CUET PG Important Formulas 2027: Check Subject-Wise Formula List

Topic

Formula

Quadratic Equation

$ax^2+bx+c=0$

Roots of Quadratic

$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

Sum of Roots

$\alpha+\beta=-\dfrac{b}{a}$

Product of Roots

$\alpha\beta=\dfrac{c}{a}$

Binomial Theorem

$(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$

$n$th term of AP

$a_n=a+(n-1)d$

Sum of $n$ terms of AP

$S_n=\dfrac{n}{2}[2a+(n-1)d]$

$n$th term of GP

$a_n=ar^{n-1}$

Inverse of Matrix

$A^{-1}=\dfrac{1}{\det(A)}\operatorname{adj}(A)$

Eigenvalue Condition

$\det(A-\lambda I)=0$

Topic

Formula

Limit

$\lim_{x\to0}\dfrac{\sin x}{x}=1$

Limit

$\lim_{x\to0}\dfrac{e^x-1}{x}=1$

Derivative of $x^n$

$\dfrac{d}{dx}(x^n)=nx^{n-1}$

Derivative of $e^x$

$\dfrac{d}{dx}(e^x)=e^x$

Derivative of $\ln x$

$\dfrac{d}{dx}(\ln x)=\dfrac{1}{x}$

Indefinite Integral

$\int x^ndx=\dfrac{x^{n+1}}{n+1}+C,\ n\ne -1$

Indefinite Integral

$\int e^x,dx=e^x+C$

Definite Integral

$\int_a^b f(x),dx=F(b)-F(a)$

Mean Value Theorem

$f'(c)=\dfrac{f(b)-f(a)}{b-a}$

Topic

Formula

Separable Differential Equation

$\dfrac{dy}{dx}=g(x)h(y)$

Solution of Separable DE

$\int\dfrac{1}{h(y)}dy=\int g(x)dx$

Linear Differential Equation

$\dfrac{dy}{dx}+Py=Q$

General Solution of Linear DE

$y=e^{-\int Pdx}\left(\int Q e^{\int Pdx}dx+C\right)$

Topic

Formula

Dot Product

$\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$

or in components

$\vec{a}\cdot\vec{b} = a_xb_x + a_yb_y + a_zb_z$

Cross Product

$|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta$

Cross Product (components)

$\vec{a}\times\vec{b} = (a_yb_z-a_zb_y)\hat{i} + (a_zb_x-a_xb_z)\hat{j} + (a_xb_y-a_yb_x)\hat{k}$

Projection of $\vec {a} $ on $\vec {b}$

$\text{comp}_{\vec{b}}(\vec{a}) = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$

Projection vector
$\text{proj}_{\vec{b}}(\vec{a}) = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2} \vec{b}$

Topic

Formula

Probability (Addition Law)

$P(A\cup B) = P(A) + P(B) - P(A\cap B)$

Conditional Probability

$P(A\mid B) = \dfrac{P(A\cap B)}{P(B)},\ P(B)\neq 0$

Mean (Arithmetic Mean)

$\bar{x} = \dfrac{1}{n}\sum_{i=1}^{n} x_i$

Variance (Population)

$\sigma^2 = \dfrac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2$

Standard Deviation

$\sigma = \sqrt{\dfrac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2}$

Topic

Formula

Objective Function

$\text{Max/Min } Z = ax + by$

Optimal Solution

Lies at a corner point of the feasible region

Concept

Formula

Taylor Expansion

$f(x) = f(a) + (x-a)f'(a) + \dfrac{(x-a)^2}{2!}f''(a) + \cdots$

Jacobian

$J = \dfrac{\partial(x,y)}{\partial(u,v)}$

Gradient

$\nabla f$

Divergence

$\nabla\cdot\vec{A}$

Curl

$\nabla\times\vec{A}$

Theorem

Formula

Gauss Divergence Theorem

$\displaystyle \iiint (\nabla\cdot\vec{A})dV = \iint \vec{A}\cdot d\vec{S}$

Green’s Theorem

$\displaystyle \oint (Pdx + Qdy) = \iint \left(\dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}\right)dA$

Stokes’ Theorem

$\displaystyle \oint \vec{A}\cdot d\vec{l} = \iint (\nabla\times\vec{A})\cdot d\vec{S}$

Concept

Formula

First-order Linear DE

$\dfrac{dy}{dx} + Py = Q$

Second-order Linear DE (homogeneous)

$a y'' + b y' + c y = 0$

Euler’s Formula

$e^{i\theta} = \cos\theta + i\sin\theta$

Concept

Formula

Newton’s Second Law

$\vec{F} = m\vec{a}$

Centripetal Force

$F = \dfrac{mv^2}{r}$

Coriolis Force

$\vec{F_c} = -2m(\vec{\omega}\times\vec{v})$

Gravitational Force

$F = \dfrac{GMm}{r^2}$

Concept

Formula

Linear Momentum

$\vec{p} = m\vec{v}$

Angular Momentum

$\vec{L} = \vec{r}\times\vec{p}$

Kinetic Energy

$K = \dfrac{1}{2}mv^2$

Moment of Inertia

$I = \sum mr^2$

Parallel Axis Theorem

$I = I_{cm} + Md^2$

Rotational Kinetic Energy

$K = \dfrac{1}{2}I\omega^2$

Concept

Formula

Continuity Equation

$A_1 v_1 = A_2 v_2$

Bernoulli’s Equation

$P + \dfrac{1}{2}\rho v^2 + \rho g h = \text{constant}$

Concept

Formula

SHM Equation

$x = A\sin(\omega t + \phi)$

Angular Frequency (spring–mass)

$\omega = \sqrt{\dfrac{k}{m}}$

Time Period

$T = \dfrac{2\pi}{\omega}$

Damped Oscillator

$x = A e^{-bt/2}\sin(\omega t)$

Concept

Formula

Wave Equation

$\dfrac{\partial^2 y}{\partial x^2} = \dfrac{1}{v^2}\dfrac{\partial^2 y}{\partial t^2}$

Wave Speed

$v = \nu \lambda$

Group Velocity

$v_g = \dfrac{d\omega}{dk}$

Concept

Formula

Lens Formula

$\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$

Magnification

$m = \dfrac{v}{u}$

YDSE Fringe Width

$\beta = \dfrac{\lambda D}{d}$

Rayleigh Criterion

$\theta = 1.22\dfrac{\lambda}{D}$

Bragg’s Law

$n\lambda = 2d\sin\theta$

Concept

Formula

Coulomb’s Law

$F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r^2}$

Gauss’s Law

$\displaystyle \oint \vec{E}\cdot d\vec{S} = \dfrac{Q}{\varepsilon_0}$

Electric Potential

$V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r}$

Capacitance

$C = \dfrac{Q}{V}$

Concept

Formula

Biot–Savart Law

$d\vec{B} = \dfrac{\mu_0}{4\pi}\dfrac{Id\vec{l}\times \hat{r}}{r^2}$

Lorentz Force

$\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$

Cyclotron Frequency

$\omega = \dfrac{qB}{m}$

Impedance (LCR in series)

$Z = \sqrt{R^2 + \left(\omega L - \dfrac{1}{\omega C}\right)^2}$

Resonance Frequency

$\omega_0 = \dfrac{1}{\sqrt{LC}}$

Concept

Formula

Mean Kinetic Energy (per molecule)

$\dfrac{3}{2}kT$

RMS Speed

$v_{\text{rms}} = \sqrt{\dfrac{3kT}{m}}$

Ideal Gas Law

$PV = nRT$

First Law of Thermodynamics

$dQ = dU + dW$

Carnot Efficiency

$\eta = 1 - \dfrac{T_2}{T_1}$

Clausius–Clapeyron Equation

$\dfrac{dP}{dT} = \dfrac{L}{T(V_2 - V_1)}$

Concept

Formula

Lorentz Factor

$\gamma = \dfrac{1}{\sqrt{1 - v^2/c^2}}$

Mass–Energy Relation

$E = mc^2$

Photoelectric Effect

$h\nu = \phi + K_{\text{max}}$

Compton Shift

$\Delta\lambda = \dfrac{h}{mc}(1-\cos\theta)$

de Broglie Wavelength

$\lambda = \dfrac{h}{p}$

Uncertainty Principle

$\Delta x\Delta p \ge \dfrac{\hbar}{2}$

Particle in a 1D Box

$E_n = \dfrac{n^2 h^2}{8mL^2}$

Radioactive Decay Law

$N = N_0 e^{-\lambda t}$

Half-Life

$T_{1/2} = \dfrac{0.693}{\lambda}$

Concept

Formula

Density of States (3D)

$g(E) \propto \sqrt{E}$

Conductivity

$\sigma = nq\mu$

Drift Velocity

$v_d = \mu E$

Diode Equation

$I = I_0\left(e^{V/\eta V_T} - 1\right)$

Concept

Formula

CE Amplifier Gain

$A_v = \dfrac{V_o}{V_i}$

Barkhausen Condition

$A\beta = 1$

OPAMP (Inverting)

$A_v = -\dfrac{R_f}{R_i}$

OPAMP (Non-Inverting)

$A_v = 1 + \dfrac{R_f}{R_i}$

De Morgan’s Theorems

$(A+B)' = A'B'$ , $(AB)' = A' + B'$

Concept

Formula

Ideal Gas Equation

$PV = nRT$

van der Waals Equation

$\left(P + \dfrac{a}{V_m^{2}}\right)(V_m - b) = RT$

Compressibility Factor

$Z = \dfrac{PV_m}{RT}$

RMS Speed

$v_{\text{rms}} = \sqrt{\dfrac{3RT}{M}}$

Mean Free Path

$\lambda = \dfrac{kT}{\sqrt{2\pi}d^{2}P}$

Concept

Formula

Raoult’s Law

$P = X_A P_A^0$

Relative Lowering of Vapour Pressure

$\dfrac{\Delta P}{P^0} = X_B$

Elevation of Boiling Point

$\Delta T_b = K_b m$

Depression of Freezing Point

$\Delta T_f = K_f m$

Osmotic Pressure

$\pi = C R T$

van’t Hoff Factor

$i = \dfrac{\text{observed}}{\text{calculated}}$

Concept

Formula

pH

$\text{pH} = -\log[H^+]$

Ionic Product of Water

$K_w = 10^{-14}$

Acid Dissociation Constant

$K_a = \dfrac{[H^+][A^-]}{[HA]}$

Henderson–Hasselbalch Equation

$\text{pH} = pK_a + \log\dfrac{[\text{salt}]}{[\text{acid}]}$

Debye–Hückel Limiting Law

$\log\gamma = -0.509z^2\sqrt{I}$

Concept

Formula

First Law

$dU = \delta Q - \delta W$

Enthalpy

$H = U + PV$

Heat Capacity Relation

$C_p - C_v = R$

Gibbs Free Energy

$G = H - TS$

Spontaneity Condition

$\Delta G < 0$

Maxwell Relation (example)

$\left(\dfrac{\partial T}{\partial V}\right)_S = -\left(\dfrac{\partial P}{\partial S}\right)_V$

Clausius–Clapeyron Equation

$\ln P = -\dfrac{\Delta H_{\text{vap}}}{RT} + C$

Concept

Formula

Gibbs Phase Rule

( F=C-P+2 )

Reduced Phase Rule

( F=C-P+1 )

Concept

Formula

Rate Law

$r = k[A]^m[B]^n$

First-Order Integrated Law

$\ln\left(\dfrac{[A]_0}{[A]}\right) = kt$

Half-Life (First Order)

$t_{1/2} = \dfrac{0.693}{k}$

Arrhenius Equation

$k = A e^{-E_a/RT}$

Concept

Formula

Nernst Equation

$E = E^0 - \dfrac{0.0591}{n}\log Q$

Gibbs Free Energy

$\Delta G = -nFE$

Conductivity

$\kappa = \dfrac{l}{RA}$

Molar Conductivity

$\Lambda_m = \dfrac{\kappa}{c}$

Concept

Formula

Freundlich Isotherm

$\dfrac{x}{m} = K P^{1/n}$

Langmuir Isotherm

$\dfrac{1}{V} = \dfrac{1}{V_m} + \dfrac{1}{K V_m P}$

Concept

Formula

de Broglie Wavelength

$\lambda = \dfrac{h}{mv}$

Uncertainty Principle

$\Delta x\Delta p \ge \dfrac{\hbar}{2}$

Hydrogen-like Energy Levels

$E_n = -\dfrac{13.6,Z^2}{n^2}\ \text{eV}$

Concept

Formula

Born–Landé Equation

$U = \dfrac{N_AMz^+z^-e^2}{4\pi\varepsilon_0r_0}\left(1-\dfrac{1}{n}\right)$

Dipole Moment

$\mu = q r$

Concept

Formula

Effective Atomic Number

$\text{EAN} = Z - \text{oxidation state} + \text{ligand electrons}$

Spin-Only Magnetic Moment

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

Stability Constant

$K = \dfrac{[ML]}{[M][L]}$

Concept

Formula

Density of Unit Cell

$\rho = \dfrac{ZM}{a^{3}N_A}$

Bragg’s Law

$n\lambda = 2d\sin\theta$

Concept

Formula

Beer–Lambert Law

$A = \varepsilon c l$

IR Stretching Frequency

$\nu = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{\mu}}$

NMR Chemical Shift

$\delta = \dfrac{\nu - \nu_0}{\nu_0}\times 10^6$

UV–Vis Transition

$\Delta E = h\nu$

Concept

Formula

SN1 Rate

$\text{Rate} = k[R-X]$

SN2 Rate

$\text{Rate} = k[R-X][Nu^-]$

E2 Rate

$\text{Rate} = k[R-X][Base]$

Optical Rotation

$[\alpha] = \dfrac{\alpha}{lc}$

Aromaticity Rule

$4n+2\ \pi\ \text{electrons}$

Concept

Formula

Radioactive Decay Law

$N = N_0 e^{-\lambda t}$

Half-Life

$t_{1/2} = \dfrac{0.693}{\lambda}$

Subject

Chapter / Unit

Important CUET PG Formulas to Revise

Mathematics

Algebra

Quadratic roots, determinants, inverse of a matrix, eigenvalue condition

Calculus

Limits, standard derivatives, standard integrals, and definite integral properties

Differential Equations

Linear DE, integrating factor, general solution

Probability & Statistics

Bayes’ theorem, mean, variance

Physics

Mechanics

Equations of motion, work–energy, and angular momentum

Oscillations & Waves

SHM equations, wave speed ( v = f\lambda )

Thermodynamics

First law, Carnot efficiency

Electricity & Magnetism

Coulomb’s law, Ohm’s law, Lorentz force

Modern Physics

Photoelectric equation, radioactive decay law

Chemistry

Physical Chemistry

Thermodynamic relations, Arrhenius equation, Nernst equation

Chemical Kinetics

Rate law, first-order reactions, half-life

Solutions

Raoult’s law, colligative properties

Organic Chemistry

pH, Henderson–Hasselbalch equation, SN1/SN2 rate laws

Inorganic Chemistry

CFSE, magnetic moment, bond order

Revision Tip

Usage

Revise twice daily in the final week

Exam Strategy

Focus

High-weightage numerical formulas only

Subject

Topic

Formula

Mathematics

Quadratic Formula

$x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$

Derivative of a Power Function

$\frac{d}{d x}\left(x^n\right)-n x^{n-1}$

Conditional Probability

$P(A \mid B)=\frac{P(A \cap B)}{P(B)}$

Physics

First Equation of Motion

$v-u+a t$

Coulomb's Law

$F=\frac{1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r^2}$

Mass-Energy Equivalence

$E-m c^2$

Chemistry

Gibbs Free Energy Equation

$\Delta G-\Delta H-T \Delta S$

Arrhenius Equation

$k-A e^{-\frac{N}{1 N}}$

Nernst Equation

$E-E^{\circ}-\frac{0.0591}{n} \log Q$