Maths Shortcuts for Physics & Chemistry

Right-triangle integer triples — instantly recognise hypotenuse and avoid square-roots in mechanics and resolution-of-vector problems.

⚡ Trick: If two legs are 3 & 4 of any scaled triangle (or 6 & 8, 9 & 12…), the hypotenuse is not something you compute — read it off.

⚡ Squares ending in 5: 25² = 2·3 | 25 = 625. (n × (n+1) followed by 25.) Use for 35² (1225), 45² (2025), 65² (4225), 75² (5625), 85² (7225), 95² (9025).

Most-used approximation in physics derivations — relativity, gravitation, optics.

• (1+x)⁻¹ ≈ 1 − x
• (1+x)½ ≈ 1 + x/2
• (1+x)⁻½ ≈ 1 − x/2
• (1+x)² ≈ 1 + 2x

Valid when θ < ~10° (~0.17 rad). For better precision: cos θ ≈ 1 − θ²/2.

⚡ Hand trick: Numerator of sin θ = √n / 2 where n = 0, 1, 2, 3, 4 for θ = 0°, 30°, 45°, 60°, 90°.

√2 ≈ 1.414 · √3 ≈ 1.732 · √5 ≈ 2.236 · √7 ≈ 2.646 · √10 ≈ 3.162

⚡ Add Sugar To Coffee — quadrant order I → II → III → IV.

⚡ Only memorise log 2 = 0.3010 and log 3 = 0.4771. Derive all others.

• log(xy) = log x + log y
• log(x/y) = log x − log y
• log xⁿ = n log x
• log_b x = log x / log b (change of base)
• logₐ a = 1, log 1 = 0
• ln x = 2.303 log₁₀ x

Constants: ln 2 = 0.693, ln 10 = 2.303, ln 3 = 1.099.

⚡ For pH 5.5 → [H⁺] = 10⁻⁵·⁵ = 10⁰·⁵ × 10⁻⁶ ≈ 3.16 × 10⁻⁶ M.

For a body starting from rest under uniform acceleration g, distances covered in 1st, 2nd, 3rd … seconds are in ratio of odd numbers.

Formula: s_nth = u + a(2n − 1)/2   (for u = 0: s_nth = a(2n−1)/2)

⚡ Cumulative distance ∝ n² (since 1+3+5+…+(2n−1) = n²).

Maximum range when 2θ = 90° → θ = 45°: R_max = u²/g.

Two angles θ and (90° − θ) give the same range (since sin 2θ = sin(180°−2θ)).

Other key results (initial speed u, angle θ):

• H = u²sin²θ / 2g  (max height)
• T = 2u·sinθ / g  (time of flight)
• H/R = tan θ / 4
• For θ = 45°: R = 4H

For radioactivity and any first-order process (also valid in Chemistry kinetics).

Key relations:

• λ = 0.693 / t₁/₂ (decay constant)
• Mean life τ = t₁/₂ / 0.693 = 1.44 × t₁/₂
• N(t) = N₀ · e^(−λt) = N₀ · (1/2)^(t/t₁/₂)

For constant acceleration along a line.

• v = u + at
• s = ut + ½at²
• v² = u² + 2as
• s = (u + v)·t / 2  (average velocity × time)
• s_nth = u + a(2n−1)/2  (distance in nth second)

⚡ For free fall: replace a with g (+ if downward chosen +ve).

• Angular velocity: ω = 2π/T = 2πf
• v = ωr
• Centripetal force: F = mv²/r = mω²r
• Banking angle: tan θ = v² / (rg)
• With friction: v_max² = rg(μ + tanθ)/(1 − μtanθ)
• Conical pendulum period: T = 2π√(L cosθ/g)

⚡ Minimum speed at top of vertical loop: v_top = √(gr).

• Spring: T = 2π√(m/k), ω = √(k/m)
• Simple pendulum: T = 2π√(L/g)
• Physical pendulum: T = 2π√(I/mgd)
• Energy: E = ½kA² = ½mω²A²
• v at displacement x: v = ω√(A²−x²)
• Max v = ωA, Max a = ω²A

⚡ For two springs in series → 1/k_eq = 1/k₁ + 1/k₂; in parallel k_eq = k₁ + k₂.

• Lens-maker: 1/f = (n − 1)(1/R₁ − 1/R₂)
• Magnification: m = v/u (mirror), m = v/u (lens with sign)
• Power: P = 1/f (in metres) = 100/f (in cm) — units: dioptres
• Snell's: n₁ sin i = n₂ sin r
• Critical angle: sin i_c = 1/n
• Prism min deviation: n = sin[(A + δ_m)/2] / sin(A/2)
• Thin prism: δ = (n − 1)A

Resistors:

• Two in parallel: R_p = R₁R₂/(R₁+R₂)
• n equal R in parallel: R_p = R/n
• n equal R in series: R_s = nR

Capacitors (opposite):

• Parallel: C_p = C₁ + C₂
• Series: 1/C_s = 1/C₁ + 1/C₂   (two: C_s = C₁C₂/(C₁+C₂))

⚡ LC resonance: ω = 1/√(LC), f = 1/(2π√(LC)).

• P = V·I (general)
• P = I²R (resistor)
• P = V²/R
• Joule heating: H = I²R·t
• Energy: 1 kWh = 3.6 × 10⁶ J

⚡ For two equal bulbs of resistance R: power in series : parallel = 1 : 4 (same V supply).

• Photon energy: E = hν = hc/λ = 1240/λ(nm) eV
• Threshold: ν₀ = φ/h, λ₀ = hc/φ
• de Broglie: λ = h/p = h/(mv) ; for electron at V volts: λ = 12.27/√V Å
• Bohr energy: E_n = −13.6 Z²/n² eV (H: Z=1)
• Bohr radius: r_n = 0.529 n²/Z Å
• Rydberg: 1/λ = R(1/n₁² − 1/n₂²), R = 1.097 × 10⁷ m⁻¹

⚡ Lyman → UV (n→1); Balmer → visible (n→2); Paschen → IR (n→3).

• String fixed both ends: f_n = nv/2L (n = 1,2,3…)
• Open pipe: f_n = nv/2L
• Closed pipe: f_n = (2n−1)v/4L (odd harmonics only)
• Doppler: f' = f·(v ± v_obs)/(v ∓ v_src)
• Beat frequency: |f₁ − f₂|
• Intensity ∝ A²f²

If a car at speed v stops in distance d under the same braking force, then at speed nv:

New stopping distance = n² × d   (since v² = u² − 2as → d ∝ u²)

⚡ Stopping time scales linearly (t ∝ v), but stopping distance scales as v². Same logic for KE = ½mv².

Powerful shortcut for resistor-network puzzles. Look for lines of symmetry perpendicular to current flow.

• Mirror symmetry: Points symmetric about the axis are at equal potential → the resistor between them carries zero current and can be deleted.
• Balanced Wheatstone bridge (P/Q = R/S): the galvanometer (middle resistor) carries no current — remove it.
• Folding: Symmetric branches can be folded so equivalent resistors stack as parallel combinations, halving the work.
• Cube of 12 equal R (across body diagonal): R_eq = 5R/6 · (across face diagonal): 3R/4 · (across edge): 7R/12.

⚡ Don't compute Kirchhoff's mesh equations until you've checked for symmetry.

Skip the full MO diagram for B₂, C₂, N₂, O₂, F₂ and their ions. Anchor on N₂ (14 e⁻, BO = 3.0). Every electron added or removed from 14 drops the bond order by 0.5.

⚡ Magnetism rule: odd electrons → always paramagnetic. Even usually diamagnetic, but memorise the two anomalies B₂ (10) and O₂ (16).

At standard room temperature (298 K), several constants collapse to clean numbers — memorise these once:

• RT = 8.314 × 298 ≈ 2477 J/mol ≈ 2.48 kJ/mol
• 2.303 RT ≈ 5705 J/mol ≈ 5.7 kJ/mol
• 2.303 RT/F = 5705/96500 = 0.0591 V  (Nernst factor)
• kT (per molecule) = 4.11×10⁻²¹ J ≈ 0.0257 eV

⚡ ΔG° = −2.303 RT log K → at 298 K: ΔG°(kJ/mol) ≈ −5.7 × log K. So K = 10 → ΔG° ≈ −5.7 kJ/mol.

For first-order kinetics and radioactive decay, learn these standard time multiples in terms of half-life:

General formula: t = (2.303/k) · log(N₀/N) = t½ · log(N₀/N)/log 2

⚡ For arbitrary fraction left f: t = t½ × log₂(1/f) = 3.32 × t½ × log₁₀(1/f).

When the options contain symbols/variables (no numbers), don't solve the physics. Just check dimensions.

• Write the target quantity's dimension: e.g. Force → [M L T⁻²], Energy → [M L² T⁻²], Velocity → [L T⁻¹].
• Reduce each option to fundamental units.
• Reject any option that doesn't match.

Common dimensions:

⚡ Inside any sin/cos/log/exp, the argument must be dimensionless. This alone eliminates many options.

For any inverse-square force (gravitation, electrostatics, intensity of a point source):

F ∝ 1/r²  ⟹  ΔF/F ≈ −2 · Δr/r  (binomial approximation)

⚡ Large change? Use the law directly. If r doubles → F becomes 1/4 (drops 75%). If r triples → F becomes 1/9 (drops ~89%).

Same idea works for: gravity g ∝ 1/r², Coulomb force, light intensity I ∝ 1/r², sound intensity.

A symmetric biconvex lens of focal length f is sliced. The new focal length depends on how it was cut:

Why: Lens-maker's formula 1/f = (n−1)(1/R₁ − 1/R₂). Cutting along axis kills one curved surface → R₂ = ∞ → f doubles. Cutting perpendicular keeps both R₁ and R₂ → f unchanged; only aperture area falls so intensity halves.

⚡ Two pieces joined back along the cut? Power adds. Two plano-convex lenses (each 2f) placed together give f again. Two semi-disc pieces joined back give f with full intensity.

Skip Lewis diagrams under exam pressure. Find hybridisation of central atom in one line.

• V = valence electrons of central atom
• M = number of monovalent atoms attached (H, F, Cl, Br, I, OH counted as 1 each)
• C = positive charge (cation, subtract)
• A = negative charge (anion, add)

⚡ For oxoanions/oxyacids (e.g. SO₄²⁻, NO₃⁻, ClO₃⁻), this short formula misses; use VSEPR pair-counting instead.

For coordination complexes, you only need to find n (unpaired electrons), then read μ off:

⚡ Each μ value starts with the digit n. Spot the option in NEET directly.

Finding n: determine oxidation state of metal → d-electron count → strong-field (low-spin) or weak-field (high-spin) using ligand series:
I⁻ < Br⁻ < SCN⁻ < Cl⁻ < F⁻ < OH⁻ < H₂O < NH₃ < en < NO₂⁻ < CN⁻ < CO (weak → strong).

The single biggest exam-strategy upgrade. Decide your precision before you start calculating.

⚡ Useful clean approximations: g ≈ 10 m/s²; π ≈ 22/7 ≈ 3.14; π² ≈ 10; 1/√2 ≈ 0.71; ln 2 ≈ 0.69; e ≈ 2.72; 1 atm ≈ 10⁵ Pa.

• Escape velocity: v_e = √(2gR) = √(2GM/R) ≈ 11.2 km/s for Earth
• Orbital velocity (near surface): v_o = √(gR) = v_e/√2 ≈ 7.9 km/s
• Time period of satellite: T = 2π√(r³/GM) (Kepler's 3rd: T² ∝ r³)
• g at height h (h ≪ R): g' = g(1 − 2h/R)
• g at depth d: g' = g(1 − d/R)
• PE: U = −GMm/r

• moles n = mass / molar mass
• n = particles / N_A  (N_A = 6.022×10²³)
• n_gas = V(L at STP) / 22.4
• Molarity M = mol solute / L solution
• Molality m = mol solute / kg solvent
• Mole fraction x_i = n_i / Σn
• Normality N = z·M  (z = n-factor)

• pH = −log[H⁺], pOH = −log[OH⁻]
• pK_a + pK_b = 14 (conjugate pair)
• K_a · K_b = K_w = 10⁻¹⁴
• Strong acid 0.01 M → pH = 2
• Weak acid: pH ≈ ½(pK_a − log C)
• Weak base: pOH ≈ ½(pK_b − log C)
• Buffer (Henderson): pH = pK_a + log([salt]/[acid])
• Half-neutralised acid: pH = pK_a

The empirical 'temperature coefficient'. Rate roughly doubles for every 10 K rise in T.

If rate doubles ΔT = 10 K, then for ΔT = 30 K: rate factor = 2³ = 8×.

For ΔT = 50 K: factor = 2⁵ = 32×.

⚡ Arrhenius form: ln(k₂/k₁) = (E_a/R)(1/T₁ − 1/T₂).

⚡ If t₁/₂ does not change with starting concentration → first-order.

• ln k = ln A − E_a/RT  (linear in 1/T)
• ln(k₂/k₁) = (E_a/R)(1/T₁ − 1/T₂)
• log(k₂/k₁) = (E_a/2.303R)(1/T₁ − 1/T₂)
• Slope of ln k vs 1/T = −E_a/R
• Catalyst: lowers E_a (does not change ΔH or K_eq)

• At standard: E = E° (when Q = 1)
• At equilibrium: E = 0, Q = K, so E° = (0.0591/n) log K
• ΔG = −nFE  (F = 96 500 C/mol)
• ΔG° = −nFE° = −RT ln K
• Hydrogen electrode: E(H₂) = −0.0591 · pH

• Relative VP lowering: Δp/p° = x_solute (Raoult)
• Boiling-point elevation: ΔTb = i·Kb·m
• Freezing-point depression: ΔTf = i·Kf·m
• Osmotic pressure: π = i·CRT (R = 0.0821 L·atm/mol·K)
• van't Hoff factor i: NaCl → 2, BaCl₂ → 3, K₄[Fe(CN)₆] → 5; glucose → 1; dimer (CH₃COOH in benzene) → 0.5

Water constants: K_b = 0.52 K·kg/mol, K_f = 1.86 K·kg/mol.

If molar solubility = s, then for a salt of formula AₓBᵧ:

• AB (e.g. AgCl): K_sp = s² → s = √K_sp
• AB₂ or A₂B (e.g. CaF₂, Ag₂CrO₄): K_sp = 4s³ → s = (K_sp/4)^(1/3)
• AB₃: K_sp = 27s⁴ → s = (K_sp/27)^(1/4)
• A₃B₂: K_sp = 108·s⁵

⚡ Common-ion effect: solubility drops drastically when one ion is already present.

• STP: 1 mol = 22.4 L at 0 °C, 1 atm
• Density of gas: d = PM/RT
• Dalton's law: P_total = ΣP_i, P_i = x_i · P_total
• Graham's law: r₁/r₂ = √(M₂/M₁) (rates of diffusion)
• RMS speed: v_rms = √(3RT/M)
• Mean speed: v_avg = √(8RT/πM)
• Most probable speed: v_mp = √(2RT/M)

Ratio v_rms : v_avg : v_mp = 1.225 : 1.128 : 1.

• q_p = ΔH, q_v = ΔU
• ΔH = ΔU + Δn_g·RT (for gases)
• Hess's law: ΔH is path-independent
• Equilibrium: ΔG° = −RT ln K = −2.303 RT log K

• Q = I × t (charge in coulombs)
• Faraday F = 96 500 C/mol e⁻
• Mass deposited: m = (M·I·t) / (n·F)  (M = molar mass, n = electrons)
• 1 F deposits 1 equivalent (1 mol of monovalent ion)

• A = absorbance (dimensionless)
• ε = molar absorptivity (L·mol⁻¹·cm⁻¹)
• b = path length (cm)
• c = concentration (mol/L)
• A = log₁₀(I₀/I), transmittance T = I/I₀, A = −log T

⚡ Chain rule: d/dx [f(g(x))] = f'(g(x))·g'(x).

• Resolution: A_x = A cosθ, A_y = A sinθ
• |A + B| = √(A² + B² + 2AB cosθ)
• Dot: A·B = AB cosθ (scalar)
• Cross: |A×B| = AB sinθ (vector perpendicular)
• Parallel vectors: cross = 0 · Perpendicular: dot = 0
• Triangle / parallelogram law for adding two vectors

⚡ Two equal vectors at θ → resultant 2A cos(θ/2) along the bisector.

• AP: aₙ = a + (n−1)d
• AP sum: S_n = n/2 · (2a + (n−1)d) = n/2 · (first + last)
• Σn = n(n+1)/2, Σn² = n(n+1)(2n+1)/6, Σn³ = [n(n+1)/2]²
• GP: aₙ = a·rⁿ⁻¹
• GP sum: S_n = a(rⁿ − 1)/(r − 1) (r ≠ 1)
• GP infinite sum: S_∞ = a/(1 − r) when |r| < 1

Question: A projectile is fired at 30°. At what other angle (same u) does it have the same range?

Question: Spin-only μ of central metal ion in high-spin [MnCl₆]⁴⁻. (Z_Mn = 25)

Question: A body falls from rest, covering 5 m in the first second. How much does it cover in the 4th second?

Q. A body is dropped freely under gravity. It covers 20 m in the first second. Distance in the third second?

• (A) 40 m
• (B) 60 m
• (C) 100 m ✓
• (D) 180 m

Q. Which diatomic species is paramagnetic with bond order 2.5?

• (A) O₂
• (B) N₂⁺ ✓
• (C) O₂²⁻
• (D) C₂

Q. Half-life of an isotope is T. Time to decay to 1/16 (6.25%) of original activity?

• (A) 2T
• (B) 3T
• (C) 4T ✓
• (D) 8T

Q. Hybridisation state and geometry of central atom in XeF₄?

• (A) sp³, tetrahedral
• (B) sp³d, trigonal bipyramidal
• (C) sp³d², square planar ✓
• (D) sp³d², octahedral

1. Scan options first. Wide gap → approximate (g → 10, π → 3, π² → 10). Tight gap → keep full precision.
2. Symbolic options? Use dimensional analysis — never compute the physics.
3. Kinematics: If the question compares states (n-th second, doubled speed), use ratios (1:3:5 odd rule, n² stopping law). Never spell out s = ut + ½at².
4. Projectile: Memorise R = 4H at 45° and complementary-angle pairs (15°-75°, 30°-60°).
5. Spin-only μ: Find n, read μ from table — never compute √(n(n+2)).
6. Hybridisation: Steric number = ½(V + M − C + A). 2→sp, 3→sp², 4→sp³, 5→sp³d, 6→sp³d².
7. Nernst @ 298 K: use 0.0591/n · log Q directly. ΔG° (kJ) ≈ −5.7 log K.
8. Half-life: t₇₅% = 2·t½  |  t₈₇.₅% = 3·t½  |  t₉₉.₉% ≈ 10·t½.
9. K_sp: AB → √K_sp · AB₂/A₂B → (K_sp/4)^⅓ · AB₃ → (K_sp/27)^¼.
10. Inverse-square: small Δr → ΔF ≈ −2·Δr%.