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All Class 12 Maths formulas

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The formulas from 11 chapters, in one place to revise from or print. Open a chapter's own sheet to see where students usually slip up.

2. Inverse Trigonometric Functions

Principal value ranges

sin⁻¹ x
x∈[−1,1],y∈[−π2,π2]\displaystyle x \in [-1, 1], \quad y \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]
cos⁻¹ x
x∈[−1,1],y∈[0,π]\displaystyle x \in [-1, 1], \quad y \in [0, \pi]
tan⁻¹ x
x∈R,y∈(−π2,π2)\displaystyle x \in \mathbb{R}, \quad y \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

Identities

Negative arguments
sin⁡−1(−x)=−sin⁡−1x,tan⁡−1(−x)=−tan⁡−1x\displaystyle \sin^{-1}(-x) = -\sin^{-1}x, \quad \tan^{-1}(-x) = -\tan^{-1}x
For cosine
cos⁡−1(−x)=π−cos⁡−1x\displaystyle \cos^{-1}(-x) = \pi - \cos^{-1}x
Complementary pairs
sin⁡−1x+cos⁡−1x=π2,tan⁡−1x+cot⁡−1x=π2\displaystyle \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}, \quad \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}
Sum of inverse tangents
tan⁡−1x+tan⁡−1y=tan⁡−1x+y1−xy\displaystyle \tan^{-1}x + \tan^{-1}y = \tan^{-1}\frac{x + y}{1 - xy}

For xy < 1.

Double angle
2tan⁡−1x=sin⁡−12x1+x2=cos⁡−11−x21+x2\displaystyle 2\tan^{-1}x = \sin^{-1}\frac{2x}{1 + x^2} = \cos^{-1}\frac{1 - x^2}{1 + x^2}

For suitable ranges of x.

3. Matrices

Operations

Product condition
Am×nBn×p=Cm×p\displaystyle A_{m \times n}B_{n \times p} = C_{m \times p}
Not commutative
AB≠BA in general\displaystyle AB \ne BA \ \text{in general}
Transpose rules
(A′)′=A,(A+B)′=A′+B′,(AB)′=B′A′\displaystyle (A')' = A, \quad (A + B)' = A' + B', \quad (AB)' = B'A'

Special matrices

Symmetric
A′=A\displaystyle A' = A
Skew-symmetric
A′=−A\displaystyle A' = -A

Diagonal entries are all zero.

Any square matrix
A=12(A+A′)+12(A−A′)\displaystyle A = \tfrac12(A + A') + \tfrac12(A - A')

Sum of a symmetric and a skew-symmetric matrix.

Inverse
AA−1=A−1A=I\displaystyle AA^{-1} = A^{-1}A = I
Inverse of a product
(AB)−1=B−1A−1\displaystyle (AB)^{-1} = B^{-1}A^{-1}

4. Determinants

Determinants

2 × 2 determinant
∣abcd∣=ad−bc\displaystyle \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc
Cofactor
Aij=(−1)i+jMij\displaystyle A_{ij} = (-1)^{i + j}M_{ij}
Scalar multiple
∣kA∣=kn∣A∣\displaystyle |kA| = k^n|A|

n is the order of A.

Product and transpose
∣AB∣=∣A∣∣B∣,∣A′∣=∣A∣\displaystyle |AB| = |A||B|, \quad |A'| = |A|
Area of a triangle
Δ=12∣x1y11x2y21x3y31∣\displaystyle \Delta = \frac12\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}

Take the absolute value.

Adjoint and inverse

Adjoint
A(adjA)=(adjA)A=∣A∣I\displaystyle A(\text{adj}A) = (\text{adj}A)A = |A|I
Inverse
A−1=1∣A∣adjA\displaystyle A^{-1} = \frac{1}{|A|}\text{adj}A

Exists only when |A| ≠ 0.

Adjoint's determinant
∣adjA∣=∣A∣n−1\displaystyle |\text{adj}A| = |A|^{n - 1}

Solving equations

Matrix method
AX=B  ⇒  X=A−1B\displaystyle AX = B \;\Rightarrow\; X = A^{-1}B

5. Continuity and Differentiability

Continuity

Condition at a point
lim⁡x→c−f(x)=lim⁡x→c+f(x)=f(c)\displaystyle \lim_{x \to c^-}f(x) = \lim_{x \to c^+}f(x) = f(c)

Differentiation rules

Chain rule
dydx=dydu⋅dudx\displaystyle \frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}
Parametric form
dydx=dy/dtdx/dt\displaystyle \frac{dy}{dx} = \frac{dy/dt}{dx/dt}
Logarithmic differentiation
y=uv  ⇒  dydx=uv[vududx+ln⁡u dvdx]\displaystyle y = u^v \;\Rightarrow\; \frac{dy}{dx} = u^v\left[\frac{v}{u}\frac{du}{dx} + \ln u\,\frac{dv}{dx}\right]

Standard derivatives

Exponential and log
ddxex=ex,ddxln⁡x=1x,ddxax=axln⁡a\displaystyle \frac{d}{dx}e^x = e^x, \quad \frac{d}{dx}\ln x = \frac1x, \quad \frac{d}{dx}a^x = a^x\ln a
Other trigonometric
ddxsec⁡x=sec⁡xtan⁡x,ddxcsc⁡x=−csc⁡xcot⁡x,ddxcot⁡x=−csc⁡2x\displaystyle \frac{d}{dx}\sec x = \sec x\tan x, \quad \frac{d}{dx}\csc x = -\csc x\cot x, \quad \frac{d}{dx}\cot x = -\csc^2x
Inverse sine and cosine
ddxsin⁡−1x=11−x2,ddxcos⁡−1x=−11−x2\displaystyle \frac{d}{dx}\sin^{-1}x = \frac{1}{\sqrt{1 - x^2}}, \quad \frac{d}{dx}\cos^{-1}x = -\frac{1}{\sqrt{1 - x^2}}
Inverse tangent
ddxtan⁡−1x=11+x2\displaystyle \frac{d}{dx}\tan^{-1}x = \frac{1}{1 + x^2}

6. Application of Derivatives

Rates and tangents

Rate of change
dydt=dydx⋅dxdt\displaystyle \frac{dy}{dt} = \frac{dy}{dx}\cdot\frac{dx}{dt}
Slope of the tangent
m=dydx∣(x0,y0)\displaystyle m = \left.\frac{dy}{dx}\right|_{(x_0, y_0)}
Tangent line
y−y0=m(x−x0)\displaystyle y - y_0 = m(x - x_0)
Normal line
y−y0=−1m(x−x0)\displaystyle y - y_0 = -\frac{1}{m}(x - x_0)

Increasing and decreasing

Increasing
f′(x)>0\displaystyle f'(x) > 0
Decreasing
f′(x)<0\displaystyle f'(x) < 0

Maxima and minima

Critical point
f′(c)=0 or f′(c) does not exist\displaystyle f'(c) = 0 \ \text{or} \ f'(c) \ \text{does not exist}
Local maximum
f′(c)=0, f′′(c)<0\displaystyle f'(c) = 0, \ f''(c) < 0
Local minimum
f′(c)=0, f′′(c)>0\displaystyle f'(c) = 0, \ f''(c) > 0

If f''(c) = 0, use the first derivative test instead.

7. Integrals

Standard integrals

Power
∫xn dx=xn+1n+1+C,n≠−1\displaystyle \int x^n\,dx = \frac{x^{n + 1}}{n + 1} + C, \quad n \ne -1
Reciprocal and exponential
∫1x dx=ln⁡∣x∣+C,∫ex dx=ex+C\displaystyle \int\frac1x\,dx = \ln|x| + C, \quad \int e^x\,dx = e^x + C
Sine and cosine
∫sin⁡x dx=−cos⁡x+C,∫cos⁡x dx=sin⁡x+C\displaystyle \int\sin x\,dx = -\cos x + C, \quad \int\cos x\,dx = \sin x + C
Squares of secant and cosecant
∫sec⁡2x dx=tan⁡x+C,∫csc⁡2x dx=−cot⁡x+C\displaystyle \int\sec^2x\,dx = \tan x + C, \quad \int\csc^2x\,dx = -\cot x + C
Tangent and cotangent
∫tan⁡x dx=ln⁡∣sec⁡x∣+C,∫cot⁡x dx=ln⁡∣sin⁡x∣+C\displaystyle \int\tan x\,dx = \ln|\sec x| + C, \quad \int\cot x\,dx = \ln|\sin x| + C
Secant and cosecant
∫sec⁡x dx=ln⁡∣sec⁡x+tan⁡x∣+C,∫csc⁡x dx=ln⁡∣csc⁡x−cot⁡x∣+C\displaystyle \int\sec x\,dx = \ln|\sec x + \tan x| + C, \quad \int\csc x\,dx = \ln|\csc x - \cot x| + C

Special forms

Inverse tangent
∫dxx2+a2=1atan⁡−1xa+C\displaystyle \int\frac{dx}{x^2 + a^2} = \frac1a\tan^{-1}\frac{x}{a} + C
Inverse sine
∫dxa2−x2=sin⁡−1xa+C\displaystyle \int\frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\frac{x}{a} + C
Log forms
∫dxx2−a2=12aln⁡∣x−ax+a∣+C\displaystyle \int\frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln\left|\frac{x - a}{x + a}\right| + C
Root forms
∫dxx2±a2=ln⁡∣x+x2±a2∣+C\displaystyle \int\frac{dx}{\sqrt{x^2 \pm a^2}} = \ln\left|x + \sqrt{x^2 \pm a^2}\right| + C
Square root of a² − x²
∫a2−x2 dx=x2a2−x2+a22sin⁡−1xa+C\displaystyle \int\sqrt{a^2 - x^2}\,dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C

Methods

Integration by parts
∫u v dx=u∫v dx−∫(u′∫v dx)dx\displaystyle \int u\,v\,dx = u\int v\,dx - \int\left(u'\int v\,dx\right)dx

Choose u using the order ILATE.

A useful exponential form
∫ex[f(x)+f′(x)] dx=exf(x)+C\displaystyle \int e^x[f(x) + f'(x)]\,dx = e^xf(x) + C

Definite integrals

Fundamental theorem
∫abf(x) dx=F(b)−F(a)\displaystyle \int_a^b f(x)\,dx = F(b) - F(a)
King's property
∫abf(x) dx=∫abf(a+b−x) dx\displaystyle \int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx
Symmetric limits
∫−aaf(x) dx=2∫0af(x) dx if f is even, and 0 if odd\displaystyle \int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx \ \text{if } f \text{ is even, and } 0 \text{ if odd}

8. Application of Integrals

Formulas

Area with the x-axis
A=∫ab∣y∣ dx\displaystyle A = \int_a^b |y|\,dx
Area with the y-axis
A=∫cd∣x∣ dy\displaystyle A = \int_c^d |x|\,dy
Between two curves
A=∫ab[f(x)−g(x)] dx\displaystyle A = \int_a^b [f(x) - g(x)]\,dx

Where f(x) ≥ g(x).

Area of a circle
πa2\displaystyle \pi a^2

For x² + y² = a². A useful check.

Area of an ellipse
πab\displaystyle \pi ab

9. Differential Equations

Types and solutions

Variable separable
dydx=f(x)g(y)  ⇒  ∫dyg(y)=∫f(x) dx\displaystyle \frac{dy}{dx} = f(x)g(y) \;\Rightarrow\; \int\frac{dy}{g(y)} = \int f(x)\,dx
Homogeneous
dydx=F(yx),put y=vx\displaystyle \frac{dy}{dx} = F\left(\frac{y}{x}\right), \quad \text{put } y = vx
Linear form
dydx+Py=Q\displaystyle \frac{dy}{dx} + Py = Q

P and Q are functions of x.

Integrating factor
I.F.=e∫P dx\displaystyle \text{I.F.} = e^{\int P\,dx}
Solution of the linear form
y⋅I.F.=∫Q⋅I.F. dx+C\displaystyle y\cdot\text{I.F.} = \int Q\cdot\text{I.F.}\,dx + C

10. Vector Algebra

Basics

Magnitude
∣a⃗∣=a12+a22+a32\displaystyle |\vec a| = \sqrt{a_1^2 + a_2^2 + a_3^2}
Unit vector
a^=a⃗∣a⃗∣\displaystyle \hat a = \frac{\vec a}{|\vec a|}
Section formula
r⃗=mb⃗+na⃗m+n\displaystyle \vec r = \frac{m\vec b + n\vec a}{m + n}

Internal division.

Dot product

Definition
a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ=a1b1+a2b2+a3b3\displaystyle \vec a\cdot\vec b = |\vec a||\vec b|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3
Perpendicular vectors
a⃗⋅b⃗=0\displaystyle \vec a\cdot\vec b = 0
Projection of a on b
a⃗⋅b⃗∣b⃗∣\displaystyle \frac{\vec a\cdot\vec b}{|\vec b|}

Cross product

Magnitude
∣a⃗×b⃗∣=∣a⃗∣∣b⃗∣sin⁡θ\displaystyle |\vec a\times\vec b| = |\vec a||\vec b|\sin\theta
Component form
a⃗×b⃗=∣i^j^k^a1a2a3b1b2b3∣\displaystyle \vec a\times\vec b = \begin{vmatrix} \hat i & \hat j & \hat k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}
Parallel vectors
a⃗×b⃗=0⃗\displaystyle \vec a\times\vec b = \vec 0
Area of a triangle
12∣a⃗×b⃗∣\displaystyle \tfrac12|\vec a\times\vec b|

a and b are two sides.

Area of a parallelogram
∣a⃗×b⃗∣\displaystyle |\vec a\times\vec b|

a and b are adjacent sides. With its diagonals, it's half the magnitude.

11. Three Dimensional Geometry

Direction cosines

Relation
l2+m2+n2=1\displaystyle l^2 + m^2 + n^2 = 1
From direction ratios
l=aa2+b2+c2\displaystyle l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}

Similarly for m and n.

Through two points
x2−x1, y2−y1, z2−z1\displaystyle x_2 - x_1,\ y_2 - y_1,\ z_2 - z_1

These are the direction ratios.

Lines

Vector form
r⃗=a⃗+λb⃗\displaystyle \vec r = \vec a + \lambda\vec b
Cartesian form
x−x1a=y−y1b=z−z1c\displaystyle \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}
Angle between lines
cos⁡θ=∣a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22∣\displaystyle \cos\theta = \left|\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}}\right|
Perpendicular lines
a1a2+b1b2+c1c2=0\displaystyle a_1a_2 + b_1b_2 + c_1c_2 = 0

Shortest distance

Skew lines
d=∣(b⃗1×b⃗2)⋅(a⃗2−a⃗1)∣b⃗1×b⃗2∣∣\displaystyle d = \left|\frac{(\vec b_1\times\vec b_2)\cdot(\vec a_2 - \vec a_1)}{|\vec b_1\times\vec b_2|}\right|
Parallel lines
d=∣b⃗×(a⃗2−a⃗1)∣∣b⃗∣\displaystyle d = \frac{|\vec b\times(\vec a_2 - \vec a_1)|}{|\vec b|}

13. Probability

Conditional probability

Conditional probability
P(E∣F)=P(E∩F)P(F),P(F)≠0\displaystyle P(E|F) = \frac{P(E \cap F)}{P(F)}, \quad P(F) \ne 0
Multiplication rule
P(E∩F)=P(E)P(F∣E)=P(F)P(E∣F)\displaystyle P(E \cap F) = P(E)P(F|E) = P(F)P(E|F)
Independent events
P(E∩F)=P(E)P(F)\displaystyle P(E \cap F) = P(E)P(F)

Total probability and Bayes

Total probability
P(A)=∑iP(Ei)P(A∣Ei)\displaystyle P(A) = \sum_i P(E_i)P(A|E_i)

The Eᵢ partition the sample space.

Bayes' theorem
P(Ei∣A)=P(Ei)P(A∣Ei)∑jP(Ej)P(A∣Ej)\displaystyle P(E_i|A) = \frac{P(E_i)P(A|E_i)}{\sum_j P(E_j)P(A|E_j)}

Random variables

Mean
E(X)=∑xiP(xi)\displaystyle E(X) = \sum x_iP(x_i)
Variance
Var(X)=E(X2)−[E(X)]2\displaystyle \text{Var}(X) = E(X^2) - [E(X)]^2
Binomial probability
P(X=r)=nCr prqn−r,q=1−p\displaystyle P(X = r) = {}^nC_r\,p^rq^{n - r}, \quad q = 1 - p
Binomial mean and variance
mean=np,variance=npq\displaystyle \text{mean} = np, \quad \text{variance} = npq

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