection of A onto B

Class 12vectors
projb⃗(a⃗)=a⃗⋅b⃗∣b⃗∣\text{proj}_{\vec{b}}(\vec{a}) = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

Magnitude of vector a in direction of vector b

#vectors#projection

Matrix Determinant (2x2)

Class 12vectors
det⁡[abcd]=ad−bc\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc

Determinant of a 2 by 2 matrix

#matrices#determinant

Matrix Inverse (2x2)

Class 12vectors
[abcd]−1=1ad−bc[d−b−ca]\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Inverse of an invertible 2 by 2 matrix

#matrices#inverse

Characteristic Equation (Eigenvalues)

Higher Edvectors
det⁡(A−λI)=0\det(A - \lambda I) = 0

Equation for finding eigenvalues λ of matrix A

#matrices#eigenvalues

probability

Permutations Formula

Class 10probability
P(n,r)=nPr=n!(n−r)!P(n, r) = {}^n P_r = \frac{n!}{(n-r)!}

Number of ordered arrangements of r elements from n

#combinatorics#permutations

Combinations Formula

Class 10probability
C(n,r)=nCr=(nr)=n!r!(n−r)!C(n, r) = {}^n C_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of unordered selections of r elements from n

#combinatorics#combinations

Addition Rule of Probability

Class 10probability
P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

Probability of union of two events

#probability#union

Multiplication Rule

Class 12probability
P(A∩B)=P(A)⋅P(B∣A)P(A \cap B) = P(A) \cdot P(B|A)

Probability of simultaneous occurrence of two events

#probability#intersection

Conditional Probability

Class 12probability
P(A∣B)=P(A∩B)P(B)(P(B)>0)P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (P(B) > 0)

Probability of event A occurring given event B has occurred

#probability#conditional

Bayes' Theorem

Higher Edprobability
P(A∣B)=P(B∣A)⋅P(A)P(B)=P(B∣A)P(A)∑P(B∣Ai)P(Ai)P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{P(B|A) P(A)}{\sum P(B|A_i) P(A_i)}

Calculates posterior probability based on prior knowledge

#probability#bayes

Binomial Distribution PMF

Class 12probability
P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Probability of k successes in n independent Bernoulli trials

#distribution#binomial

Poisson Distribution PMF

Higher Edprobability
P(X=k)=λke−λk!P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval with average rate λ

#distribution#poisson

Expected Value (Discrete)

Class 12probability
E[X]=∑ixi⋅P(X=xi)E[X] = \sum_{i} x_i \cdot P(X = x_i)

Weighted average of all possible values of discrete random variable

#statistics#mean#expectation

Variance & Standard Deviation

Class 12probability
Var(X)=E[X2]−(E[X])2,σ=Var(X)\text{Var}(X) = E[X^2] - (E[X])^2, \quad \sigma = \sqrt{\text{Var}(X)}

Measure of dispersion of a random variable

#statistics#variance

Standard Normal Distribution (PDF)

Higher Edprobability
f(z)=12πe−z2/2,z=x−μσf(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}, \quad z = \frac{x - \mu}{\sigma}

Probability density function of standard normal variable

#distribution#normal#z-score

trigonometry

Pythagorean Identity

Class 10trigonometry
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

Fundamental trigonometric identity relating sine and cosine

#identities#pythagorean

Secant & Tangent Identity

Class 10trigonometry
1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta

Pythagorean identity for tangent and secant

#identities#pythagorean

Cosecant & Cotangent Identity

Class 10trigonometry
1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta

Pythagorean identity for cotangent and cosecant

#identities#pythagorean

Sine Addition & Subtraction

Class 12trigonometry
sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

Sine of sum or difference of two angles

#compound#addition

Cosine Addition & Subtraction

Class 12trigonometry
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Cosine of sum or difference of two angles

#compound#addition

Tangent Addition & Subtraction

Class 12trigonometry
tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

Tangent of sum or difference of two angles

#compound#addition

Double Angle (Sine)

Class 12trigonometry
sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta

Sine of double angle

#double-angle

Double Angle (Cosine)

Class 12trigonometry
cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

Cosine of double angle formulas

#double-angle

Half Angle Formulas

Class 12trigonometry
sin⁡(θ2)=±1−cos⁡θ2,cos⁡(θ2)=±1+cos⁡θ2\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1-\cos\theta}{2}}, \quad \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1+\cos\theta}{2}}

Sine and cosine of half angle

#half-angle

Law of Sines

Class 10trigonometry
asin⁡A=bsin⁡B=csin⁡C=2R\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Relates sides of a triangle to the sines of its angles

Interactive Triangle Solver (SAS):
Side c (opposite to ∠C):—
#triangles#sines

Law of Cosines

Class 10trigonometry
c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

Relates sides of a triangle to cosine of an angle

Interactive Triangle Solver (SAS):
Side c (opposite to ∠C):—
#triangles#cosines

logarithms

Product Rule for Logs

Class 10logarithms
log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)

Logarithm of a product equals sum of logarithms

#product#logarithm

Quotient Rule for Logs

Class 10logarithms
log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)

Logarithm of a quotient equals difference of logarithms

#quotient#logarithm

Power Rule for Logs

Class 10logarithms
log⁡b(xk)=klog⁡b(x)\log_b(x^k) = k \log_b(x)

Exponent comes out as a multiplier

#power#exponent#logarithm

Change of Base Formula

Class 12logarithms
log⁡b(x)=log⁡k(x)log⁡k(b)=ln⁡xln⁡b=log⁡10xlog⁡10b\log_b(x) = \frac{\log_k(x)}{\log_k(b)} = \frac{\ln x}{\ln b} = \frac{\log_{10} x}{\log_{10} b}

Converting logarithm to a different base k

#base#conversion

Identity & Zero Properties

Class 10logarithms
log⁡b(b)=1,log⁡b(1)=0\log_b(b) = 1, \quad \log_b(1) = 0

Basic identity properties of logarithms

#identity#logarithm

Inverse Properties

Class 12logarithms
blog⁡bx=x,log⁡b(bx)=xb^{\log_b x} = x, \quad \log_b(b^x) = x

Logarithms and exponentials are inverse functions

#inverse#exponential

Natural Log & Euler Constant

Higher Edlogarithms
ln⁡(x)=log⁡e(x),e=lim⁡n→∞(1+1n)n≈2.71828\ln(x) = \log_e(x), \quad e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n \approx 2.71828

Definition of natural logarithm and base e

#natural#euler

Exponential Laws

Class 10logarithms
am⋅an=am+n,aman=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}

Fundamental laws of exponent arithmetic

#exponential#powers

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