JEE Main Formulas 2027 - JEE Main is a very competitive entrance exam for engineering education. To pass this exam, you should know the basic concepts of Mathematics, Physics and Chemistry with their formulas. JEE Mains is conducted by the National Test Agency (NTA). The JEE Mains 2027 exam is an assessment of a candidate’s proficiency in Physics, Chemistry and Mathematics. Since there are many formulas in these three subjects, candidates need a way to recall them for revision purposes. JEE Main important formulas in Math, Physics, and Chemistry are integral to calculating answers for numerical questions according to the JEE Main 2027 syllabus.

JEE Main Formulas 2027 for Physics, Chemistry, Maths

Candidates must make a handy note of all important formulas to revise frequently. Candidates must have a good command of each topic and the formulas to crack the JEE Main 2027 exam to ace the test. Through this article, students can find the provided JEE Main formulas for all three subjects. Memorizing some crucial formulas will enable you to solve the questions faster and accurately, which will be beneficial for your JEE Main score. All the formulas listed below are as per the JEE Main 2027 syllabus . The complete list of all JEE Formulas PDF for each subject is given below.

JEE Main 2027 Exam Pattern

The JEE Main 2027 Exam Pattern is such that the exam is divided into two main sections:

Section A: The quizzes contained in this section are 20 MCQs for each course. What the MCQ requires is four options, and only one of them is the right answer.

Section B: This section has 5 numerical value questions for each of the subject areas, and the candidate only has to answer all five of these. These numerical value questions have to be answered accurately, sometimes to the second decimal place.

JEE Main Formulas for Physics 2027

Aspirants preparing for JEE Mains must remember that, along with concepts, one needs to revise and remember the formulas, which are very important while solving any problems. As JEE Main Physics formulas are given below, these formulas need to be memorised daily, as direct questions and formulas are asked in exams. Students can also solve JEE Main chapter-wise PYQs.

Physics and Measurement

Relative density $=\frac{\text { density of object }}{\text { density of water at } 4^{\circ} \mathrm{c}}$
Absolute Error for $\mathrm{n}^{\text {th }}$ reading $=\Delta a_n=a_m-a_n=$ true value - measured value
Mean absolute error

$\Delta \bar{a}=\frac{\left|\Delta a_1\right|+\left|\Delta a_2\right|+\ldots\left|\Delta a_n\right|}{n}$
$\begin{aligned} & \text { Relative error }=\frac{\Delta \bar{a}}{a_m} \\ & \Delta \bar{a}=\text { mean absolute error } \\ & a_m=\text { mean value }\end{aligned}$
Percentage error $=\frac{\Delta \bar{a}}{a_m} \times 100 \%$

Kinematics

Some important Formulas of differentiation

$\begin{aligned}
& \frac{d}{d x}\left(x^n\right)=n x^{n-1} \\
& \frac{d}{d x} \sin x=\cos x \\
& \frac{d}{d x} \cos x=-\sin x \\
& \frac{d}{d x} \tan x=\sec ^2 x \\
& \frac{d}{d x} \cot x=-\csc ^2 x \\
& \frac{d}{d x} \sec x=\sec x \tan x \\
& \frac{d}{d x} \csc x=-\csc x \cot x \\
& \frac{d}{d x} e^x=e^x \\
& \frac{d}{d x} a^x=a^x \ln a \\
& \frac{d}{d x} \ln |x|=\frac{1}{x}
\end{aligned}$
Some important Formulas of integration

$\begin{aligned}
& \int x^n d x=\frac{x^{n+1}}{n+1}+C \\
& \int \frac{d x}{x}=\ln |x|+C \\
& \int e^x d x=e^x+C \\
& \int a^x d x=\frac{1}{\ln a} a^x+C \\
& \int \ln x d x=x \ln x-x+C \\
& \int \sin x d x=-\cos x+C \\
& \int \cos x d x=\sin x+C \\
& \int \tan x d x=-\ln |\cos x|+C \\
& \int \cot x d x=\ln |\sin x|+C \\
& \int \sec x d x=\ln |\sec x+\tan x|+C \\
& \int \csc x d x=-\ln |\csc x+\cot x|+C \\
& \int \sec 2 x d x=\tan x+C \\
& \int \csc c^2 x d x=-\cot x+C \\
& \int \sec x \tan x d x=\sec x+C \\
& \int \csc x \cot x d x=-\csc x+C \\
& \int \frac{d x}{\sqrt{a^2-x^2}}=\sin ^{-1} \frac{x}{a}+C \\
& \int \frac{d x}{a^2+x^2}=\frac{1}{a} \tan ^{-1} \frac{x}{a}+C \\
& \int \frac{d x}{x \sqrt{x^2-a^2}}=\frac{1}{a} \sec ^{-1} \frac{|x|}{a}+C
\end{aligned}$
$\vec{A} \times \vec{B}=A B \sin \theta$
$\begin{aligned} \text { Speed } & =\frac{\text { Change in distance }}{\text { change in time }} \\ v & =\frac{\text { distance }}{\text { time }}\end{aligned}$
Average speed $=\frac{\text { total distance covered }}{\text { total time taken }}$
$\begin{aligned} & \text { Average Velocity }=\frac{\text { Total Displacement }}{\text { Total time taken }}, \\ & \vec{V}_{\text {avg }}=\frac{\vec{S}_{\text {net }}}{t}\end{aligned}$
$\vec{a}=\frac{\text { change in velocity }}{\text { time taken }}=\frac{\vec{v}_f-\vec{v}_i}{t}$
$v=u+a t$
$s=u t+\frac{1}{2} a t^2$
$v^2-u^2=2 a s$
Average angular velocity-

$\omega_{a v g}=\frac{\Delta \theta}{\Delta t}$
Time of flight

$T=\frac{2 U \sin \theta}{g \cos \beta}$
Range along incline plane

$R=\frac{2 u^2 \cdot \sin (\alpha-\beta) \cdot \cos \alpha}{g \cos ^2 \beta}$

Laws of motion

Inertial mass $(k g)=\frac{F}{a}$
Recoiling of Gun $\begin{aligned} & F=V_{\text {rel }}\left(\frac{d m}{d t}\right)=V(\mathrm{mn}) \\ & F=m n v \\ & F=\text { force required to hold the gun } \\ & n=\text { no.of bullets }\end{aligned}$
Force in non-uniform Circular Motion: $\begin{aligned} & F_c=m a_c=\frac{m v^2}{r} \quad\left(\vec{F}_c \perp \vec{v}\right) \\ & \mathrm{F}_{\mathrm{t}}=\mathrm{ma}_{\mathrm{t}} \\ & F_{\text {net }}=m \sqrt{a_c^2+a_t^2} \\ & \mathrm{~m}=\text { mass } \\ & \mathrm{a}_{\mathrm{c}}=\text { centripetal acceleration } \\ & \mathrm{a}_{\mathrm{t}}=\text { tangential acceleration } \\ & \mathrm{F}_{\mathrm{c}}=\text { centripetal force }\end{aligned}$

Work, Energy,y and Power

Work $=$ force $\times$ displacement $\times \cos \theta$
Kinetic Energy $=0.5 \times m \times v^2$
Potential Energy $=m \times g \times h$
Law of conservation of Energy $\Delta K+\Delta U=\Delta E=W_{f n c}$
Average power-

$P_{a v}=\frac{\Delta w}{\Delta t}=\frac{\int_0^t p \cdot d t}{\int_0^t d t}$
Instantaneous power-

$P=\frac{d w}{d t}=P=\vec{F} \cdot \vec{v}$
Where, $\vec{F} \rightarrow$ force

$\vec{v} \rightarrow \text { velocity }$
Ia perfectly inelastic collision:

When the colliding bodies are moving in the same direction

$\begin{aligned}
& m_1 u_1+m_2 u_2=\left(m_1+m_2\right) v \\
& v=\frac{m_1 u_1+m_2 u_2}{\left(m_1+m_2\right)}
\end{aligned}$

Loss in kinetic energy

$\begin{aligned}
& \Delta K \cdot E=\left(\frac{1}{2} m_1 u_1^2+\frac{1}{2} m_2 u_2^2\right)-\left(\frac{1}{2}\left(m_1+m_2\right) V^2\right) \\
& \Delta K \cdot E=\frac{1}{2}\left(\frac{m_1 m_2}{m_1+m_2}\right)\left(u_1-u_2\right)^2
\end{aligned}$

When the colliding bodies are moving in the opposite direction
$\begin{aligned}
& m_1 u_1+m_2\left(-u_2\right)=\left(m_1+m_2\right) v \\
& v=\frac{m_1 u_1-m_2 u_2}{m_1+m_2}
\end{aligned}$

Loss in kinetic energy

$\begin{aligned}
& \Delta K \cdot E=\left(\frac{1}{2} m_1 u_1^2+\frac{1}{2} m_2 u_2^2\right)-\left(\frac{1}{2}\left(m_1+m_2\right) V^2\right) \\
& \Delta K \cdot E=\frac{1}{2}\left(\frac{m_1 m_2}{m_1+m_2}\right)\left(u_1+u_2\right)^2
\end{aligned}$
Centre of mass:

For a system of N discrete particles

$\begin{aligned}
x_{c m} & =\frac{m_1 x_1+m_2 x_2 \ldots \ldots \ldots}{m_1+m_2 \ldots \ldots} \\
y_{c m} & =\frac{m_1 y_1+m_2 y_2+m_3 y_3 \ldots \ldots \ldots}{m_1+m_2+m_3 \ldots \ldots} \\
z_{c m} & =\frac{m_1 z_1+m_2 z_2+m_3 z_3 \ldots \ldots \ldots}{m_1+m_2+m_3 \ldots \ldots}
\end{aligned}$
Velocity of the centre of mass

$\vec{v}_{C M}=\frac{m_1 \vec{v}_1+m_2 \vec{v}_2 \ldots \ldots \ldots}{m_1+m_2 \ldots \ldots \ldots}$
Acceleration of the centre of mass

$\vec{a}_{C M}=\frac{m_1 \vec{a}_1+m_2 \vec{a}_2 \ldots \ldots \ldots}{m_1+m_2 \ldots \ldots .}$
Moment of inertia of a particle

$I=m r^2$
Radius of gyration (K): $K=\sqrt{\frac{I}{M}}$

Gravitation

Gravitational force $\vec{F}=G\left[\frac{M m}{r^2}\right]^r$
Acceleration due to gravity $g=\frac{G M}{R^2}$
Gravitational field Intensity:
$\vec{I}=\frac{\vec{F}}{m}$

$\vec{I} \rightarrow G$. field Intensity
$m \rightarrow$ mass of object
$\vec{f} \rightarrow$ Gravitational Force
Gravitational Potential: $\begin{aligned} & V=-\int \vec{I} \cdot \overrightarrow{d r} \\ & V \rightarrow \text { Gravitational potential } \\ & I \rightarrow \text { Field Intensity } \\ & d r \rightarrow \text { small distance }\end{aligned}$

Mechanical Properties of Solids

The magnitude of stress, $\sigma=\frac{F}{A}$
Unit of stress: $N / m^2$ or Pascal(Pa)
Dimension of stress: $\left[M L^{-1} T^{-2}\right]$
Volume stress $=\frac{F}{A}=P$
Longitudinal strain $=\frac{\Delta L}{L}$
Hooke’s law:" $\begin{gathered}\text { Stress } \alpha \text { Strain } \\ \Rightarrow \text { Stress }=E(\text { Strain }) \\ \Rightarrow E=\frac{\text { Stress }}{\text { Strain }}\end{gathered}$
Young's Modulus(Y): $Y=\frac{\text { longitudinal stress }}{\text { longitudinal strain }}=\frac{F / A}{\Delta L / L}=\frac{F l}{A \Delta L}$
Modulus of rigidity(G): $G=\frac{\text { shearing stress }}{\text { shearing strain }}=\frac{F / A}{x / L}=\frac{F l}{A x}=\frac{F}{A \phi}$
Bulk Modulus(B):
$\begin{gathered}
\text { Volumestress }=\frac{F}{A}=\text { Pressure } \\
\qquad B=-\frac{P}{\Delta V / V}
\end{gathered}$

where $\mathrm{P}=$ increase in pressure, $\mathrm{V}=$ original volume, $\Delta V=$ change in volume

Mechanical Properties of Fluids

Pressure $P=\frac{F}{A}$
Buoyant force $F_B=\rho V g$
Where $\mathrm{F}_{\mathrm{B}}=$ Buoyant force

$\rho=\text { density of the fluid }$
V= Volume of the solid body immersed in the liquid or Volume of the fluid displaced
Relative density of a body

$R. D=\frac{\text { density of body }}{\text { density of water }}$
Bernoulli's equation $P+\rho g h+\frac{1}{2} \rho v^2=\text { constant }$

$P \rightarrow$ Pressure energy per unit volume $\rho g h \rightarrow$ Potential Energy per unit volume $\frac{1}{2} \rho v^2 \rightarrow$ Kinetic Energy per unit volume
Velocity gradient $=\frac{\text { chane in velocity }}{\text { change in height }}$

Kinetic Theory of Gases

Boyle's law $V \propto \frac{1}{P}$
Charle's law $V \propto T$
GRAHAM’S LAW OF DIFFUSION
$r \propto \frac{1}{\sqrt{\rho}} \propto \frac{1}{\sqrt{M}} \propto V_{r m s}$
Where $r=$ rate of diffusion of gas
$\rho=$ Density of the gas
M = Molecular weight of the gas
$V_{r m s}=$ Root mean square velocity
Ideal gas equation $P V=n R T$
Degree of freedom

$f=3 N-R$
Where
$N=$ no. of particle
$R=$ no. of relation

Thermodynamics

$\Delta W=P \Delta V=P\left(V_f-V_i\right)$
Heat transfer -

$\begin{aligned}
& \Delta Q=m L_{\text {(for change of state) }} \\
& \Delta Q=m s \Delta T \text { (for change in temperature) }
\end{aligned}$
First law of thermodynamics $\Delta Q=\Delta U+\Delta W$
Efficiency of Heat Engine $\eta=\frac{\text { Work done }}{\text { Heat input }}=\frac{W}{Q_1}$
Entropy $d S=\frac{\text { Heat absorbed by system }}{\text { Absolute temperature }}$ or $d S=\frac{d Q}{T}$

Oscillations

General equation of SHM
1. For Displacement:-
$x=A \operatorname{Sin}(w t+\phi) ;$ where $\phi$ is initial phase or epoch and $(\omega t+\phi)$ is called as phase.
Various displacement equations:-
(1) $x=A$ Sin $\omega t \Rightarrow$ when particle starts from mean position towards right.
(2) $x=-$ ASinwt $\Rightarrow$ when particle starts from mean position towards left.
(3) $x=A C o s w t \Rightarrow$ when particle starts from right extreme position towards left
(4) $x=-$ ACoswt $\Rightarrow$ when particle starts from the left extreme position towards the right.
2. For Velocity (v):-

$$
\begin{aligned}
x & =A \operatorname{Sin}(\omega t+\phi) \\
\Rightarrow v & =\frac{d x}{d t}=A \omega \operatorname{Cos}(\omega t+\phi)=A \omega \operatorname{Sin}\left(\omega t+\phi+\frac{\pi}{2}\right)
\end{aligned}
$$

3. For Acceleration:-

$$
\begin{aligned}
x & =A \operatorname{Sin}(\omega t+\phi) \\
\Rightarrow v & =\frac{d x}{d t}=A \omega \operatorname{Cos}(\omega t+\phi)=A \omega \operatorname{Sin}\left(\omega t+\phi+\frac{\pi}{2}\right) \\
\Rightarrow a & =\frac{d v}{d t}=-A \omega^2 \operatorname{Sin}(\omega t+\phi)=A \omega^2 \operatorname{Sin}(\omega t+\phi+\pi)=-\omega^2 x
\end{aligned}
$$
Differential equation of SHM $\begin{aligned} & \frac{d v}{d t}=-\omega^2 x \\ & \Rightarrow \frac{d}{d t}\left(\frac{d x}{d t}\right)=-\omega^2 x \\ & \Rightarrow \frac{d^2 x}{d t^2}+\omega^2 x=0\end{aligned}$
Kinetic Energy $K=\frac{1}{2} m v^2$
Spring Force $F=-k x$

Electric Charges and Fields

Coulomb's Law $F=\frac{K Q_1 Q_2}{r^2}$
Electric Field Intensity $E=\frac{F}{q_0}$
Linear charge distribution $\lambda=\frac{q}{L}=\frac{C}{m}=C m^{-1}$
Surface charge distribution $\sigma=\frac{Q}{A}=\frac{C}{m^2}=C m^{-2}$
Volume Charge distribution $\rho=\frac{Q}{V}=\frac{C}{m^3}=C m^{-3}$
Electric Dipole $(\vec{P})=q(2 \vec{l})$
Torque on dipole $\tau=Q E d \sin \theta$
Electric flux $\phi=\int \vec{E} \cdot \vec{d} A$
Gauss's law $\phi=\oint E \cdot d A$

Electrostatic Potential and Capacitance

Electric Potential $V=\frac{W_{e x t}}{q_0}$
Electrostatic Potential energy (U) $U=\frac{K Q q}{r}$
Capacitance $C=\frac{Q}{V}$
Series combination $C=\frac{c_1 c_2}{c_1+c_2}$
Parallel combination $\mathbf{C}=c_1+c_2+c_3$
Energy stored in capacitor $U=\frac{1}{2} \frac{Q^2}{C}=\frac{1}{2} Q V=\frac{1}{2} C V^2$

Current Electricity

$I=\frac{Q}{t}$
Current Density $\bar{j}=\frac{\Delta i}{\Delta A}$
Drift Velocity $v_d=\frac{-e \vec{E}}{m} \tau$
Ohm’s law $V=I R$
Mobility $\mu=\frac{v_d}{E}$
Resistance $R=\rho \frac{l}{A}$
Specific Resistance $(\rho)$ $\rho=\frac{m}{n e^2 \tau}$
Series grouping of Resistance $R_{e q}=R_1+R_2+R_3+\cdots+R_n$
Parallel Grouping Of Resistance $\frac{1}{R_{e q}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots+\frac{1}{R_n}$
Kirchhoff's first law $i_1+i_3=i_2+i_4$
Kirchoff's second law $-i_1 R_1+i_2 R_2-E_1-i_3 R_3+E_2+E_3-i_4 R_4=0$

Moving Charges and Magnetism

Biot-Savart Law $d B=K \frac{I d l \sin \theta}{r^2}$
Ampere's Circuital Law $\oint \vec{B} d \vec{l}=\mu_0 \sum i$
Magnetic Feild of Toroid $\mathrm{B}=\frac{\mu_0 \mathrm{NI}}{2 \pi r}$
Lorentz Force $\vec{F}=q[\vec{E}+(\vec{v} \times \vec{B})]$
Current sensitivity (S_i) $S_i=\frac{\alpha}{i}=\frac{N B A}{C}$
Voltage sensitivity (S_V) $S_V=\frac{\alpha}{V}=\frac{\alpha}{i R}=\frac{S_i}{R}=\frac{N B A}{R C}$

Magnetism and Matter

Magnetic flux $\phi_B=\int \vec{B} \cdot d \vec{S}$
Magnetic Intensity $H=\frac{B_0}{\mu_0}$

Electromagnetic Induction

Faraday's Law: Rate of change of magnetic Flux= $\varepsilon=\frac{-d \phi}{d t}$ where $d \phi \rightarrow \phi_2-\phi_{1=}$ change in flux
Induced Current $I=\frac{\varepsilon}{R}=\frac{-N}{R} \frac{d \phi}{d t}$
Induced Charge $d q=i . d t=\frac{-N}{R} \frac{d \phi}{d t} . d t$
Induced Power $P=\frac{\varepsilon^2}{R}=\frac{N^2}{R}\left(\frac{d \phi}{d t}\right)^2$
Induced Electric Field $\varepsilon=\oint \vec{E}_{i n} \cdot \overrightarrow{d l}=\frac{-d \phi}{d t}$
Energy Stored In An Inductor $U=\frac{1}{2} L I^2$

Electromagnetic Waves

Maxwell's equations 1. $\oint \mathbf{E} \cdot \mathrm{d} \mathbf{A}=\mathrm{Q} / \varepsilon_0$
(Gauss's Law for electricity)
2. $\oint \mathbf{B} \cdot \mathrm{d} \mathbf{A}=0$
(Gauss's Law for magnetism)
3. $\oint \mathbf{E} \cdot \mathrm{d} \mathbf{l}=\frac{-\mathrm{d} \phi_{\mathrm{B}}}{\mathrm{d} t}$
(Faraday's Law)
4. $\oint \mathbf{B} \cdot \mathrm{d} \mathbf{l}=\mu_0 i_{\mathrm{c}}+\mu_0 \varepsilon_0 \frac{\mathrm{~d} \phi_E}{\mathrm{~d} t}$
(Ampere-Maxwell Law)
Intensity (I) $I=\frac{\text { Total EM energy }}{\text { Surface area } \times \text { Time }}=\frac{\text { Total energy density } \times \text { Volume }}{\text { Surface area } \times \text { Time }}$
Wavelength of EM Wave
$\lambda=\frac{\lambda_o}{\mu}$

$\lambda_o=$ Wavelength in vacuum
$\mu$ = Refractive index of medium

Ray Optics and Optical Instruments

Mirror formula $f=\frac{1}{u}+\frac{1}{v}$
lateral magnification $m_v=\frac{\text { height of image }}{\text { height of object }}=\frac{h_i}{h_0}$
Refractive index $\mu=\frac{c}{v}$
Magnifying power $m=\frac{\text { Visual angle with instrument }(\beta)}{\text { Visual angle when object is placed at least distance of distinct vision }(\alpha)}$

Dual Nature of Matter and Radiation

Work function $\phi=h \nu_o=\frac{h c}{\lambda_o}$
$E=h \nu=\frac{h c}{\lambda}$
Momentum of the photon $p=m \times c=\frac{E}{c}=\frac{h \nu}{c}=\frac{h}{\lambda}$
Einstein's Photoelectric Equation K.E. $=h v-\Phi$
Photon Flux $\phi=\frac{\text { Intensity }}{\text { Energy of each photon }}=\frac{I}{E}=\frac{n}{A}$
De Broglie’s Equation $\lambda=\frac{h}{p}$

JEE Main Formulas for Chemistry 2027

While studying chemistry, they need to revise and practice the chemical equations and symbols. Chemistry is a tough subject, but when candidates practice chemical equations and review the properties, formulas, and symbols, they will have command over the subject. Candidates can check the JEE Main Chemistry formulas below

JEE Main Important Formulas of Physical Chemistry

Some Basic Concepts in Chemistry

Boyle's Law: $P_1 V_1=P_2 V_2$ (at constant T and n )
Charles's Law: $\frac{V_1}{T_1}=\frac{V_2}{T_2}($ at constant P and n$)$
Avogadro's Law: $\frac{V}{n}=$ constant
Average Atomic Mass $=\frac{\Sigma(\text { Mass of Isotopes })_i \times(\% \text { abundance })_i}{100}$
Mole $=\frac{\mathrm{W}}{\mathrm{M}}=\frac{(\mathrm{Wt} . \text { of substance in gm. })}{(\text { Molar mass of substance }(\mathrm{G} . \mathrm{m} . \mathrm{m}))}$
Mass $\%$ of an element $=\frac{\text { Mass of that element in one mole of the compound }}{\text { Molar mass of the compound }} \times 100$
Equivalent Weight $=\frac{\text { Molecular weight }}{\mathrm{n}-\text { factor }(\mathrm{x})}$

Atomic Structure

Frequency $\nu=\frac{1}{\mathrm{~T}}$
Wave number $(\bar{\nu})$ $\bar{\nu}=\frac{1}{\lambda}$
$E=h \nu=\frac{h c}{\lambda}$
Line Spectrum of Hydrogen-like atoms

$\frac{1}{\lambda}=R Z^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$
Bohr radius of nth orbit:

$\mathrm{r}_{\mathrm{n}}=0.529 \frac{\mathrm{n}^2}{\mathrm{Z}} \mathrm{~A}^0$
Velocity of electron in the nth orbit:

$\mathrm{V}_{\mathrm{n}}=\left(2.18 \times 10^6\right) \frac{\mathrm{Z}}{\mathrm{n}} \mathrm{~m} / \mathrm{s}$

where Z is the atomic number
Total energy of an electron in the nth orbit:

$\mathrm{E}_{\mathrm{n}}=-13.6 \frac{\mathrm{Z}^2}{\mathrm{n}^2} \mathrm{eV}=-2.18 \times 10^{-18} \frac{\mathrm{Z}^2}{\mathrm{n}^2} \mathrm{~J}$

where Z is the atomic number
Heisenberg Uncertainty Principle: $\Delta x. \Delta P \geq \frac{h}{4 \pi}$
$\mathrm{E}_{\mathrm{n}}=-\frac{1312 \times \mathrm{Z}^2}{\mathrm{n}^2} \mathrm{~kJ} / \mathrm{mol}$

Chemical Thermodynamics

Expansion Work $=\mathrm{P} \times \Delta \mathrm{V}=-\mathrm{P}_{\text {ext. }}\left[\mathrm{V}_2-\mathrm{V}_1\right]$
$\mathrm{P}=$ external pressure And $\Delta \mathrm{V}=$ increase or decrease in volume.
Work done in a reversible isothermal process

$\begin{aligned}
& \mathrm{W}=-2.303 \mathrm{nRT} \log _{10} \frac{\mathrm{~V}_2}{\mathrm{~V}_1} \\
& \mathrm{~W}=-2.303 \mathrm{nRT} \log _{10} \frac{\mathrm{P}_1}{\mathrm{P}_2}
\end{aligned}$
Work done in an irreversible isothermal process
Work $=-\mathrm{P}_{\text {ext. }}\left(\mathrm{V}_2-\mathrm{V}_1\right)$
That is, Work $=-\mathrm{P} \times \Delta \mathrm{V}$
$W=\Delta E=n C_V \Delta T$
Enthalpy: $H=U+p V$
First Law of Thermodynamics: $\Delta U=q+W$
$\Delta \mathrm{G}=\Delta \mathrm{H}-\mathrm{T} \Delta(\mathrm{S})$
$\Delta G=-n F E$

Equilibrium

For a reaction:
$\mathrm{mA}+\mathrm{nB} \rightleftharpoons \mathrm{pC}+\mathrm{qD}$ $\frac{\mathrm{K}_{\mathrm{f}}}{\mathrm{K}_{\mathrm{b}}}=\frac{[\mathrm{C}]^{\mathrm{p}}[\mathrm{D}]^{\mathrm{q}}}{[\mathrm{A}]^{\mathrm{m}}[\mathrm{B}]^{\mathrm{n}}}=\mathrm{K}_{\mathrm{c}}$
$\mathrm{pH}=-\log _{10}\left[\mathrm{H}^{+}\right]$
$\mathrm{k}_{\mathrm{w}}=\left[\mathrm{H}^{+}\right]\left[\mathrm{OH}^{-}\right]=10^{-14}$
$\mathrm{pH}=\mathrm{pK}_{\mathrm{a}}+\log _{10} \frac{[\text { Salt }]}{\text { Acid }}$
$\mathrm{pOH}=\mathrm{pK}_{\mathrm{b}}+\log _{10} \frac{[\text { Salt }]}{[\text { Base }]}$

ELECTROCHEMISTRY

$\mathrm{W}=\frac{\text { Eit }}{96500}$
$\begin{aligned}
& \frac{E_1}{E_2}=\frac{M_1}{M_2} \text { or } \frac{W_1}{W_2}=\frac{Z_1}{Z_2} \\
& E_1=\text { equivalent weight } \\
& E_2=\text { equivalent weight }
\end{aligned}$
W or M = mass deposited
$\begin{aligned} & E_{\text {cell }} \text { or } E M F=\left[E_{\text {red }}(\text { cathode })-E_{\text {red }}(\text { anode })\right] \\ & E_{\text {eell }}^{\circ} \text { or } E M F^{\circ} \\ & =\left[E_{\text {red }}^{\circ}(\text { cathode })-E_{\text {red }}^{\circ}(\text { anode })\right]\end{aligned}$
$\mathrm{E}=\mathrm{E}^{\circ}-\frac{\mathrm{RT}}{\mathrm{nF}} \ln Q$
$\mathrm{xA}+\mathrm{yB} \xrightarrow{\mathrm{ne}^{-}} \mathrm{mC}+\mathrm{nD}$
The emf can be calculated as

$\text { Ecell }=\mathrm{E}^{\circ} \text { cell }-\frac{0.059}{\mathrm{n}} \log \frac{[\mathrm{C}]^{\mathrm{m}}[\mathrm{D}]^{\mathrm{n}}}{[\mathrm{~A}]^x[\mathrm{~B}]^{\mathrm{y}}}$
$\wedge_{\mathrm{m}}=\kappa \times \frac{1000}{\mathrm{c}}$
$\wedge_{\text {eq }}=\frac{1000 \times \kappa}{\mathrm{N}}$

Solutions

Mass $\%$ of a component $=\frac{\text { Mass of the component in the solution }}{\text { Total mass of the solution }} \times 100$
Volume $\%$ of a component $=\frac{\text { Volume of the component }}{\text { Total volume of solution }} \times 100$
Mass by Volume $\%$ of a component $=\frac{\text { Mass of the component }}{\text { Total volume of solution }} \times 100$
Parts per million $=\frac{\text { Number of parts of the component }}{\text { Total number of parts of all components of the solution }} \times 10^6$
Mole fraction of a component $=\frac{\text { Number of moles of the component }}{\text { Total number of moles of all the components }}$
Molarity: $(M)=\frac{\text { No. of Moles of Solutes }}{\text { Volume of Solution in Liters }}$
Molality: $(m)=\frac{\text { No. of Moles of Solutes }}{\text { Mass of solvent in } \mathrm{kg}}$
$\left(P_T\right)=P_A^o X_A+P_B^o X_B$ ($\begin{aligned} & P_A=P_A^o X_A \\ & P_B=P_B^o X_B \\ & P_T=P_A+P_B\end{aligned}$)
$\Delta \mathrm{T}_{\mathrm{b}}=\mathrm{K}_{\mathrm{b}} \times \frac{\mathrm{w}}{\mathrm{M}} \times \frac{1000}{\mathrm{~W}}$
$\Delta T_f=K_f \times \frac{w}{M} \times \frac{1000}{W}$
$\Pi=C R T$
$\begin{aligned} & \mathrm{i}=\frac{\text { Observed number of solute particles }}{\text { Number of particles initially taken }} \\ & \mathrm{i}=\frac{\text { Observed value of colligative property }}{\text { Theoretical value of colligative property }}\end{aligned}$

Chemical kinetics

Unit of average velocity $=\frac{\text { Unit of concentration }}{\text { Unit of time }}=\frac{\text { mole }}{\text { litre second }}=$ mole litre $^{-1}$ second $^{-1}$
$\mathrm{aA}+\mathrm{bB} \rightarrow \mathrm{cC}+\mathrm{dD}$
Rate w.r.t. $[\mathrm{A}]=-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}} \times \frac{1}{\mathrm{a}}$
Rate w.r.t. $[B]=-\frac{d[B]}{d t} \times \frac{1}{b}$
Rate w.r.t. $[\mathrm{C}]=-\frac{\mathrm{d}[\mathrm{C}]}{\mathrm{dt}} \times \frac{1}{\mathrm{c}}$
Rate w.r.t. $[\mathrm{D}]=-\frac{\mathrm{d}[\mathrm{D}]}{\mathrm{dt}} \times \frac{1}{\mathrm{~d}}$
$\mathrm{R} \propto[\mathrm{A}]^{\mathrm{p}}[\mathrm{B}]^{\mathrm{q}}$
Unit of Rate Constant-

$\begin{aligned}
& \text { The differential rate expression for } \mathrm{n}^{\text {th }} \text { order reaction is as follows: } \\
& \qquad-\frac{\mathrm{dx}}{\mathrm{dt}}=\mathrm{k}(\mathrm{a}-\mathrm{x})^{\mathrm{n}} \\
& \text { or } \quad \mathrm{k}=\frac{\mathrm{dx}}{(\mathrm{a}-\mathrm{x})^{\mathrm{n}} \mathrm{dt}}=\frac{(\text { concentration })}{(\text { concentration })^{\mathrm{n}} \text { time }}=(\text { conc. })^{1-\mathrm{n}} \text { time }^{-1}
\end{aligned}$
For the first-order reaction,

$k=\frac{2.303}{t} \log \frac{[\mathrm{R}]_0}{[\mathrm{R}]}$
$t_{1 / 2}=\frac{0.693}{k}$
For any general nth order reaction, it is evident that,

$\mathrm{t}_{\frac{1}{2}} \propto[\mathrm{~A}]_0^{1-\mathrm{n}}$
It is to be noted that the above formula is applicable for any general nth-order reaction except $\mathrm{n}=1$.
Arrhenius Equation: $\mathrm{k}=\mathrm{Ae}^{-\mathrm{Ea} / \mathrm{RT}}$
$\log \frac{\mathrm{K}_2}{\mathrm{~K}_1}=\frac{\mathrm{Ea}}{2.303 \mathrm{R}}\left[\frac{1}{\mathrm{~T}_1}-\frac{1}{\mathrm{~T}_2}\right]$

JEE Main Important Formulas of Inorganic Chemistry

Coordination Compounds

$\mathrm{EAN}=Z-O+2 L$
Where:
$\mathbf{Z}=$ Atomic number of the central metal atom/ion
$\mathbf{O}=$ Oxidation state of the metal atom/ion
L = Number of ligands (or donor atoms) $\times$ number of electrons donated per ligand
Crystal Field Stabilisation Energy (CFSE):
Octahedral:

$\mathrm{CFSE}=(-0.4 x+0.6 y) \Delta_0$

Tetrahedral:

$\mathrm{CFSE}=(-0.6 x+0.4 y) \Delta_t$

where $x=t_2 g$ electrons, $y=$ e_g electrons

d- & f-Block Elements

Magnetic Moment:

$\mu=\sqrt{n(n+2)} \mathrm{BM}$

Chemical Bonding and Molecular Structure

Formal Charge:

$\text { F.C. }=V-N-\frac{B}{2}$

( $\mathrm{V}=$ valence electrons, $\mathrm{N}=$ non-bonding, $\mathrm{B}=$ bonding electrons)
Bond Order (Molecular Orbital Theory):

$\text { Bond Order }=\frac{\left(N_b-N_a\right)}{2}$
Dipole Moment:

$\mu=q \times d$

( $q=$ charge,$d=$ distance between charges)

JEE Main Important Formulas: Inorganic Chemistry

Some Basic Principles of Organic Chemistry

Application of Inductive Effect

The decreasing -I effect or increasing +I effect order is as follows:

$\begin{aligned}
& -\mathrm{NH}_3+>-\mathrm{NO}_2>-\mathrm{SO}_2 \mathrm{R}>-\mathrm{CN}>-\mathrm{SO}_3 \mathrm{H}>-\mathrm{CHO}>-\mathrm{CO}>-\mathrm{COOH}>-\mathrm{F}>-\mathrm{COCl}>-\mathrm{CONH}_2>-\mathrm{Cl}>-\mathrm{Br}>-\mathrm{I}>-\mathrm{OR}>-\mathrm{OH}>-\mathrm{NR}_2>-\mathrm{NH}_2> \\
& -\mathrm{C}_6 \mathrm{H}_5>-\mathrm{CH}=\mathrm{CH}_2>-\mathrm{H}
\end{aligned}$
Degree of Unsaturation (DU or IHD):

$\mathrm{DU}=\frac{2 C+2-H+N-X}{2}$

( $\mathrm{C}=$ carbon, $\mathrm{H}=$ hydrogen, $\mathrm{N}=$ nitrogen, $\mathrm{X}=$ halogen)

Hydrocarbons

Alkanes: $C_n H_{2 n+2}$
Alkenes: $C_n H_{2 n}$
Alkynes: $C_n H_{2 n-2}$

Carboxylic Acids and Derivatives

Method of Preparation of Carboxylic Acid

JEE Main formulas for Maths 2027

Candidates must go through all the formulas and practice the mathematical problems. Without formulas, you cannot solve any problem, even if you know how to solve it. Revising the formulas daily is very important. Here we have provided the Mathematics formulas for JEE Mains.

Sets, Relations, and Functions

Properties of union $\mathrm{A} \cup \mathrm{B}=\mathrm{B} \cup \mathrm{A} \quad$ (Commutative Property)
$(\mathrm{A} \cup \mathrm{B}) \cup \mathrm{C}=\mathrm{A} \cup(\mathrm{B} \cup \mathrm{C})$ (Associative property)
$\mathrm{A} \cup \varphi=\mathrm{A}$ (Law of identity element, $\varphi$ is the identity of Null Set)
$\mathrm{A} \cup \mathrm{A}=\mathrm{A}$ (Idempotent law)
$\mathrm{U} \cup \mathrm{A}=\mathrm{U}($ Law of U$)$
If A is a subset of B , then $\mathrm{A} \cup \mathrm{B}=\mathrm{B}$
Properties of intersection $\mathrm{A} \cap \mathrm{B}=\mathrm{B} \cap \mathrm{A}$ (Commutative law).
$(\mathrm{A} \cap \mathrm{B}) \cap \mathrm{C}=\mathrm{A} \cap(\mathrm{B} \cap \mathrm{C})$ (Associative law).
$\mathrm{A} \cap \phi=\phi$,
$\mathrm{A} \cap \mathrm{U}=\mathrm{A}$ (Law of $\phi$ and U ).
$\mathrm{A} \cap \mathrm{A}=\mathrm{A}$ (Idempotent law)
If A is subset of B , then $\mathrm{A} \cap \mathrm{B}=\mathrm{A}$
Properties of Difference of Sets 1. In general A - B does not equal B - A
2. $\mathrm{A}-\mathrm{A}=\phi$
3. $\mathrm{A}-\phi=\mathrm{A}$
4. $\mathrm{A}-\mathrm{U}=\phi$
5. If A is a subset of B , then $\mathrm{A}-\mathrm{B}=\phi$
Symmetric Difference of Sets ( A Δ B ) $A \Delta B=(A-B) \cup(B-A)$
Properties of Compliment $\begin{aligned} & A \cup A^{\prime}=U \\ & A \cap A^{\prime}=\varphi \\ & \left(A^{\prime}\right)^{\prime}=A \\ & U^{\prime}=\varphi \text { and } \varphi^{\prime}=U \\ & A-B=A \cap B^{\prime}\end{aligned}$

Complex Numbers and Quadratic Equations

Equality of Complex Numbers $a+i b=c+i d$
Addition of Two Complex Numbers $z_1+z_2=(a+i b)+(c+i d)=(a+c)+i(b+d)$
Difference of Two Complex Numbers $z_1-z_2=(a+i b)-(c+i d)=(a-c)+i(b-d)$
Multiplication of Two Complex Numbers $z_1 \times z_2=(a+i b)(c+i d)$
Division of Two Complex Numbers $\frac{z_1}{z_2}=\frac{a+i b}{c+i d} \cdot \frac{c-i d}{c-i d}$
Modulus $|z|=\sqrt{x^2+y^2}$

Matrices and Determinants

Addition of matrices $A=\left[a_{i j}\right]_{m \times n}, B=\left[b_{i j}\right]_{m \times n}$ Then, $A+B=\left[a_{i j}+b_{i j}\right]_{m \times n}$ for all $i, j$
Subtraction of matrices $A=\left[a_{i j}\right]_{m \times n}, B=\left[b_{i j}\right]_{m \times n}$ Then, $A-B=\left[a_{i j}-b_{i j}\right]_{m \times n}$ for all $i, j$
Multiplication of Determinant $\Delta_1=\left|\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|$ and $\Delta_2=\left|\begin{array}{ccc}\alpha_1 & \beta_1 & \gamma_1 \\ \alpha_2 & \beta_2 & \gamma_2 \\ \alpha_3 & \beta_3 & \gamma_3\end{array}\right|$ then $\Delta_1 \times \Delta_2=\left|\begin{array}{lll}a_1 \alpha_1+b_1 \beta_1+c_1 \gamma_1 & a_1 \alpha_2+b_1 \beta_2+c_1 \gamma_2 & a_1 \alpha_3+b_1 \beta_3+c_1 \gamma_3 \\ a_2 \alpha_1+b_2 \beta_1+c_2 \gamma_1 & a_2 \alpha_2+b_2 \beta_2+c_2 \gamma_2 & a_2 \alpha_3+b_2 \beta_3+c_2 \gamma_3 \\ a_3 \alpha_1+b_3 \beta_1+c_3 \gamma_1 & a_3 \alpha_2+b_3 \beta_2+c_3 \gamma_2 & a_3 \alpha_3+b_3 \beta_3+c_3 \gamma_3\end{array}\right|$

Sequence and Series

Series $\mathrm{S}_{\mathrm{n}}=a_1+a_2+a_3+\ldots \ldots \ldots .+a_n=\sum_{\mathrm{r}=1}^{\mathrm{n}} a_r=\sum a_r$
General Term of an AP $\begin{aligned} & a_1=a+(1-1) d=a \\ & a_2=a+(2-1) d=a+d \\ & a_3=a+(3-1) d=a+2 d \\ & a_4=a+(4-1) d=a+3 d\end{aligned}$
Arithmetic Mean $A=\frac{a_1+a_2+a_3+\ldots . .+a_n}{n}$.
.General Term of a GP $\begin{aligned} & a_1=a=a r^{1-1} \quad\left(1^{\text {st }} \text { term }\right) \\ & a_2=a r=a r^{2-1} \quad\left(2^{\text {nd }} \text { term }\right) \\ & a_3=a r^2=a r^{3-1} \quad\left(3^{\text {rd }} \text { term }\right)\end{aligned}$
Geometric Mean $G=\sqrt[n]{a_1 \cdot a_2 \cdot a_3 \cdot \ldots . . \cdot a_n}$
Sum of n-term of a GP $S_n=a\left(\frac{r^n-1}{r-1}\right)$
.Sum of an infinite GP $\mathrm{S}_{\infty}=\frac{\mathrm{a}}{1-\mathrm{r}}$
Harmonic Mean
$H=\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\ldots .+\frac{1}{a_n}}$

Trigonometry

Trigonometric Functions of Acute Angles $\begin{aligned} & \text { Sine } \quad \sin t=\frac{\text { opposite }}{\text { hypotenuse }} \\ & \text { Cosine } \quad \cos t=\frac{\text { adjacent }}{\text { hypotenuse }} \\ & \text { Tangent } \quad \tan t=\frac{\text { opposite }}{\text { adjacent }}\end{aligned}$
Trigonometric Identities $\begin{aligned} & \sin ^2 t+\cos ^2 t=1 \\ & 1+\tan ^2 t=\sec ^2 t \\ & 1+\cot ^2 t=\csc ^2 t \\ & \tan t=\frac{\sin t}{\cos t}, \quad \cot t=\frac{\cos t}{\sin t}\end{aligned}$
Trigonometric Ratio for Compound Angles $\begin{aligned} & \cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta \\ & \cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta \\ & \sin (\alpha-\beta)=\sin \alpha \cos \beta-\cos \alpha \sin \beta \\ & \sin (\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \beta\end{aligned}$$\begin{aligned} \tan (\alpha+\beta) & =\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta} \\ \tan (\alpha-\beta) & =\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \tan \beta} \\ \cot (\alpha+\beta) & =\frac{\cot \alpha \cot \beta-1}{\cot \alpha+\cot \beta} \\ \cot (\alpha-\beta) & =\frac{\cot \alpha \cot \beta+1}{\cot \beta-\cot \alpha}\end{aligned}$
Product into Sum/Difference 1. $2 \cos \alpha \cos \beta=[\cos (\alpha-\beta)+\cos (\alpha+\beta)]$
2. $2 \sin \alpha \sin \beta=[\cos (\alpha-\beta)-\cos (\alpha+\beta)]$
3. $2 \sin \alpha \cos \beta=[\sin (\alpha+\beta)+\sin (\alpha-\beta)]$
4. $2 \cos \alpha \sin \beta=[\sin (\alpha+\beta)-\sin (\alpha-\beta)]$
Sum/Difference into Product 1. $\sin \alpha+\sin \beta=2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)$
2. $\sin \alpha-\sin \beta=2 \sin \left(\frac{\alpha-\beta}{2}\right) \cos \left(\frac{\alpha+\beta}{2}\right)$
3. $\cos \alpha-\cos \beta=-2 \sin \left(\frac{\alpha+\beta}{2}\right) \sin \left(\frac{\alpha-\beta}{2}\right)$
4. $\cos \alpha+\cos \beta=2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)$
Reduction Formula $\begin{aligned} \sin ^2 \theta & =\frac{1-\cos (2 \theta)}{2} \\ \cos ^2 \theta & =\frac{1+\cos (2 \theta)}{2} \\ \tan ^2 \theta & =\frac{1-\cos (2 \theta)}{1+\cos (2 \theta)}\end{aligned}$
Triple Angle Formula 1. $\sin 3 \mathrm{~A}=3 \sin \mathrm{~A}-4 \sin ^3 \mathrm{~A}$
2. $\cos 3 \mathrm{~A}=4 \cos ^3 \mathrm{~A}-3 \cos A$
3. $\tan 3 \mathrm{~A}=\frac{3 \tan \mathrm{~A}-\tan ^3 \mathrm{~A}}{1-3 \tan ^2 \mathrm{~A}}$
Half Angle Formula 1. $\sin \left(\frac{\alpha}{2}\right)= \pm \sqrt{\frac{1-\cos \alpha}{2}}$
2. $\quad \cos \left(\frac{\alpha}{2}\right)= \pm \sqrt{\frac{1+\cos \alpha}{2}}$
3. $\tan \left(\frac{\alpha}{2}\right)= \pm \sqrt{\frac{1-\cos \alpha}{1+\cos \alpha}}$

Co-ordinate Geometry

Internal division $\mathbf{x}=\frac{\mathbf{m x}_2+\mathbf{n} \mathbf{x}_1}{\mathbf{m}+\mathbf{n}}, \mathbf{y}=\frac{\mathbf{m y}_2+\mathbf{n y}_1}{\mathbf{m}+\mathbf{n}}$
External Division $\mathbf{x}=\frac{\mathbf{m x}_2-\mathbf{n x}_1}{\mathbf{m}-\mathbf{n}}, \quad \mathbf{y}=\frac{\mathbf{m y}_2-\mathbf{n y}_1}{\mathbf{m}-\mathbf{n}}$
Parametric Form for $(x-h)^2+(y-k)^2=\mathbf{r}^2$

Limit, Continuity And Differentiability

Sum law for limits : $\lim _{x \rightarrow a}(f(x)+g(x))=\lim _{x \rightarrow a} f(x)+\lim _{x \rightarrow a} g(x)=L+M$
Difference law for limits : $\lim _{x \rightarrow a}(f(x)-g(x))=\lim _{x \rightarrow a} f(x)-\lim _{x \rightarrow a} g(x)=L-M$
Constant multiple law for limits : $\lim _{x \rightarrow a} c f(x)=c \cdot \lim _{x \rightarrow a} f(x)=c L$
Product law for limits : $\lim _{x \rightarrow a}(f(x) \cdot g(x))=\lim _{x \rightarrow a} f(x) \cdot \lim _{x \rightarrow a} g(x)=L \cdot M$
Quotient law for limits : $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}=\frac{L}{M}$ for $M \neq 0$
Binomial Expansion for any index
$(1+x)^n=1+n x+\frac{n(n-1)}{2!} x^2+\frac{n(n-1)(n-2)}{3!} x^3 \ldots$

where, $|x|<1$
DIFFERENTIATION $\frac{d}{d x}($ constant $)=0$ $\frac{d}{d x}\left(\mathbf{x}^{\mathbf{n}}\right)=\mathbf{n} \mathbf{x}^{\mathbf{n}-\mathbf{1}}$ $\frac{d}{d x}\left(\mathbf{a}^{\mathbf{x}}\right)=\mathbf{a}^{\mathbf{x}} \log _{\mathrm{e}} \mathbf{a}$

$\frac{d}{d x}\left(\mathbf{e}^{\mathbf{x}}\right)=\mathbf{e}^{\mathbf{x}} \log _{\mathbf{e}} \mathbf{e}=\mathbf{e}^{\mathbf{x}}$

$\frac{d}{d x}\left(\log _{\mathbf{a}}|\mathbf{x}|\right)=\frac{1}{\mathbf{x} \log _{\mathbf{e}} \mathbf{a}}, \quad \mathbf{x} \neq \mathbf{0}$
$\frac{d}{d x}\left(\log _{\mathrm{e}}|\mathbf{x}|\right)=\frac{1}{\mathbf{x}}, \quad \mathbf{x} \neq \mathbf{0}$
Sum Rule $\frac{d}{d x}(f(x)+g(x))=\frac{d}{d x}(f(x))+\frac{d}{d x}(g(x))$
Difference Rule $\frac{d}{d x}(f(x)-g(x))=\frac{d}{d x}(f(x))-\frac{d}{d x}(g(x))$
Constant Multiple Rule $\frac{d}{d x}(k f(x))=k \frac{d}{d x}(f(x))$
Product Rule $\frac{d}{d x}(f(x) g(x))=g(x) \cdot \frac{d}{d x}(f(x))+f(x) \cdot \frac{d}{d x}(g(x))$

Integral Calculus

Rules of integration (a) $\int \mathrm{kf}(x) d x=k \int f(x) d x$ for any constant $k$.
(b) $\int(f(x)+g(x)) d x=\int f(x) d x+\int g(x) d x$
(c) $\int(f(x)-g(x)) d x=\int f(x) d x-\int g(x) d x$
Trigonometric Functions 1. $\frac{d}{d x}(-\cos x)=\sin x \Rightarrow \int \sin x d x=-\cos x+C$
2. $\frac{d}{d x}(\sin x)=\cos x \Rightarrow \int \cos x d x=\sin x+C$
3. $\frac{\mathrm{d}}{\mathrm{dx}}(\tan \mathrm{x})=\sec ^2 \mathrm{x} \Rightarrow \int \sec ^2 \mathrm{x} \mathrm{dx}=\tan \mathrm{x}+\mathrm{C}$
4. $\frac{\mathrm{d}}{\mathrm{dx}}(-\cot \mathrm{x})=\csc ^2 \mathrm{x} \Rightarrow \int \csc ^2 \mathrm{x} \mathrm{dx}=-\cot \mathrm{x}+\mathrm{C}$
5. $\frac{d}{d x}(\sec x)=\sec x \tan x \Rightarrow \int \sec x \tan x d x=\sec x+C$
6. $\frac{\mathrm{d}}{\mathrm{dx}}(-\csc \mathrm{x})=\csc \mathrm{x} \cot \mathrm{x} \Rightarrow \int \csc \mathrm{x} \cot \mathrm{x} \mathrm{dx}=-\csc \mathrm{x}+\mathrm{C}$

Differential Equations

(i) $\frac{d y}{d x}=\sin 2 x+\cos x$
(ii) $\mathrm{k} \frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}=\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^2\right]^{3 / 2}$

Vector Algebra

Unit Vector $\hat{\mathbf{a}}=\frac{\overrightarrow{\mathbf{a}}}{|\overrightarrow{\mathbf{a}}|}$
Properties of vector Subtraction 1. $\vec{a}-\vec{b} \neq \vec{b}-\vec{a}$
2. $(\vec{a}-\vec{b})-\vec{c} \neq \vec{a}-(\vec{b}-\vec{c})$
3. For any two vectors $\vec{a}$ and $\vec{b}$
(a) $|\vec{a}+\vec{b}| \leq|\vec{a}|+|\vec{b}|$
(b) $|\vec{a}+\vec{b}| \geq||\vec{a}|-|\vec{b}||$
(c) $|\vec{a}-\vec{b}| \leq|\vec{a}|+|\vec{b}|$
(d) $|\vec{a}-\vec{b}| \geq||\vec{a}|-|\vec{b}||$
Linear Combinations of Vectors $\vec{r}=\lambda_1 \vec{a}_1+\lambda_2 \vec{a}_2+\lambda_3 \vec{a}_3+\ldots \ldots+\lambda_n \vec{a}_n$
Properties of Dot (Scalar) Product 1. $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{a}} \quad$ ( commutative )
2. $\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}})=\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{c}} \quad$ (distributive)
3. $\quad(m \overrightarrow{\mathbf{a}}) \cdot \overrightarrow{\mathbf{b}}=m(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}})=\overrightarrow{\mathbf{a}} \cdot(m \overrightarrow{\mathbf{b}})$; where $m$ is a scalar and $\vec{a}, \vec{b}$ are any two vectors
4. $\quad(l \overrightarrow{\mathbf{a}}) \cdot(m \overrightarrow{\mathbf{b}})=\operatorname{lm}(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}})$; where $l$ and $m$ are scalars
Angle between two vectors $\begin{aligned} & & \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}} & =|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}| \cos \theta \\ \Rightarrow & & \cos \theta & =\frac{\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}}{|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|} \\ \Rightarrow & & \theta & =\cos ^{-1}\left(\frac{\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}}{|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}|}\right)\end{aligned}$
Vector Projection Formula Projection of $\overrightarrow{\mathbf{a}}$ on $\overrightarrow{\mathbf{b}}=\frac{\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}}{|\overrightarrow{\mathbf{b}}|}=\overrightarrow{\mathbf{a}} \cdot \frac{\overrightarrow{\mathbf{b}}}{|\overrightarrow{\mathbf{b}}|}=\overrightarrow{\mathbf{a}} \cdot \hat{\mathbf{b}}$ Projection of $\overrightarrow{\mathbf{b}}$ on $\overrightarrow{\mathbf{a}}=\frac{\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}}{|\overrightarrow{\mathbf{a}}|}=\overrightarrow{\mathbf{b}} \cdot \frac{\overrightarrow{\mathbf{a}}}{|\overrightarrow{\mathbf{a}}|}=\overrightarrow{\mathbf{b}} \cdot \hat{\mathbf{a}}$
Cross Product of Two Vectors If $\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{b}}=b_1 \hat{\mathbf{i}}+b_2 \hat{\mathbf{j}}+b_3 \hat{\mathbf{k}}$, then their cross product given by
$\vec{a} \times \vec{b}=\left|\begin{array}{lll}
\hat{i} & \hat{j} & \hat{k} \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3
\end{array}\right|$

Statistics and Probability

Mean $\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}$
Mean of the Ungrouped Data $\bar{x}=\frac{x_1+x_2+x_3+\ldots \cdots+x_n}{n}=\frac{1}{n} \sum_{i=1}^n x_i$
Mean of Ungrouped Frequency Distribution $\bar{x}=\frac{f_1 x_1+f_2 x_2+f_3 x_3+\ldots \ldots+f_n x_n}{f_1+f_2+f_3+\ldots \ldots+f_n}=\frac{\sum_{i=1}^n f_i x_i}{\sum_{i=1}^n f_i}$
Median of Ungrouped Data. If $n$ is odd :

$\text { Median }=\left(\frac{n+1}{2}\right)^{t h} \text { observation }$
If $\mathbf{n}$ is even :

$\text { Median }=\frac{\text { Value of }\left(\frac{n}{2}\right)^{t h} \text { observation }+ \text { Value of }\left(\frac{n}{2}+1\right)^{t h} \text { observation }}{?}$
Median of Continuous Frequency Distribution: Median $=l+\frac{\left(\frac{N}{2}-c f\right)}{f} \times h$ where,
l = lower limit of median class,
N = number of observations,
cf $=$ cumulative frequency of class preceding the median class,
$\mathrm{f}=$ frequency of median class,
$\mathrm{h}=$ class size (width) (assuming class size to be equal).
Mode $=l+\left(\frac{f_1-f_0}{2 f_1-f_0-f_2}\right) \times h$
where
$\mathrm{l}=$ lower limit of the modal class,
$\mathrm{h}=$ size of the class interval (assuming all class sizes to be equal),
$\mathrm{f}_1=$ frequency of the modal class,
$\mathrm{f}_0=$ frequency of the class preceding the modal class,
$f_2=$ frequency of the class succeeding the modal class.
Standard Deviation $\sigma=\sqrt{\frac{1}{n} \sum_{i=1}^n\left(x_i-\bar{x}\right)^2}$

JEE Main 2027 Syllabus of Physics, Chemistry, Maths

The JEE Main 2027 syllabus for Physics, Chemistry, and Mathematics is based on the NCERT Class 11 and 12 curriculum. It includes important topics such as Mechanics, Electricity and Modern Physics under Physics, Physical, Organic and Inorganic Chemistry under Chemistry and Algebra, Calculus and Coordinate Geometry under math. The syllabus is designed to test the conceptual understanding, application and problem solving skills. To score high, one needs to study NCERT books thoroughly and solve JEE Main mock tests and previous years’ question papers. For detailed Syllabus Information, visit the JEE Main 2027 Syllabus Page.

Tips to Learn the Formula for JEE Main 2027

Students often find it challenging to learn formulas for the JEE Main, but with the right approach, they can effectively remember them. Given below are some points to remember:

1. Students must try to understand why a formula works. For example, derivations in Physics or Maths often follow a logical pattern.

2. Then break down formulas into chapters or topics.

3. To learn these formulas easily, try to make a formula notebook.

4. Sometimes students must try to make Mnemonics and short tricks, as it helps in quick revision.

5. Try to solve as many questions and revise

6. Try to use diagrams and flowcharts.

JEE Main Important Formula Revision Tips

Below are some tips on how to revise formulas for JEE Main

1. Know the derivation: Understanding how the formula was derived will help you remember it while taking the examination.

2. Solve PYQs: Practicing previous years’ questions will help you know which formulas will be asked often.

3. Do spaced repetitions: You should revise formulas on a daily basis, then on alternate days, then on a weekly basis.

4. Learn the formulas related to each other: It will be good to revise formulas that are related to each other (like kinematic equations).

5. Self-test often: Cover the sheet and try recalling formulas.

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