A system of units is a complete, self-consistent set of units built from a chosen set of base quantities. The confusion that costs marks on NEET: treating radian and steradian as if they carry dimensions, when they are dimensionless ratios that merely have unit names.
What NCERT says. A system of units selects a small number of base (fundamental) quantities — length, mass, time, electric current, temperature, amount of substance, luminous intensity — and defines one unit for each. Every other unit is derived from these seven by multiplication, division, or exponentiation. The four historically important systems are CGS (centimetre–gram–second), FPS (foot–pound–second), MKS (metre–kilogram–second), and the modern SI, which extends MKS with four more base quantities (NCERT Class 11 Physics Chapter 1, page 2).
The SI supplementary-unit trap. Before 1995, radian and steradian were classified as "supplementary units" — a third category alongside base and derived. The 20th CGPM (1995) abolished that category and reclassified both as derived units. They remain dimensionless: radian = arc length ÷ radius (length ÷ length), steradian = subtended area ÷ radius² (length² ÷ length²). NEET distractors exploit the fact that these quantities have named units, tempting you to assign them dimensions they do not possess.
How NEET tests this. The typical question presents plane angle and solid angle side by side and asks about their dimensional status. Distractors include "plane angle has dimensions of length" or "solid angle has dimensions of area." The correct answer: both are dimensionless; both have units (rad, sr) but zero dimensional formula.
Watch-out. When a question says "supplementary units," it is testing whether you know the historical label and its modern reclassification — not whether you assign dimensions. Dimensionless means [M⁰ L⁰ T⁰].
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
Which of the following is NOT a base quantity in the SI system?
How many base units does the SI system define?
In the CGS system, the base units of length, mass, and time are respectively:
Plane angle (radian) and solid angle (steradian) are:
Before the 20th CGPM resolution in 1995, radian and steradian were classified as:
The dimensional formula of solid angle is:
Which of the following pairs of physical quantities have different dimensional formulas?
A student claims: "Since radian is a unit, plane angle must be a dimensional quantity." Which statement correctly identifies the flaw?
Given
- Q = (intercepted area on sphere) / (radius of sphere)² - Area has dimensions [L²] - Radius has dimensions [L]
Required
- Dimensional formula of Q - Whether Q has a named SI unit
Concept
Solid angle is defined as the ratio of the surface area subtended at the centre of a sphere to the square of the sphere's radius. Since both numerator and denominator have the same dimensions, the result is dimensionless.
Formula
[Q] = [Area] / [Radius]² = [L²] / [L²]
Substitution
[Q] = L² / L² = L⁰ = [M⁰ L⁰ T⁰]
Calculation
The length dimensions cancel exactly. No arithmetic is needed — this is a dimensional cancellation. All quantities here (area, radius) are geometric definitions with exact relationships, not measured values requiring significant-figure treatment.
Final answer
Q has dimensional formula **[M⁰ L⁰ T⁰]** (dimensionless). Q is the solid angle, and its named SI unit is the **steradian (sr)**. Note: The numbers 2 (as exponents) are exact mathematical powers — they do not contribute to any significant-figure count. This problem involves no measured values.
Common trap
Students see "area" in the numerator and write [L²], then forget to divide by [L²] from the radius squared — concluding Q has dimensions of area. Others see the named unit "steradian" and assume it must carry dimensions. A named unit ≠ a dimensional unit. (Trap: unit-vs-dimension confusion for radian/steradian.)
Similar NEET-style question
"The ratio of the circumference of a circle to its diameter is π. What is the dimensional formula of π?" Answer: [M⁰ L⁰ T⁰] — it is the ratio of two lengths (circumference [L] / diameter [L]), hence dimensionless, with no named unit. ---
The SI is the internationally accepted system of units, based on seven base units: metre (length), kilogram (mass), second (time), ampere (electric current), kelvin (thermodynamic temperature), mole (amount of substance) and candela (luminous intensity).
-- NCERT Class 11 Physics, Ch. 1, p. 2The maximum relative error in a power expression is the sum of the absolute exponents weighted by the relative errors of the bases. Negative exponents (divisions) still take the |.| value because we want the worst-case error.
| Symbol | Quantity | SI Unit |
|---|---|---|
| Z | Result | (combined) |
| p, q, r | Exponents (signed) | (dimensionless) |
| A, B, C | Measured quantities | (measured) |
When two measured quantities are multiplied or divided, the maximum RELATIVE errors add. The absolute error in the result is then Delta_Z = Z * (relative-error sum).
| Symbol | Quantity | SI Unit |
|---|---|---|
| Z | Result of product/quotient | (combined unit) |
| A | First measured quantity | (measured) |
| B | Second measured quantity | (measured) |
| Delta_A/A | Relative error in A | (dimensionless) |
| Delta_B/B | Relative error in B | (dimensionless) |
When two quantities are added or subtracted, the maximum absolute errors of the inputs simply add to give the maximum absolute error of the output. The relative error is NOT what adds in this case.
| Symbol | Quantity | SI Unit |
|---|---|---|
| Z | Result of sum/difference | (same as A,B) |
| A | First measured quantity | (measured) |
| B | Second measured quantity | (measured) |
| Delta_Z | Maximum absolute error in Z | (same as A,B) |
| Delta_A | Maximum absolute error in A | (same as A) |
| Delta_B | Maximum absolute error in B | (same as B) |
These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.
Category: Similar Terms
Student gets the time exponent wrong by 1 (e.g. T⁻¹ vs T⁻²) when manipulating dimensional formulas.
Question asks for dimensions of a derived combination (e.g. E/G, F = αt² + βt) where time exponent matters.
Write each base quantity's dimensional formula explicitly, then combine. Common errors: dividing forces forgets sub of T exponents; energy/length includes implicit time. Always check final units against expected SI.
Category: Similar Terms
Student sums relative errors of all measured quantities without weighting by the exponent. For ρ = m/(πr²L), the relative error contribution of r is 2 × Δr/r, NOT Δr/r — the exponent of r in the formula carries through as a multiplicative factor.
Question gives a derived quantity formula with mixed-power dependencies; asks for the max relative error. Distractors omit the power factor.
Always write the full general rule: for Z = A^p B^q C^r, ΔZ/Z = |p|·ΔA/A + |q|·ΔB/B + |r|·ΔC/C. Identify the powers (1, 2, 3, ½) before adding.
Category: Similar Terms
Student conflates random errors (statistical, unpredictable, reduced by averaging) with instrumental errors (consistent bias from the apparatus) or with systematic errors (consistent bias from the method). Each has a distinct definition and different mitigation.
Question describes an error source and asks for its taxonomic category. Distractors include cognate categories.
Memorise the 5-category taxonomy: PERSONAL (observer-side), INSTRUMENTAL (apparatus calibration), LEAST-COUNT (instrument resolution floor), RANDOM (statistical, reduced by repeated trials), SYSTEMATIC (method-level bias, NOT reduced by averaging).
Category: Similar Terms
Student swaps which is the input vs the output: least count = pitch / N (where N is the number of circular-scale divisions). Distractors offer the ratio inverted or the wrong unit.
Question gives one of (pitch, N, least count) and asks for another; distractors offer the inverted ratio or off-by-factor-of-10.
Anchor on the definition: least count is the SMALLEST measurement the instrument can resolve. It is always SMALLER than the pitch. So pitch = LC × N (and not LC = pitch × N).
Category: Similar Terms
Student applies the 'fewest significant figures' rule (which governs multiplication and division) to a sum or difference. Subtraction of two measured numbers must instead reflect the FEWEST decimal places.
Question involves addition/subtraction of measured numbers with very different magnitudes or decimal-place counts (e.g. 9.99 - 0.0099). Distractors offer answers rounded by sig-fig rule rather than decimal-place rule.
Memorise: multiplication/division → fewest SIGNIFICANT FIGURES; addition/subtraction → fewest DECIMAL PLACES. Always identify which arithmetic operation is being performed before applying any rule.
Category: Similar Terms
Student treats radian/steradian as having dimensions because they have unit names.
Question asks about dimensions of plane angle, solid angle, or comparison.
Plane angle (radian) and solid angle (steradian) are DIMENSIONLESS — they're ratios (arc/radius for radian; surface-area/r² for steradian). They have unit NAMES for clarity but no dimensions.
Category: Similar Terms
Confusing whether N or N+1 is the smaller count when (N+1) divisions of vernier match N divisions of main scale.
Question gives '(N+1) divisions of vernier coincide with N divisions of main' or similar phrasing.
Always interpret carefully: N+1 vernier divisions span the SAME LENGTH as N main divisions. So 1 VSD = (N/(N+1)) MSD; vernier constant = 1 MSD - 1 VSD = 1 MSD / (N+1). Result smaller than 1 MSD.
Root cause: formula misuse
Use Delta_Z = Delta_A + Delta_B for sums/differences (absolute errors add). Use Delta_Z/Z = Delta_A/A + Delta_B/B for products/quotients (relative errors add). They are NOT interchangeable — the rule is dictated by whether the operation is additive or multiplicative.
Distractor option uses the wrong rule (e.g. quotes a small relative error for a sum where absolute errors should add).
Root cause: concept gap
For multiplication/division, the result has the fewest significant figures of any input. For addition/subtraction, the result has the fewest decimal places of any input. Decimal places ≠ significant figures.
Distractor truncates a sum to too few significant figures by applying the multiplication rule.
The quantities which have the same dimensions as those of solid angle are :
The dimensions [MLT–2A–2] belong to the
Plane angle and solid angle have
miscounts power of T
Off-by-one on time exponent (e.g. -1 vs -2)
forgets power of two on radius
Default to summing all relative errors with weight 1
linear sum no powers
Adding all relative errors with weight 1
treats radian as dimensional
Confuses 'has unit' with 'has dimensions'
instrumental misclassified as random
Both feel 'unavoidable'; need to recognise instrumental ≠ random
swaps pitch and LC
Confusing which side of the formula is asked
no rounding
Calculator answer kept verbatim
applies mult rule to subtraction
Default to 'fewest sig figs' without distinguishing subtraction's decimal-places rule
swap N and N+1
Confusing which side has the larger count
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