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NEET 2027 · Physics · Class 11 · Ch. 1 Units and Measurement

SI Conventions, Angles & Systems of Units

The last of the chapter: radian and steradian, the seven base-unit definitions, writing rules, the older systems, and the remaining NCERT estimation exercises. Low yield per topic — but cheap marks when they appear.

PRIORITY 3 + 4TOPICS 10, 12–15 OF 1520 QUESTIONS2 NCERT EXERCISESFINAL FILE OF 8

01Chapter coverage map

This is the eighth and final file for the chapter. Several Priority 3 and 4 topics were folded into earlier files where they belonged naturally rather than being split off into thin standalone sets. The table below shows where every one of the fifteen topics ended up, so nothing has to be hunted for.

eight files, 160 worked questions, complete chapter coverage
#TopicPriorityWhere it is covered
1Dimensional formulae of physical quantitiesP1P1-01
2SI units and derived unitsP1P1-02
3Errors in measurement and propagationP1P1-03
4Unit conversion between systemsP1P1-04
5Least count: vernier and screw gaugeP2P2-05
6Same-dimension pairs, dimensionless quantitiesP2P1-01, §04
7Significant figures: counting rulesP2P2-07
8Dimensional analysis: checking and derivingP2P2-08
9Significant figures in arithmetic, roundingP3P2-07, §05–06
10Practical units and SI prefixesP3P1-02, §05–06 and this file
11Limitations of dimensional analysisP3P2-08, §06
12Order of magnitude and scientific notationP3P2-07, §06 and this file
13Radian, steradian, plane and solid angleP4this file
14Definitions of the seven base unitsP4this file
15CGS, FPS and MKS systemsP4this file

02Concept in plain language

Everything on this page is recall rather than reasoning. That makes it low-yield per hour of study, but it also makes it the cheapest marks in the chapter: there is nothing to work out, only something to know.

Angles are dimensionless but not unitless

Plane angle is an arc divided by a radius; solid angle is an area divided by a radius squared. Both are ratios of like quantities, so both are dimensionless — and both still have units, the radian and the steradian. NEET tests that pairing more often than it tests any calculation involving them.

The base units are defined by constants, not objects

Since May 2019 every SI base unit is fixed by declaring some constant of nature to have an exact value. What is examinable is the pairing — which constant defines which unit — not the digits. NCERT footnotes this explicitly.

The older systems explain the conversion factors

CGS, FPS and MKS differ only in their choice of length and mass units; all three share the second. That shared second is why the time ratio is 1 in every SI-to-CGS conversion, and the gram-to-kilogram factor of 1000 is where the 10³ in those conversions comes from. This section is not decoration — it is the origin of the arithmetic in P1-04.

How to study this file

Read it twice and then work the questions. There is no derivation to master here and no technique to practise. The whole value is in recognising a definition or a convention on sight, which is a matter of exposure rather than effort.

03Angles, systems and conventions

BOTH ARE RATIOS OF LIKE QUANTITIES — SO BOTH ARE DIMENSIONLESS plane angle O ds r dθ = ds / r radian (rad) · full circle = 2π rad solid angle O dA r dΩ = dA / r² steradian (sr) · full sphere = 4π sr
Fig. 1 — Plane angle uses a length ratio, solid angle an area-to-radius-squared ratio. Both cancel completely, which is why both are dimensionless.
FROM THREE COMPETING SYSTEMS TO ONE, AND THEN TO CONSTANTS OF NATURECGScm · g · sGauss, 1832FPSft · lb · sBritish systemMKSm · kg · sGiorgi, 1901SI7 base unitsBIPM, 1971SI revised7 constantsCGPM, Nov 2018CGS, FPS and MKS differed only in which units they chose for length, mass and time. SI added four more base units and made everything decimal.The 2019 revision changed nothing about the sizes of the units — only what defines them. No SI unit now rests on a physical artefact.
Fig. 2 — From three competing systems to one. The 2019 revision changed what defines the units, not their sizes.
FIVE WRITING RULES NEET HAS TESTEDWRONGRIGHT10 kgs10 kgnever pluralise a symbol5 Kg5 kglower case unless named after a person5kg5 kgleave a space before the symbol1 mμm1 nmonly one prefix at a time10⁶ kkg10⁶ kg = 1 Mgbuild mass prefixes on the gramThe litre is the one deliberate exception: its symbol L is capitalised so it cannot be mistaken for the digit 1.
Fig. 3 — Five writing conventions NEET has tested, each shown as a wrong form and its correction.

04Radian and steradian

the two are structurally identical; only the power of r differs
Plane angleSolid angle
Definitiondθ = ds / rdΩ = dA / r²
Unitradian (rad)steradian (sr)
DimensionsM⁰ L⁰ T⁰M⁰ L⁰ T⁰
Full measure2π rad in a circle4π sr in a sphere
Half measureπ rad in a semicircle2π sr in a hemisphere
Conversion1 rad ≈ 57.3° ; 1° = π/180 rad—
Where it appearsω in rad s⁻¹, arc length s = rθlumen = cd sr, lux = cd sr m⁻²

Standard angle conversions

30° = π/6 · 45° = π/4 · 60° = π/3 · 90° = π/2 · 180° = π · 360° = 2π

Sanity check for every conversion: an angle in radians is a smaller number than the same angle in degrees, because one radian is about 57°.

05The seven base-unit definitions

Learn this as seven pairs. The constant's unit usually tells you which base unit it fixes, provided the earlier links in the chain are already in place — which is why the second comes first.

NCERT's footnote: these values need not be remembered or asked in a test — the pairings are what matter
Base quantityUnitFixed byValue (need not be memorised)
Timesecond (s)caesium-133 hyperfine transition frequency ΔνCs9 192 631 770 Hz
Lengthmetre (m)speed of light in vacuum c299 792 458 m s⁻¹
Masskilogram (kg)Planck constant h6.626 070 15 × 10⁻³⁴ J s
Electric currentampere (A)elementary charge e1.602 176 634 × 10⁻¹⁹ C
Temperaturekelvin (K)Boltzmann constant k1.380 649 × 10⁻²³ J K⁻¹
Amount of substancemole (mol)Avogadro constant NA6.022 140 76 × 10²³ mol⁻¹
Luminous intensitycandela (cd)luminous efficacy Kcd at 540 × 10¹² Hz683 lm W⁻¹

Two points NCERT makes that get examined

On the mole: when the mole is used, the elementary entities must be specified — atoms, molecules, ions, electrons, or a specified group of particles. A mole of oxygen atoms is not a mole of oxygen molecules.

On the kilogram: the platinum–iridium prototype was retired in 2019. No SI unit now depends on a manufactured object, which is the whole point of the revision.

06Writing conventions in full

small rules, but they have appeared as full one-mark questions
RuleWrongRight
Symbols are never pluralised10 kgs, 5 Ns10 kg, 5 N
No full stop after a symbol unless it ends a sentence5 m. long5 m long
Symbols are lower case unless the unit is named after a person5 Kg, 3 Sec5 kg, 3 s
Symbols of units named after people are capitalised, but the written name is not5 Newtons, 3 pascal (symbol)5 newtons, 3 Pa
A space separates the number from the symbol5kg5 kg
Except for the degree symbol of angle30 °30°
Only one prefix at a time1 mμm1 nm
Mass prefixes build on the gram, not the kilogram1 kkg1 Mg (= 1 tonne)
A prefix attaches to the whole unit before any power is takencm² read as c(m²)cm² = (cm)² = 10⁻⁴ m²
The litre is the deliberate exception on case5 l (looks like 5 1)5 L

07CGS, FPS, MKS and SI

all three older systems share the second — which is why time ratios vanish in most conversions
SystemLengthMassTimeNote
CGScentimetregramsecondForce unit dyne, energy unit erg. Proposed by Gauss, 1832.
FPS (British)footpoundsecondForce unit poundal — note that the pound is the mass unit.
MKSmetrekilogramsecondForce unit newton. The direct ancestor of SI.
SImetrekilogramsecondMKS extended to seven base units, adding A, K, mol and cd. Decimal throughout.

Why this section is not just history

The SI-to-CGS conversion factor for any quantity is 103a + 2b, where a and b are its mass and length exponents. The 3 comes from kilogram-to-gram and the 2 from metre-to-centimetre. Knowing which units each system chose is knowing where every conversion factor in P1-04 comes from.

Check it: energy has a = 1, b = 2, giving 10⁷ — and indeed 1 J = 10⁷ erg.

The institutional timeline

1832 Gauss proposes an absolute system based on millimetre, milligram and second. 1875 The Metre Convention establishes the BIPM. 1901 Giorgi proposes MKS with a fourth electrical base unit. 1971 The BIPM's SI scheme of symbols and abbreviations is adopted. November 2018 The CGPM votes the redefinition. 20 May 2019 It takes effect.

08Exceptions and traps

Dimensionless is not unitless

Radian and steradian are dimensionless and have units. Strain and refractive index are dimensionless and have none. The two properties are independent, and options are written to exploit the confusion.

2π and 4π are easy to swap

2π radians in a full circle, 4π steradians in a full sphere. A hemisphere is 2π sr, which numerically matches the full circle in radians — the same number describing different things.

Only two base units are named after people

The kelvin and the ampere. Metre, kilogram, second, mole and candela are not, which is why their symbols are lower case. The symbol's case is the giveaway.

Celsius is derived, kelvin is base

A difference of 1 °C equals a difference of 1 K exactly, so specific heat may be quoted either way. An absolute temperature must be in kelvin.

Pound is mass, poundal is force

Each older system has a mass unit and a separate force unit: gram and dyne in CGS, pound and poundal in FPS, kilogram and newton in MKS. Mixing the two within a system is a standard distractor.

Centi breaks the pattern

Most prefixes step in thousands, but deca, hecto, deci and centi do not. Centi being 10⁻² rather than 10⁻³ is behind most cm-to-m conversion errors in the whole chapter.

09Numbers to remember

the Earth's-mass entry is worth pausing on — its mantissa exceeds 5, so the order is 25, not 24
ItemValue
Full circle2π rad = 360°
Full sphere4π sr ≈ 12.57 sr
Hemisphere2π sr
1 radian≈ 57.3°
1 degreeπ/180 rad ≈ 0.01745 rad
Number of SI base units7 (plus rad and sr, both dimensionless)
Base units named after people2 — kelvin and ampere
SI scheme adoptedBIPM, 1971
Redefinition voted / effectiveCGPM November 2018 / 20 May 2019
Metre Convention1875
Order of magnitude, Sun's mass (2 × 10³⁰ kg)30
Order of magnitude, Earth's mass (6 × 10²⁴ kg)25 — note 6 > 5
Mean density of the Sun (NCERT Ex. 1.17)≈ 1.4 × 10³ kg m⁻³
Atomic volume of a mole of hydrogen (Ex. 1.14)≈ 3.2 × 10⁻⁷ m³
Molar volume ÷ atomic volume (Ex. 1.15)≈ 7 × 10⁴
Density of water / air10³ / ≈ 1.2 kg m⁻³

10Scientists to remember

definition and history items draw on these names directly
NameWhat to attach to the name
Carl Friedrich GaussProposed the first absolute system of units in 1832 — the ancestor of CGS.
Giovanni GiorgiProposed the MKS system with a fourth electrical base unit in 1901; the proposal that grew into SI.
William Thomson, Lord KelvinThe kelvin, one of only two base units named after a person.
André-Marie AmpèreThe ampere, the other base unit named after a person.
Anders CelsiusThe degree Celsius — a derived unit, the same size as the kelvin but offset by 273.15.
Blaise PascalThe pascal; also the unit of stress and of energy density.
Bureau International des Poids et Mesures (BIPM)Established under the Metre Convention of 1875; developed the SI scheme adopted in 1971.
Conférence Générale des Poids et Mesures (CGPM)The conference that voted the redefinition in November 2018, effective 20 May 2019.
Amedeo AvogadroNA, now the fixed constant defining the mole.
Max Planckh, now the fixed constant defining the kilogram.

11Twenty worked questions

About the labels

Items tagged PYQ here are NCERT chapter-end exercises. Every numerical result on this page — the angle conversions, the molar-volume ratio and the Sun's density — was recomputed independently before printing.

Q01Solid angle

The solid angle subtended by a complete sphere at its centre is:

  1. 4π sr
  2. 2π sr
  3. π sr
  4. 8π sr
Show full solution
Given
dΩ = dA / r², from NCERT Fig. 1.1(b).
Asked
Total solid angle at the centre of a sphere.
Concept
Add up the whole surface. The total solid angle is the total surface area divided by the square of the radius — and every r cancels, as it must for a dimensionless quantity.
Formula
Ω = A / r²
Baby steps
  1. Surface area of a sphere: A = 4πr².
  2. Ω = A/r² = 4πr²/r².
  3. = 4π sr, about 12.57 steradians.
  4. The radius cancels completely — the result is the same for any sphere.
Answer
(a) 4π sr
Why not others
(a)Correct.
(b)2π is the plane-angle answer, for a full circle in radians.
(c)π sr would be a quarter of the sphere.
(d)This doubles the surface area.
Shortcut
Two totals to keep side by side: 2π rad in a full circle, 4π sr in a full sphere. The 2 and the 4 come straight from the circumference and surface-area formulae.
Where it goes wrong
A hemisphere subtends 2π sr, which is numerically equal to the full circle in radians. The two numbers coincide but describe different things.
Q02Plane angle

One radian is approximately equal to:

  1. 180°
  2. 90°
  3. 45°
  4. 57.3°
Show full solution
Given
π radians = 180°.
Asked
One radian expressed in degrees.
Concept
A radian is the angle for which the arc equals the radius. Since a full circle is 2π radians and 360°, one radian is 360/2π degrees.
Formula
1 rad = 180°/π
Baby steps
  1. 2π rad = 360°.
  2. 1 rad = 360/(2π) = 180/π.
  3. = 180/3.1416.
  4. ≈ 57.3°.
Answer
(d) 57.3°
Why not others
(a)180° is π radians.
(b)90° is π/2 radians.
(c)45° is π/4 radians.
(d)Correct.
Shortcut
Carry the pair together: 1 rad ≈ 57.3° and 1° = π/180 ≈ 0.01745 rad. One is the reciprocal of the other.
Where it goes wrong
A radian is a little under 60°, so an angle in radians is always a smaller number than the same angle in degrees. That check catches an inverted conversion instantly.
Q03Angles

The dimensional formula of plane angle is:

  1. [L]
  2. [L⁻¹]
  3. [M⁰ L⁰ T⁰]
  4. [T⁻¹]
Show full solution
Given
dθ = ds / r.
Asked
Dimensional formula of plane angle.
Concept
Both the arc and the radius are lengths, so the ratio has no dimensions at all.
Formula
[θ] = [ds]/[r] = L/L
Baby steps
  1. [ds] = L (an arc length).
  2. [r] = L (a radius).
  3. [θ] = L/L = M⁰ L⁰ T⁰.
  4. Solid angle is the same: L²/L² is equally dimensionless.
Answer
(c) [M⁰ L⁰ T⁰]
Why not others
(a)The two lengths cancel; nothing survives.
(b)This is the dimension of a wave number, not an angle.
(c)Correct — and yet the angle still has the unit radian.
(d)This is a frequency.
Shortcut
Angular velocity, however, is T⁻¹, because the dimensionless angle is divided by a time. Rad s⁻¹ reduces to s⁻¹.
Where it goes wrong
Dimensionless is not unitless. The radian is a real unit attached to a quantity with no dimensions — the two properties are independent.
Q04Angles

Which statement about the radian and the steradian is correct?

  1. They are two of the seven SI base units
  2. They are dimensionless derived units with special names
  3. They have the dimensions of length and area respectively
  4. They are base quantities alongside mass and time
Show full solution
Given
The status of rad and sr in the SI.
Asked
The correct description.
Concept
They were formerly a separate class called 'supplementary units'. In the modern SI that class was abolished and both were reclassified as dimensionless derived units.
Formula
rad = m/m ; sr = m²/m²
Baby steps
  1. The seven base units are m, kg, s, A, K, mol and cd — the radian is not among them.
  2. Both rad and sr are ratios of like quantities, so both are dimensionless.
  3. Both nonetheless have names and symbols, so they are derived units with special names.
  4. NCERT describes them as two more units defined alongside the seven, and notes explicitly that both are dimensionless.
Answer
(b) They are dimensionless derived units with special names
Why not others
(a)There are exactly seven base units and neither is one of them.
(b)Correct.
(c)Both cancel completely; neither retains a dimension.
(d)Plane and solid angle are derived, not base, quantities.
Shortcut
If an older question offers 'supplementary units' as an option, that is the pre-1995 terminology for exactly these two. The physics is unchanged.
Where it goes wrong
The lumen (cd sr) and the lux (cd sr m⁻²) both contain the steradian. Because sr is dimensionless, the lumen is dimensionally just the candela.
Q05Solid angle

An area of 4 m² on the surface of a sphere of radius 2 m subtends, at the centre, a solid angle of:

  1. 1 sr
  2. 2 sr
  3. 0.5 sr
  4. 4 sr
Show full solution
Given
A = 4 m², r = 2 m.
Asked
The solid angle subtended.
Concept
Direct substitution into the definition. Note that the radius is squared.
Formula
Ω = A / r²
Baby steps
  1. r² = 2² = 4 m².
  2. Ω = A/r² = 4/4.
  3. = 1 sr.
  4. One steradian is the solid angle cutting out an area r² on a sphere of radius r.
Answer
(a) 1 sr
Why not others
(a)Correct.
(b)This uses r instead of r².
(c)This inverts the ratio.
(d)This omits the division by r² entirely.
Shortcut
One steradian is defined so that the area cut out equals r². Read the definition and the arithmetic is immediate.
Where it goes wrong
Forgetting to square the radius. The plane-angle formula uses r to the first power; the solid-angle formula uses r².
Q06Plane angle

An angle of 30° expressed in radians is:

  1. π/3
  2. π/2
  3. π/4
  4. π/6
Show full solution
Given
1° = π/180 rad.
Asked
30° in radians.
Concept
Multiply the number of degrees by π/180 and simplify.
Formula
θrad = θdeg × π/180
Baby steps
  1. 30 × π/180.
  2. = 30π/180.
  3. = π/6 rad ≈ 0.524 rad.
  4. Check: 0.524 is well under 1, and 30° is well under 57.3°. ✓
Answer
(d) π/6
Why not others
(a)π/3 is 60°.
(b)π/2 is 90°.
(c)π/4 is 45°.
(d)Correct.
Shortcut
The standard set, worth knowing without calculation: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π.
Where it goes wrong
Multiplying by 180/π instead of π/180. The sanity check — radians give a smaller number than degrees — catches it every time.
Q07Base-unit definitions

In the SI, the metre is defined by fixing the numerical value of:

  1. the Planck constant h
  2. the caesium frequency ΔνCs
  3. the speed of light in vacuum c
  4. the elementary charge e
Show full solution
Given
NCERT Table 1.1.
Asked
The defining constant for the metre.
Concept
The metre is whatever length makes the speed of light come out to exactly 299 792 458 m s⁻¹. The second must already be defined for this to work, which is why the caesium frequency comes first in the chain.
Formula
c = 299 792 458 m s⁻¹ (exact, by definition)
Baby steps
  1. c is declared to have an exact numerical value with no uncertainty.
  2. Its unit is m s⁻¹.
  3. The second is already fixed by ΔνCs.
  4. So fixing c defines the metre.
Answer
(c) the speed of light in vacuum c
Why not others
(a)h defines the kilogram.
(b)ΔνCs defines the second.
(c)Correct. A consequence is that the speed of light can no longer be measured — it is fixed by definition, and measurements now refine the metre instead.
(d)e defines the ampere.
Shortcut
The unit inside the constant tells you which unit it defines, provided the others in the chain are already fixed. c is in m s⁻¹, and the second comes first, so c gives the metre.
Where it goes wrong
NCERT footnotes that these numerical values need not be memorised. Learn the seven pairings, not the digits.
Q08Base-unit definitions

The SI second is defined in terms of:

  1. the Planck constant
  2. the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom
  3. the elementary charge
  4. the Avogadro constant
Show full solution
Given
NCERT Table 1.1.
Asked
The defining constant for the second.
Concept
An atomic transition is used because every caesium-133 atom in the universe behaves identically, which no manufactured clock can guarantee.
Formula
ΔνCs = 9 192 631 770 Hz (exact)
Baby steps
  1. The caesium-133 hyperfine transition has a fixed, universal frequency.
  2. That frequency is declared to be exactly 9 192 631 770 Hz.
  3. Since Hz = s⁻¹, fixing it defines the second.
  4. The second is defined first, because five of the other six definitions depend on it.
Answer
(b) the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom
Why not others
(a)The Planck constant defines the kilogram.
(b)Correct.
(c)The elementary charge defines the ampere.
(d)The Avogadro constant defines the mole.
Shortcut
The second sits at the head of the chain. c, h, e and Kcd all have the second buried in their units, so nothing else could be defined until the second was.
Where it goes wrong
Answering 'a caesium atomic clock'. The clock is the instrument; the definition rests on the transition frequency itself, which is a property of the atom.
Q09Base-unit definitions

One mole is defined as the amount of substance containing exactly:

  1. 6.022 140 76 × 10²³ elementary entities
  2. 6.023 × 10²³ atoms of carbon-12
  3. 1.602 176 634 × 10¹⁹ elementary entities
  4. 9.192 631 770 × 10⁹ elementary entities
Show full solution
Given
NCERT Table 1.1.
Asked
The modern definition of the mole.
Concept
Since 2019 the mole is defined by a fixed count, not by a mass of carbon-12. That fixed number is the Avogadro constant.
Formula
NA = 6.022 140 76 × 10²³ mol⁻¹ (exact)
Baby steps
  1. The mole contains exactly NA elementary entities.
  2. NA is fixed at 6.022 140 76 × 10²³ mol⁻¹.
  3. The entities must be specified — atoms, molecules, ions, electrons or any other particle or specified group of particles.
  4. So the answer is 6.022 140 76 × 10²³.
Answer
(a) 6.022 140 76 × 10²³ elementary entities
Why not others
(a)Correct.
(b)The old definition was tied to 12 g of carbon-12; the modern one is a fixed count and is not restricted to carbon.
(c)That is the elementary charge in coulombs.
(d)That is the caesium frequency.
Shortcut
NCERT emphasises one practical point: when the mole is used, the elementary entities must be specified. A mole of oxygen atoms and a mole of oxygen molecules are different amounts of matter.
Where it goes wrong
6.023 × 10²³ is the familiar school value but is no longer exact. The defined value is 6.022 140 76 × 10²³.
Q10Base-unit definitions

The candela is the SI unit of:

  1. luminous flux
  2. illuminance
  3. radiant power
  4. luminous intensity
Show full solution
Given
NCERT Table 1.1: the candela is defined using monochromatic radiation of frequency 540 × 10¹² Hz with Kcd = 683 lm W⁻¹.
Asked
Which quantity the candela measures.
Concept
Three closely related photometric quantities have three different units. The candela measures intensity in a given direction; the others measure the total and the per-unit-area versions.
Formula
cd (intensity) · lm = cd sr (flux) · lx = lm m⁻² (illuminance)
Baby steps
  1. Candela: luminous intensity in a given direction — a base unit.
  2. Lumen: luminous flux, cd × sr — the total light emitted.
  3. Lux: illuminance, lm m⁻² — light falling on unit area.
  4. So the candela measures luminous intensity.
Answer
(d) luminous intensity
Why not others
(a)Luminous flux uses the lumen.
(b)Illuminance uses the lux.
(c)Radiant power uses the watt, a radiometric rather than photometric quantity.
(d)Correct — and it is the only photometric base unit.
Shortcut
The frequency 540 × 10¹² Hz is chosen because it is green light, where the human eye is most sensitive. Photometric units are weighted by human vision; radiometric ones are not.
Where it goes wrong
The candela is the base unit most often mistaken for a derived one, because it feels less fundamental than the metre or the kilogram. It is base.
Q11Systems of units

Which of the following was not a base unit of the CGS system?

  1. centimetre
  2. second
  3. kilogram
  4. gram
Show full solution
Given
The CGS system, as described in NCERT §1.2.
Asked
The unit that does not belong.
Concept
The name is the answer: C, G and S stand for centimetre, gram and second.
Formula
CGS: cm · g · s
Baby steps
  1. CGS takes the centimetre for length.
  2. It takes the gram for mass — not the kilogram.
  3. It takes the second for time.
  4. So the kilogram is the odd one out; it belongs to MKS and SI.
Answer
(c) kilogram
Why not others
(a)The C in CGS.
(b)The S in CGS.
(c)Correct — the kilogram is the MKS and SI mass unit.
(d)The G in CGS.
Shortcut
Three systems, three acronyms that spell out their own base units: CGS = cm, g, s; FPS = foot, pound, second; MKS = metre, kilogram, second.
Where it goes wrong
The factor of 1000 between the gram and the kilogram is exactly what generates the 10³ in every SI-to-CGS conversion. This question and the conversion file are the same fact seen twice.
Q12Systems of units

The base units for length, mass and time in the FPS system are:

  1. foot, pound, minute
  2. foot, pound, second
  3. furlong, pound, second
  4. foot, poundal, second
Show full solution
Given
NCERT §1.2 on earlier systems of units.
Asked
The FPS base units.
Concept
FPS is the British system, and its acronym names its base units directly.
Formula
FPS: ft · lb · s
Baby steps
  1. F for foot — length.
  2. P for pound — mass.
  3. S for second — time.
  4. So the answer is foot, pound, second.
Answer
(b) foot, pound, second
Why not others
(a)The time unit is the second, shared by all three older systems.
(b)Correct.
(c)The furlong is a length unit but not the FPS base unit.
(d)The poundal is the FPS unit of force, not mass — the analogue of the newton and the dyne.
Shortcut
All three older systems share the second, which is why the time ratio is 1 in every SI-to-CGS conversion and why energy and power convert by the same factor of 10⁷.
Where it goes wrong
Option (d) is the one worth thinking about. Pound is mass, poundal is force. Mixing up the mass and force units of a system is a standard distractor.
Q13Systems of units

The SI scheme developed by the Bureau International des Poids et Mesures in 1971 was revised in November 2018 by the:

  1. General Conference on Weights and Measures (CGPM)
  2. Bureau International des Poids et Mesures (BIPM)
  3. National Institute of Standards and Technology (NIST)
  4. International Union of Pure and Applied Chemistry (IUPAC)
Show full solution
Given
NCERT §1.2.
Asked
The body responsible for the 2018 revision.
Concept
Two organisations appear in the chapter and they do different jobs. The BIPM developed and maintains the scheme; the CGPM is the diplomatic conference that votes changes into force.
Formula
BIPM develops · CGPM decides
Baby steps
  1. The SI, with its standard scheme of symbols and abbreviations, was developed by the BIPM in 1971.
  2. It was recently revised by the General Conference on Weights and Measures in November 2018.
  3. That revision took effect on 20 May 2019.
  4. So the answer is the CGPM.
Answer
(a) General Conference on Weights and Measures (CGPM)
Why not others
(a)Correct — NCERT names this body explicitly.
(b)The BIPM developed the 1971 scheme but did not vote the 2018 revision.
(c)NIST is the United States national metrology institute, not an international body.
(d)IUPAC governs chemical nomenclature, not units.
Shortcut
Remember the sequence with three dates: Metre Convention 1875, SI scheme 1971, redefinition voted 2018 and effective 20 May 2019.
Where it goes wrong
The two French acronyms are easy to swap. BIPM is the permanent bureau; CGPM is the conference that meets and decides.
Q14Prefixes

The SI prefix that denotes a factor of 10⁻² is:

  1. milli
  2. deci
  3. micro
  4. centi
Show full solution
Given
The standard SI prefix table.
Asked
Which prefix equals 10⁻².
Concept
Most prefixes step in factors of a thousand. Deci, centi, deca and hecto are the exceptions, and centi is by far the most used of them.
Formula
d 10⁻¹ · c 10⁻² · m 10⁻³
Baby steps
  1. deci = 10⁻¹.
  2. centi = 10⁻².
  3. milli = 10⁻³.
  4. So 10⁻² is centi.
Answer
(d) centi
Why not others
(a)milli is 10⁻³.
(b)deci is 10⁻¹.
(c)micro is 10⁻⁶.
(d)Correct.
Shortcut
Because centi is 10⁻² rather than 10⁻³, 1 cm² = 10⁻⁴ m² and 1 cm³ = 10⁻⁶ m³. This single irregular prefix generates most conversion errors in the chapter.
Where it goes wrong
Assuming every prefix is a power of 1000. Deca, hecto, deci and centi break the pattern, and they are the four that appear in everyday measurement.
Q15Writing conventions

Which of the following is written correctly according to SI conventions?

  1. 5 Kgs
  2. 5 kgs
  3. 5 kg
  4. 5 Kg
Show full solution
Given
SI rules for writing unit symbols.
Asked
The correctly written form.
Concept
Three rules apply at once: symbols are never pluralised, symbols are lower case unless the unit is named after a person, and a space separates the number from the symbol.
Formula
number · space · symbol, no plural, correct case
Baby steps
  1. The symbol for kilogram is kg, all lower case, because 'kilogram' is not a person's name.
  2. Unit symbols are never given an s for the plural.
  3. A space is left between the numerical value and the symbol.
  4. So the correct form is 5 kg.
Answer
(c) 5 kg
Why not others
(a)Wrong case and a spurious plural.
(b)The plural s is not permitted.
(c)Correct.
(d)K is the symbol for kelvin; kg must be lower case.
Shortcut
Capital letters appear only for units named after people: N for Newton, Pa for Pascal, K for Kelvin, A for Ampère. The written-out name stays lower case even then: '5 newtons', not '5 Newtons'.
Where it goes wrong
The litre is the deliberate exception. Its symbol L is capitalised even though Litre is not a person, so that it cannot be confused with the digit 1.
Q16Writing conventions

Which of the following is a correctly formed unit symbol?

  1. 1 mμm
  2. 1 nm
  3. 1 μμm
  4. 1 kkm
Show full solution
Given
SI rules on the use of prefixes.
Asked
The correctly formed symbol.
Concept
Only one prefix may be attached to a unit. Compound prefixes were abolished precisely because they are ambiguous and hard to read.
Formula
one prefix per unit, always
Baby steps
  1. mμm would mean milli-micro-metre = 10⁻³ × 10⁻⁶ = 10⁻⁹ m.
  2. That value has its own single prefix: nano.
  3. So it must be written 1 nm.
  4. μμm and kkm are compound prefixes for the same reason and are equally invalid.
Answer
(b) 1 nm
Why not others
(a)A compound prefix; write nm instead.
(b)Correct.
(c)A compound prefix; this is 10⁻¹² m, which is 1 pm.
(d)A compound prefix; this is 10⁶ m, which is 1 Mm.
Shortcut
If you find yourself writing two prefixes, add their exponents and look up the single prefix for the total.
Where it goes wrong
The kilogram is the one unit that already contains a prefix. Multiples of mass are built on the gram, so 10⁶ kg is a megagram (or tonne), never a kilo-kilogram.
Q17PYQ · NCERT Ex. 1.15Estimation

One mole of an ideal gas at STP occupies 22.4 L. Taking the size of a hydrogen molecule to be about 1 Å, the ratio of the molar volume to the atomic volume of a mole of hydrogen is closest to:

  1. 7 × 10⁴
  2. 7 × 10²
  3. 7 × 10⁶
  4. 7 × 10³
Show full solution
Given
Molar volume 22.4 L; molecular diameter about 1 Å, so radius 0.5 Å.
Asked
Ratio of molar volume to the actual volume of the molecules.
Concept
Work out the volume of a single molecule, multiply by Avogadro's number to get the volume the molecules themselves occupy, and divide the molar volume by it.
Formula
Vatomic = NA × (4/3)πr³
Baby steps
  1. r = 0.5 Å = 0.5 × 10⁻¹⁰ m.
  2. Volume of one molecule = (4/3)πr³ ≈ 5.24 × 10⁻³¹ m³.
  3. For a mole: × 6.022 × 10²³ ≈ 3.15 × 10⁻⁷ m³.
  4. Molar volume = 22.4 L = 22.4 × 10⁻³ m³.
  5. Ratio = 22.4 × 10⁻³ / 3.15 × 10⁻⁷ ≈ 7 × 10⁴.
Answer
(a) 7 × 10⁴
Why not others
(a)Correct.
(b)Two orders of magnitude too small.
(c)Two orders too large.
(d)One order too small.
Shortcut
This ratio is the physical reason gases are so compressible: the molecules themselves occupy about one part in 10⁵ of the container, and the rest is empty space.
Where it goes wrong
Using the diameter as the radius. The molecule's size is given as about 1 Å, so the radius is 0.5 Å. Cubing doubles that error eightfold.
Q18Named units

Which of the following units is not named after a scientist?

  1. newton
  2. kelvin
  3. pascal
  4. metre
Show full solution
Given
Four SI units.
Asked
The one not named after a person.
Concept
Most derived units carry a scientist's name, and so do two of the base units. The metre, kilogram, second, mole and candela do not.
Formula
named units → capitalised symbol
Baby steps
  1. Newton → Isaac Newton.
  2. Kelvin → William Thomson, Lord Kelvin.
  3. Pascal → Blaise Pascal.
  4. Metre comes from the Greek metron, meaning a measure — not a person's name.
Answer
(d) metre
Why not others
(a)Named after Isaac Newton; symbol N.
(b)Named after Lord Kelvin; symbol K.
(c)Named after Blaise Pascal; symbol Pa.
(d)Correct.
Shortcut
Only two base units are named after people: the kelvin and the ampere. The other five — metre, kilogram, second, mole, candela — are not, and their symbols are lower case.
Where it goes wrong
The symbol tells you: a capitalised symbol means a person's name. K and A are capitals; m, kg, s, mol and cd are not.
Q19Order of magnitude

The mass of the Sun is about 2.0 × 10³⁰ kg. Its order of magnitude is:

  1. 29
  2. 31
  3. 30
  4. 32
Show full solution
Given
M = 2.0 × 10³⁰ kg.
Asked
Order of magnitude.
Concept
Round the mantissa to 1 if it is at most 5, or to 10 if it exceeds 5. The surviving power of ten is the order of magnitude.
Formula
a ≤ 5 → order = b
Baby steps
  1. Written as a × 10b, a = 2.0 and b = 30.
  2. Since 2.0 ≤ 5, round a down to 1.
  3. The number becomes 1 × 10³⁰.
  4. Order of magnitude = 30.
Answer
(c) 30
Why not others
(a)One order too low.
(b)This applies the b + 1 rule, which requires a > 5.
(c)Correct.
(d)Two orders too high.
Shortcut
Useful astronomical orders: Sun's mass 10³⁰ kg, Earth's mass 10²⁴ kg, Sun's radius 10⁹ m, Earth's radius 10⁶ m, Sun–Earth distance 10¹¹ m.
Where it goes wrong
Had the mass been 6.0 × 10³⁰ kg, the mantissa would exceed 5 and the order would be 31. The mantissa always has to be checked.
Q20PYQ · NCERT Ex. 1.17Estimation

The Sun has mass 2.0 × 10³⁰ kg and radius 7.0 × 10⁸ m. Its mean mass density is closest to:

  1. 1.4 × 10¹ kg m⁻³
  2. 1.4 × 10³ kg m⁻³
  3. 1.4 × 10⁵ kg m⁻³
  4. 1.4 × 10⁻³ kg m⁻³
Show full solution
Given
M = 2.0 × 10³⁰ kg, R = 7.0 × 10⁸ m.
Asked
Mean mass density of the Sun.
Concept
Treat the Sun as a sphere and divide mass by volume. The point of the NCERT exercise is the comparison that follows: the answer lands in the range of solids and liquids, not gases, even though the Sun is a plasma.
Formula
ρ = M / [(4/3)πR³]
Baby steps
  1. R³ = (7.0 × 10⁸)³ = 3.43 × 10²⁶ m³.
  2. V = (4/3)πR³ = 4.19 × 3.43 × 10²⁶ ≈ 1.44 × 10²⁷ m³.
  3. ρ = 2.0 × 10³⁰ / 1.44 × 10²⁷.
  4. ≈ 1.4 × 10³ kg m⁻³ — slightly denser than water.
  5. So the answer is 1.4 × 10³ kg m⁻³, in the range of solids and liquids rather than gases.
Answer
(b) 1.4 × 10³ kg m⁻³
Why not others
(a)Three orders of magnitude too small.
(b)Correct.
(c)Two orders too large.
(d)Six orders too small — this would be a rarefied gas.
Shortcut
Anchor densities for comparison: air about 1.2 kg m⁻³, water 10³ kg m⁻³, rock about 3 × 10³, iron 7.9 × 10³. The Sun sits just above water.
Where it goes wrong
Cubing 7.0 × 10⁸ gives 3.43 × 10²⁶, not 3.43 × 10²⁴. Multiply the exponent by three; do not add it three times to itself incorrectly. Under exam pressure this is where the error appears.