JEE Mains + Advanced 2026 — Full Prep

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⚡ JEE Mains 2026 — Session 1

Estimated Jan 2026. Every second counts.

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Physics

Mechanics · Electrostatics · Modern Physics · Optics · SHM & Waves · Thermodynamics

01 Kinematics & Laws of Motion 🔥 High Weightage ▼
Equations of Motion
v = u + at
s = ut + ½at²
v² = u² + 2as
s_n = u + a(2n−1)/2
u=initial vel, a=acceleration, s=displacement, t=time
Projectile Motion
R = u²sin2θ / g
H = u²sin²θ / 2g
T = 2u sinθ / g
R_max at θ = 45°
Horizontal: ax=0, Vertical: ay=−g. Velocity at H: v=u cosθ
Newton's Laws
F = ma
F = dp/dt
Impulse J = FΔt = Δp
Action-reaction: equal magnitude, opposite direction, on different bodies
Friction
f_s(max) = μₛ N
f_k = μ_k N
tan θ = μₛ (incline slip)
μₛ > μ_k always. Rolling friction ≪ sliding friction
⚠ JEE Traps
  • Always draw FBD before writing equations — never skip it
  • Pseudo force = −ma_frame (non-inertial frames like elevator, car)
  • For inclined plane: resolve forces along & perpendicular to incline
  • Constraint relations: for pulley systems, differentiate geometric relation to get accelerations
Relative Motion
  • V_AB = V_A − V_B (use vector subtraction)
  • Rain-man problems: use resultant velocity direction
  • Minimum relative speed = 0 when they move together
  • Relative acceleration in same direction = a_A − a_B
Trick Formulas
  • Atwood: a = (m₁−m₂)g / (m₁+m₂)
  • Range on incline: R = 2u²cos θ sin(θ−α) / g cos²α
  • Spring: equilibrium extension x₀ = mg/k
  • Time to reach ground from height h: t = √(2h/g)
Atwood machineWedge problems Monkey-gunRiver crossing Conical pendulum
02 Work, Energy & Power 🔥 High Weightage ▼
Work Done
W = Fd cosθ
W = ∫F·ds (variable F)
W_net = ΔKE
Area under F-x graph = work. Conservative force: W independent of path
Energy
KE = ½mv²
PE_grav = mgh
PE_spring = ½kx²
ME = KE + PE = const
Collisions
e = v_sep / v_approach
Elastic: e=1, KE conserved
Inelastic: e < 1
Perfectly inelastic: e=0
Momentum always conserved. KE loss = ½μ(u₁−u₂)²(1−e²)
Power
P = W/t = F·v
P_avg = ΔW/Δt
η = P_out/P_in × 100%
1 HP = 746 W. 1 kWh = 3.6 × 10⁶ J
💡 Key Insight

Equal mass elastic collision: velocities exchange completely. In head-on elastic: v₁' = ((m₁−m₂)u₁+2m₂u₂)/(m₁+m₂), v₂' = ((m₂−m₁)u₂+2m₁u₁)/(m₁+m₂)

03 Rotational Mechanics 🔥 High Weightage ▼
Moment of Inertia
Ring: I = MR²
Disc: I = MR²/2
Solid sphere: I = 2MR²/5
Hollow sphere: I = 2MR²/3
Rod (center): I = ML²/12
Rod (end): I = ML²/3
Theorems
Parallel axis: I = I_cm + Md²
Perp axis: I_z = I_x + I_y
τ = Iα
L = Iω = mvr
Perp. axis theorem: laminar (2D) bodies only
Rolling Motion
v_cm = Rω (pure rolling)
KE = ½mv²(1 + k²/R²)
a = g sinθ / (1+k²/R²)
Sphere(2/5) reaches bottom before disc(1/2) before ring(1) — smaller k²/R² wins
Angular Kinematics
ω = ω₀ + αt
θ = ω₀t + ½αt²
ω² = ω₀² + 2αθ
Exactly mirrors linear kinematics. Torque is rotational analogue of force
04 Electrostatics & Capacitors 🔥 High Weightage ▼
Coulomb & Field
F = kq₁q₂/r² (k=9×10⁹)
E = kq/r²
V = kq/r
E = −dV/dr
Gauss's Law
∮E·dA = Q_enc/ε₀
E_inside conductor = 0
E_sheet = σ/2ε₀
E_between plates = σ/ε₀
ε₀ = 8.85×10⁻¹² C²/N·m²
Capacitors
C = Q/V = ε₀A/d
With dielectric: C' = κC
Series: 1/C_eq = Σ1/Cᵢ
Parallel: C_eq = ΣCᵢ
U = ½CV² = Q²/2C
Important Results
V inside hollow sphere = kQ/R
Energy density u = ε₀E²/2
Dipole: p = qd
Torque τ = p × E
Charge on conductor: only on outer surface. Inside = 0
Key Constants
  • k = 1/4πε₀ = 9 × 10⁹ N·m²/C²
  • e = 1.6 × 10⁻¹⁹ C
  • ε₀ = 8.85 × 10⁻¹² F/m
  • μ₀ = 4π × 10⁻⁷ T·m/A
Current Electricity
  • V = IR; P = I²R = V²/R = VI
  • Kirchhoff KCL: ΣI_in = ΣI_out
  • KVL: Σ(EMF) = Σ(IR) in a loop
  • Wheatstone: P/Q = R/S (balanced)
05 Modern Physics ⭐ Important ▼
Photoelectric Effect
E = hf = hc/λ
KE_max = hf − φ
eV_stop = KE_max
h=6.63×10⁻³⁴J·s. φ=work function. Threshold: f₀=φ/h
Bohr's Model & de Broglie
E_n = −13.6/n² eV (H atom)
r_n = 0.529n² Å
λ = h/mv = h/p
mvr = nh/2π
Nuclear Physics
N = N₀e^(−λt)
t₁/₂ = 0.693/λ
E = Δm·c²
BE/A = binding energy per nucleon
Hydrogen Spectrum
1/λ = R(1/n₁²−1/n₂²)
Lyman(UV): n₁=1. Balmer(visible): n₁=2. Paschen(IR): n₁=3. Brackett: n₁=4. R=1.097×10⁷m⁻¹
06 SHM, Waves & Thermodynamics ⭐ Important ▼
Simple Harmonic Motion
x = A cos(ωt + φ)
ω = √(k/m) = 2π/T
v = ω√(A²−x²)
E_total = ½kA² (constant)
v is max at x=0, zero at x=±A. KE+PE=constant
Pendulums
Simple: T = 2π√(L/g)
Spring: T = 2π√(m/k)
Spring parallel: k_eq = k₁+k₂
Spring series: 1/k_eq = 1/k₁+1/k₂
Waves & Sound
v = fλ = ω/k
v_string = √(T/μ)
v_sound = √(γP/ρ)
Beat freq = |f₁ − f₂|
Doppler: f'=f(v±v_o)/(v∓v_s)
Thermodynamics
ΔU = Q − W (1st law)
η_Carnot = 1 − T₂/T₁
PVᵞ = const (adiabatic)
PV = nRT
KE_avg = (3/2)kT
γ=Cp/Cv. Isothermal:PV=const. Isochoric:W=0. Isobaric:W=PΔV
🧪

Chemistry

Physical · Inorganic · Organic — Complete JEE Coverage

01 Physical Chemistry — Core Concepts 🔥 High Weightage ▼
Mole Concept
n = mass / M
N = n × Nₐ (6.022×10²³)
V = n × 22.4 L at STP
Molar volume at STP = 22.4 L. At NTP (25°C): 24.5 L
Chemical Equilibrium
Kc = [P]^p[Q]^q / [A]^a[B]^b
Kp = Kc(RT)^Δn
ΔG° = −RT ln K
Le Chatelier: system opposes external change. K>1: product favoured
Electrochemistry
E_cell = E_cathode − E_anode
E = E° − (0.059/n)log Q
ΔG = −nFE
m = ZIt (Faraday's law)
F=96500 C/mol. Z=M/nF. SHE=0.00V reference electrode
Chemical Kinetics
rate = k[A]^m[B]^n
1st order: t₁/₂ = 0.693/k
Zero order: t₁/₂ = [A]₀/2k
Arrhenius: k = Ae^(−Ea/RT)
Colligative Properties
  • ΔTb = i·Kb·m (boiling point elevation)
  • ΔTf = i·Kf·m (freezing point depression)
  • Osmotic pressure π = iMRT
  • Raoult's law: P_A = x_A · P°_A
Thermochemistry
  • ΔH = ΔU + ΔnᵍRT
  • ΔG = ΔH − TΔS (spontaneity)
  • Hess's law: ΔH is path independent
  • Bond enthalpy: ΔH = ΣBE_broken − ΣBE_formed
02 Organic Chemistry — Reactions & Mechanisms 🔥 High Weightage ▼
Named Reactions (Must Know)
  • Aldol: α-H + carbonyl → β-hydroxy carbonyl
  • Cannizzaro: no α-H aldehyde → disproportionation
  • Sandmeyer: ArN₂⁺ + CuX → ArX
  • Reimer-Tiemann: phenol+CHCl₃/NaOH → salicylaldehyde
  • Friedel-Crafts: Ar + RX/AlCl₃ → alkylbenzene
  • Kolbe: sodium phenoxide + CO₂ → salicylic acid
  • Hoffmann bromamide: RCONH₂ + Br₂/NaOH → RNH₂
Reagent → Product
  • LiAlH₄: reduces esters, acids, amides → alcohol/amine
  • NaBH₄: reduces ketones/aldehydes only (mild)
  • O₃ / Zn-H₂O: ozonolysis cleaves C=C
  • Br₂/CCl₄ (dark): electrophilic addition to alkene
  • HBr + peroxide: anti-Markovnikov (free radical)
  • KMnO₄ (cold, alk): cis-diol formation (Baeyer)
  • PCC: primary alcohol → aldehyde (no further oxidation)
Stability Order
  • Carbocation: 3° > 2° > 1° > methyl
  • Carbanion: methyl > 1° > 2° > 3° (opposite!)
  • Free radical: 3° > 2° > 1° > methyl
  • Allylic/Benzylic > tertiary (resonance wins)
IUPAC Priority
  • COOH > CHO > C=O > OH > NH₂ > C≡C > C=C
  • E/Z: same side higher priority = Z (zusammen)
  • R: clockwise arrangement of 1>2>3 priorities
  • Number chain from end nearer to principal group
⚠ Isomerism Checklist

Structural: chain, position, functional group, metamerism, tautomerism. Stereoisomers: E/Z (geometric) and R/S (optical). Optical isomers need a chiral centre (4 different groups). Geometric: restricted rotation — C=C or ring.

03 Inorganic — Periodic Table, s/p/d-Block & Coordination ⭐ Important ▼
Periodic Trends
  • IE: left→right ↑, top→bottom ↓ (N>O anomaly)
  • EA: Cl > F (size anomaly for fluorine)
  • EN: F > O > N > Cl (Pauling scale)
  • Atomic radius: increases down group, decreases across period
Coordination Chemistry
  • Strong field ligands: CN⁻, CO, en, NH₃, py (low spin, large CFSE)
  • Weak field: I⁻ < Br⁻ < Cl⁻ < F⁻ < OH⁻ < H₂O
  • EAN rule: 18 electron rule for stable complexes
  • IUPAC: ligands (alphabetical) + central atom(ox. state)
s-Block Anomalies
  • Li resembles Mg (diagonal relationship)
  • Li₃N: only alkali metal forming stable nitride
  • Na₂O₂ + H₂O → NaOH + H₂O₂
  • Be resembles Al (diagonal); amphoteric oxide
p-Block Highlights
  • PCl₅: sp³d (trigonal bipyramidal)
  • SF₆: sp³d² (octahedral)
  • XeF₂: linear (3 lp); XeF₄: square planar (2 lp)
  • H₂SO₄ conc: dehydrating agent; fuming H₂SO₄ = oleum
📐

Mathematics

Calculus · Algebra · Co-ordinate Geometry · Trigonometry · 3D & Vectors

01 Calculus — Limits, Derivatives & Integrals 🔥 High Weightage ▼
Standard Limits
lim(x→0) sinx/x = 1
lim(x→0) tanx/x = 1
lim(x→0) (1+x)^(1/x) = e
lim(x→∞) (1+1/x)^x = e
lim(x→0) (aˣ−1)/x = ln a
Key Derivatives
d/dx(xⁿ) = nxⁿ⁻¹
d/dx(eˣ) = eˣ
d/dx(ln x) = 1/x
d/dx(sin x) = cos x
d/dx(tan x) = sec²x
d/dx(sin⁻¹x) = 1/√(1−x²)
Standard Integrals
∫xⁿdx = xⁿ⁺¹/(n+1) + C
∫eˣdx = eˣ + C
∫sin x dx = −cos x + C
∫sec²x dx = tan x + C
∫1/(x²+a²)dx = (1/a)tan⁻¹(x/a)
∫1/√(a²−x²)dx = sin⁻¹(x/a)
Definite & Applications
∫ₐᵇf dx = F(b) − F(a)
Area = ∫ₐᵇ|f(x)|dx
IBP: ∫uv = u∫v − ∫(u'∫v)
LIATE order for IBP: Log, Inverse-trig, Algebraic, Trig, Exponential
⚠ L'Hôpital — Use ONLY for 0/0 or ∞/∞

Differentiate numerator and denominator separately (NOT as a quotient). Apply repeatedly if still indeterminate. Check form first — many students waste time applying it to non-indeterminate forms.

02 Algebra — Sequences, Complex Numbers, Matrices & P&C 🔥 High Weightage ▼
AP / GP / HP
AP: aₙ = a+(n−1)d
AP: Sₙ = n/2[2a+(n−1)d]
GP: aₙ = arⁿ⁻¹
GP: Sₙ = a(rⁿ−1)/(r−1)
GP∞ = a/(1−r), |r| < 1
AM ≥ GM ≥ HM. AM·HM = GM² (2 positive numbers)
Complex Numbers
|z|² = a²+b²
arg(z) = tan⁻¹(b/a)
De Moivre: (cosθ+isinθ)ⁿ = cosnθ+isinnθ
ω³ = 1; 1+ω+ω² = 0
Modulus-argument form: z = r(cosθ + i sinθ)
Matrices
|AB| = |A|·|B|
A⁻¹ = adj(A)/|A|
Cramer's: x=Dx/D, y=Dy/D
|A|=0 → singular (no inverse). Consistent if D≠0 or D=Dx=Dy=0
Permutation & Combination
ⁿPᵣ = n!/(n−r)!
ⁿCᵣ = n!/[r!(n−r)!]
Derangements: D_n = n!Σ(−1)^k/k!
Stars & Bars: distributing n identical into r distinct = C(n+r−1,r−1)
Probability
  • P(A|B) = P(A∩B)/P(B) (conditional)
  • Bayes': P(A|B) = P(B|A)P(A)/P(B)
  • Independent: P(A∩B) = P(A)·P(B)
  • Binomial: P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ
Binomial Theorem
  • (a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ (r=0 to n)
  • T_(r+1) = ⁿCᵣ aⁿ⁻ʳ bʳ (general term)
  • Sum of coefficients: put a=b=1 → 2ⁿ
  • Middle term: n even → (n/2+1)th; n odd → 2 middle terms
03 Coordinate Geometry — Conics & Straight Lines 🔥 High Weightage ▼
Straight Line
Slope: m = (y₂−y₁)/(x₂−x₁)
Dist point to line: |ax₁+by₁+c|/√(a²+b²)
Angle: tan θ = |m₁−m₂|/|1+m₁m₂|
Section: x=(mx₂+nx₁)/(m+n)
Circle
x² + y² = r²
Tangent at (x₁,y₁): xx₁+yy₁=r²
Length tangent: √(h²+k²−r²)
General: x²+y²+2gx+2fy+c=0, centre=(−g,−f), r=√(g²+f²−c)
Parabola y²=4ax
Focus: (a,0), Vertex: (0,0)
Directrix: x = −a
Tangent: ty = x + at²
Parametric: (at², 2at)
Ellipse & Hyperbola
Ellipse: x²/a²+y²/b²=1, e=c/a < 1
Sum focal dist = 2a
Hyperbola: x²/a²−y²/b²=1, e > 1
Asymptotes: y = ±(b/a)x
c²=a²−b² (ellipse), c²=a²+b² (hyperbola)
Trigonometry
  • sin2A = 2sinAcosA
  • cos2A = cos²A−sin²A = 1−2sin²A
  • sin(A+B) = sinAcosB + cosAsinB
  • Cosine rule: a²=b²+c²−2bc cosA
3D & Vectors
  • a⃗·b⃗ = |a||b|cosθ (scalar product)
  • |a⃗×b⃗| = |a||b|sinθ (vector product ⊥ both)
  • [a b c] = a·(b×c) = volume of parallelepiped
  • Direction cosines: l²+m²+n² = 1
04 Statistics & Mathematical Reasoning Moderate ▼
Central Tendency
Mean: x̄ = Σfᵢxᵢ / Σfᵢ
Median: middle value(s)
Mode = 3Median − 2Mean
Empirical relation — useful to cross-check
Dispersion
Variance σ² = Σfᵢ(xᵢ−x̄)²/N
SD σ = √(σ²)
CV = (σ/x̄) × 100%
Lower CV = more consistent data
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