1🗓️ How to Prepare
NEET weightage: 1–2 questions almost every year — usually on dimensional formulas, error propagation or significant figures. These are the fastest marks on the paper (under 40 seconds each) if the rules are automatic. Target: never drop a mark here.
Day 1 — SI & Sig Figs
7 base units, prefixes ladder, all significant-figure and rounding rules. Do 15 counting drills.
Day 2 — Errors
Absolute → relative → percentage error. The three combination rules. 10 propagation problems.
Day 3 — Dimensions
Memorise the dimensional table below. Practise checking and deriving formulas by homogeneity.
Day 4 — Mixed Drill
Problem Types T1–T10 below, one of each. Review every trap in the Traps section.
Day 5 — Mock 🧪
Timed 20-question chapter test. Anything wrong goes straight into your error notes.
2🧠 Concepts
1 · The seven SI base units memorise
Length — metre (m) · Mass — kilogram (kg) · Time — second (s) · Electric current — ampere (A) · Temperature — kelvin (K) · Amount of substance — mole (mol) · Luminous intensity — candela (cd).
Two supplementary (dimensionless) units: plane angle — radian, solid angle — steradian. Every other unit is derived from these.
2 · Significant figures — the five rules
- All non-zero digits are significant.
- Zeros between non-zero digits are significant (2.05 → 3 sf).
- Leading zeros are never significant (0.0025 → 2 sf).
- Trailing zeros without a decimal point are not significant (2500 → 2 sf), but with a decimal point they are (2500. or 2.500 × 10³ → 4 sf).
- Changing units never changes the number of significant figures.
3 · Arithmetic with significant figures
Multiplication / division: the result keeps as many significant figures as the least precise input.
Addition / subtraction: the result keeps as many decimal places as the least precise input. (Different rule — NEET loves this distinction.)
4 · Rounding rule for the digit 5
If the digit dropped is 5 (with nothing after it): round to make the preceding digit even. 2.745 → 2.74, but 2.735 → 2.74.
5 · Errors — the vocabulary
- Systematic error — same direction every time (zero error, calibration fault). Can be corrected.
- Random error — scatters both ways; reduced by averaging many readings.
- Least count error — from instrument resolution.
- Accuracy = closeness to true value; precision = closeness of readings to each other. An instrument can be precise but inaccurate.
6 · Absolute, relative, percentage error
Mean value a̅ from n readings → absolute error of each reading Δai = |a̅ − ai| → mean absolute error Δa̅ = (ΣΔai)/n. Result quoted as a̅ ± Δa̅. Relative error = Δa̅/a̅; percentage error = (Δa̅/a̅) × 100%.
7 · Principle of homogeneity core idea
Every term in a physically correct equation must have the same dimensions. You can add or equate only like dimensions. Arguments of sin, cos, log, ex must be dimensionless.
8 · Uses and limits of dimensional analysis
Can: check equations, convert units (n₁u₁ = n₂u₂), derive form of relations up to a constant.
Cannot: find dimensionless constants (½, 2π), handle sums of terms, or derive relations involving more than three unknowns / trig & exponential functions.
3🧮 Formula Bank
| Situation | Formula | Note |
|---|---|---|
| Mean value | a̅ = (a₁+a₂+…+aₙ)/n | best estimate |
| Mean absolute error | Δa̅ = Σ|a̅−aᵢ| / n | always positive |
| Relative error | Δa̅ / a̅ | no unit |
| Percentage error | (Δa̅/a̅)×100% | — |
| Z = A + B or Z = A − B | ΔZ = ΔA + ΔB | absolute errors add — even for subtraction |
| Z = AB or Z = A/B | ΔZ/Z = ΔA/A + ΔB/B | relative errors add |
| Z = Aᵖ Bᵠ / Cʳ | ΔZ/Z = p·ΔA/A + q·ΔB/B + r·ΔC/C | powers multiply the error; r adds (never subtracts) |
| Unit conversion | n₁u₁ = n₂u₂ → n₂ = n₁[M₁/M₂]ᵃ[L₁/L₂]ᵇ[T₁/T₂]ᶜ | bigger unit → smaller number |
| Least count (vernier) | LC = 1 MSD − 1 VSD | usually 0.1 mm |
| Least count (screw gauge) | LC = pitch / no. of circular divisions | usually 0.01 mm |
| Density from measurement | Δρ/ρ = Δm/m + 3·Δr/r | sphere: r appears cubed |
| g by pendulum | Δg/g = ΔL/L + 2·ΔT/T | from g = 4π²L/T² |
4📋 Formula Sheet — one glance before the exam
5✍️ Derivations
D1 · Error in a product (why relative errors add)
Z = AB. Measured: (A ± ΔA)(B ± ΔB) = AB ± AΔB ± BΔA ± ΔAΔB.
Drop the tiny ΔAΔB term, divide by Z = AB: ΔZ/Z = ΔA/A + ΔB/B. Maximum error takes both + signs.
D2 · Error in a power
Z = Aⁿ = A·A·…·A (n times). Apply the product rule n times: ΔZ/Z = n·ΔA/A. A power amplifies the error n-fold — this is why timing 20 oscillations beats timing 1.
D3 · Unit conversion n₁u₁ = n₂u₂
A physical quantity Q is the same regardless of unit: Q = n₁u₁ = n₂u₂. With u = MᵃLᵇTᶜ, n₂ = n₁[M₁/M₂]ᵃ[L₁/L₂]ᵇ[T₁/T₂]ᶜ. Example: 1 N = 1 kg·m·s⁻² = (10³ g)(10² cm)s⁻² = 10⁵ dyne.
D4 · Pendulum period by dimensional analysis
Assume T ∝ mᵃ lᵇ gᶜ → [T] = [M]ᵃ[L]ᵇ[LT⁻²]ᶜ. Match: a = 0, b + c = 0, −2c = 1 → c = −½, b = ½.
T = k√(l/g) — dimensional analysis gives the form; the constant k = 2π must come from experiment or full theory.
6🧩 Problem Types — the 10 ways NEET asks this chapter
T1 · Count the significant figures
“How many sig figs in 0.06070?” → leading zeros no, captive zero yes, trailing zero after decimal yes → 4.
Rewrite in scientific notation if unsure: 6.070 × 10⁻².
T2 · Round the result of arithmetic
Multiplication → least sig figs; addition → least decimal places. They test whether you mix the two rules.
T3 · Percentage error propagation
Given % errors in A, B, C and Z = AᵖBᵠ/Cʳ → answer = p(%A) + q(%B) + r(%C). All plus.
T4 · Match dimensional formulas
Column matching of quantities and [MᵃLᵇTᶜ]. Learn the table in Section 8 cold.
T5 · Check an equation by homogeneity
Substitute dimensions in every term; any mismatch means wrong. Correct dimensions ≠ guaranteed correct equation.
T6 · Derive a relation by dimensions
Assume Q ∝ xᵃyᵇzᶜ, equate powers of M, L, T (as in derivation D4).
T7 · Convert units of a derived quantity
Use n₁u₁ = n₂u₂. Classic: N → dyne (10⁵), J → erg (10⁷), pressure SI → CGS.
T8 · Least count / vernier / screw gauge reading
Reading = main scale + (coinciding division × LC) − zero error (with its sign).
T9 · Identify dimensionless quantities
Strain, refractive index, relative density, angles, Reynolds number, all pure ratios.
T10 · Dimensions of constants in an equation
e.g. F = at + bt² → [a] = [F]/[T] = MLT⁻³, [b] = MLT⁻⁴. Van der Waals: [a] = ML⁵T⁻², [b] = L³.
7📈 Graphs
G1 · Accuracy vs precision — the four targets
Precision = tight cluster (small random error). Accuracy = centred on true value (small systematic error). A zero-error instrument gives the second target: beautifully precise, consistently wrong.
G2 · Why timing 20 oscillations beats timing 1
The stopwatch least count Δt is fixed; measured time is nT, so ΔT = Δt/n. Percentage error in T falls as 1/n — the single most-quoted “good practice” in this chapter.
G3 · Homogeneity as a graph — every term lands on the same dimension line
ut and ½at² both reduce to [L] — they may be added. A stray v term ([LT⁻¹]) floats off the line: the equation is dimensionally wrong, no calculation needed.
8📏 Units & Dimensions — the table to memorise
| Quantity | SI unit | Dimensional formula |
|---|---|---|
| Velocity | m s⁻¹ | [M⁰LT⁻¹] |
| Acceleration | m s⁻² | [M⁰LT⁻²] |
| Force | newton (N) | [MLT⁻²] |
| Work · Energy · Heat · Torque | joule (J) / N·m | [ML²T⁻²] — same for all four |
| Power | watt (W) | [ML²T⁻³] |
| Momentum · Impulse | kg m s⁻¹ / N·s | [MLT⁻¹] — same |
| Pressure · Stress · Elastic moduli · Energy density | pascal (Pa) | [ML⁻¹T⁻²] — same for all |
| Density | kg m⁻³ | [ML⁻³] |
| Frequency · Angular velocity · Velocity gradient · Decay constant | s⁻¹ / rad s⁻¹ | [T⁻¹] — same |
| Angular momentum · Planck's constant | kg m² s⁻¹ / J·s | [ML²T⁻¹] — same |
| Surface tension · Force constant (spring) | N m⁻¹ | [MT⁻²] — same |
| Coefficient of viscosity | Pa·s (poiseuille) | [ML⁻¹T⁻¹] |
| Gravitational constant G | N m² kg⁻² | [M⁻¹L³T⁻²] |
| Universal gas constant R | J mol⁻¹ K⁻¹ | [ML²T⁻²K⁻¹mol⁻¹] |
| Boltzmann constant kB | J K⁻¹ | [ML²T⁻²K⁻¹] |
| Specific heat capacity | J kg⁻¹ K⁻¹ | [L²T⁻²K⁻¹] — no M! |
| Strain · Refractive index · Relative density · Angle | — | dimensionless [M⁰L⁰T⁰] |
9🔢 Standard Values & Prefixes
| Constant / value | Value to remember |
|---|---|
| Speed of light c | 3 × 10⁸ m s⁻¹ (exact: 299 792 458) |
| g (standard) | 9.8 m s⁻² (use 10 for estimates) |
| Planck's constant h | 6.63 × 10⁻³⁴ J·s |
| Electron charge e | 1.6 × 10⁻¹⁹ C |
| Avogadro number NA | 6.022 × 10²³ mol⁻¹ |
| Gas constant R | 8.314 J mol⁻¹ K⁻¹ |
| Boltzmann kB | 1.38 × 10⁻²³ J K⁻¹ |
| G | 6.67 × 10⁻¹¹ N m² kg⁻² |
| 1 light year | 9.46 × 10¹⁵ m (a distance, not time) |
| 1 astronomical unit (AU) | 1.496 × 10¹¹ m |
| 1 parsec | 3.08 × 10¹⁶ m = 3.26 ly |
| 1 fermi / 1 angstrom | 10⁻¹⁵ m / 10⁻¹⁰ m |
| SI prefixes ladder | |||||
|---|---|---|---|---|---|
| tera T 10¹² | giga G 10⁹ | mega M 10⁶ | kilo k 10³ | centi c 10⁻² | milli m 10⁻³ |
| micro µ 10⁻⁶ | nano n 10⁻⁹ | pico p 10⁻¹² | femto f 10⁻¹⁵ | deci d 10⁻¹ | atto a 10⁻¹⁸ |
10⚡ Shortcuts
Z = A²B³/√C → % error = 2(%A) + 3(%B) + ½(%C). Write the powers, multiply, add. 15-second question.
Whether A multiplies or divides, its relative error adds. If an option subtracts errors, it is wrong.
Work = energy = heat = torque. Pressure = stress = modulus = energy density. Frequency = angular velocity = decay constant. h = angular momentum. Learn the families, not 40 separate formulas.
Rewrite the number as a.bcd × 10ⁿ — the digits you had to write are exactly the significant ones.
Worth memorising outright; derivable from n₁u₁ = n₂u₂ but asked often enough to hard-code.
[constant] = [the term it sits in] ÷ [whatever multiplies it]. No memorising needed — rearrange and read.
11⚠️ Traps
Z = A − B → ΔZ = ΔA + ΔB. And since Z itself is small, the relative error of a difference explodes. Never design an answer around subtracting errors.
Using "least sig figs" for addition (or "least decimal places" for multiplication) is the most common wrong answer in T2-type questions.
Trailing zeros count only with a decimal point. 0.030 has 2 (the leading zeros never count, the trailing one does).
s = ut + at² passes the homogeneity check but is physically wrong (missing ½). Dimensions are necessary, not sufficient.
Radian is a unit; [angle] = M⁰L⁰T⁰. "Unitless" and "dimensionless" are not synonyms — strain is both, angle is only dimensionless.
g = 9.8 m s⁻² is acceleration [LT⁻²]; G = 6.67 × 10⁻¹¹ has [M⁻¹L³T⁻²]. Matching questions bait this constantly.
Despite the names, both measure distance. "1 ly of time" options are distractors.
Corrected reading = observed − zero error. A negative zero error therefore adds. Screw-gauge questions hinge on this sign.
In n₁u₁ = n₂u₂ conversions to CGS, remember M₁ = kg = 10³ g — forgetting the 10³ shifts the answer by three orders.
[c] = L²T⁻²K⁻¹ — the "per kg" removes M. Frequently mis-marked as containing M.
Each Δaᵢ is a modulus. If your propagation gives a negative error, a sign rule was broken upstream.
Averaging cannot fix a systematic (calibration/zero) error — the average is precisely wrong. Fix systematic errors by correction, not repetition.
12🧵 Mnemonics
Metre, Kilogram, Second, Ampere, Kelvin, Mole, Candela — say it as one word: "MKS-AKM-C".
Tera Giga Mega Kilo · deci centi milli · µ n p f — each big step is ×10³ after kilo.
Sum/difference → absolute add · product/quotient → relative add · powers → multiply the relative error by the exponent.
Work, Heat, Energy, Torque — all joules by dimension.
Pressure, Stress, Moduli, Energy density — all pascals.
Whatever sits inside sin, cos, log or eˣ must be stripped of dimensions.
13🛠️ Directions & Graphs Repair — fixing the classic misreads
The recurring wrong turns in this chapter are not about knowing formulas — they are about reading instruments and error statements in the right direction. Repair kit:
Reading = MSR + (VSD that coincides × LC). You scan the vernier scale for the aligned line — not the main scale. Then apply zero correction with sign: corrected = observed − zero error.
Zero line of circular scale below reference line → positive zero error → subtract. Above → negative → add. Draw the arrow on the diagram before computing.
They are the same percentage (r = d/2 divides value and error alike). But in V = 4/3·πr³ the % error triples: %V = 3 × %r. Read which quantity the question actually gives.
Timing n oscillations divides the clock's error by n. It does nothing to a wrong length measurement — each source of error must be shrunk by its own method (see graph G2).
Work term-by-term, left to right, reducing each to [MᵃLᵇTᶜ] (graph G3). The moment two terms disagree, stop — the answer is "dimensionally incorrect", regardless of the rest.
NEET options often list 4%, 6%, 8%, 11% for Z = A²B/C with %A=1, %B=2, %C=3 — the right move is 2(1)+2+3 = 7%… if 7% is missing, re-read the powers; the paper-setter's exponents override your assumption.
14🚨 Exceptions — the odd ones out NEET loves
Statement-type questions from this chapter are almost always built on one of these exceptions. Each one breaks a pattern you would otherwise assume.
The base unit of mass is the kilogram, not the gram. Consequence: prefixes attach to gram (mg, µg), never doubled onto kg — there is no "kilokilogram" and no µkg.
Radian and steradian are real SI units, yet [angle] = M⁰L⁰T⁰. The reverse case — a dimensional quantity without a unit — does not exist.
Strain, refractive index, relative density, coefficient of friction, Poisson's ratio, e (emissivity) — pure ratios. Distinguish from angle, which is dimensionless but has a unit.
Work and torque share [ML²T⁻²] but one is a scalar (J) and one a vector (N·m — never written as J). Identical dimensions do not mean identical quantities or units-in-use.
And the fermi (10⁻¹⁵ m) and angstrom (10⁻¹⁰ m) are lengths too, though they sound like scientists' surnames only.
It cannot derive s = ut + ½at² (two terms) and cannot find ½ or 2π (dimensionless). A dimensionally consistent equation can still be physically wrong — necessary, never sufficient.
The full set is 7: M, L, T, A (current), K (temperature), mol, cd. Writing charge's dimension needs A: [q] = [AT] — current is the base, charge is derived (the reverse of intuition).
Averaging n readings legitimately yields more decimal places than a single reading — random errors partially cancel. But it never fixes a systematic error (E12 in Traps).
The lone rule with no "opposite case": ΔZ = ΔA + ΔB for both A+B and A−B. Any option showing ΔA − ΔB is automatically wrong.
They coexist with SI by convention (1 L = 10⁻³ m³, 1 t = 10³ kg) — "non-SI" does not mean "not allowed". But CGS-only units (dyne, erg, poise) must be converted in SI answers.