Correct Answer: C — 7

The SI system has 7 base units: metre (length), kilogram (mass), second (time), ampere (electric current), kelvin (thermodynamic temperature), mole (amount of substance), and candela (luminous intensity). There are also 2 supplementary units — radian and steradian — but they are dimensionless.

Correct Answer: C — Force

Force is a derived quantity with dimensional formula [MLT⁻²]. All other options (luminous intensity — cd, amount of substance — mol, thermodynamic temperature — K) are base quantities in SI.

Correct Answer: C — 4

In 0.06900: The leading zeros (0.0) are not significant. The digits 6, 9, 0, 0 are significant (4 total). Trailing zeros after a decimal point are significant — they indicate the precision of measurement.

Correct Answer: B — [M⁰ L T⁻¹]

Velocity = displacement / time = [L]/[T] = [M⁰ L T⁻¹]. Mass does not appear in velocity, hence M⁰.

Correct Answer: A — Centimetre, Gram, Second

CGS = Centimetre–Gram–Second. MKS = Metre–Kilogram–Second. FPS (British) = Foot–Pound–Second.

Correct Answer: C

In multiplication/division: retain as many significant figures as are in the original number with the least significant figures. The rule of fewest decimal places applies only to addition/subtraction.

Correct Answer: B — [M L⁻¹ T⁻²]

Pressure = Force / Area = [MLT⁻²] / [L²] = [M L⁻¹ T⁻²]

Correct Answer: A — 2.74

When the digit to be dropped is exactly 5: if the preceding digit is even, the digit is simply dropped. Here, preceding digit is 4 (even) → 2.74. If it were 2.735, the preceding digit 3 is odd → round up to 2.74.

Correct Answer: C — Torque

Energy = [ML²T⁻²]. Torque = Force × distance = [MLT⁻²][L] = [ML²T⁻²]. Both have the same dimensions, though they are different physical quantities.

Correct Answer: B — Need not be physically correct

A dimensionally correct equation confirms dimensional consistency but does NOT guarantee correctness — dimensionless constants like 2, π, ½ are not revealed by dimensional analysis. However, a dimensionally incorrect equation is definitely wrong.

Correct Answer: B — Radian

Plane angle dθ = arc length (ds) / radius (r) → unit is radian (rad). Steradian is the unit for solid angle. Both are dimensionless quantities.

Correct Answer: A — 1

In 0.007: the zeros before 7 are not significant. Only the digit 7 is significant. So the number has 1 significant figure.

Correct Answer: A — Impulse and momentum

Impulse = Force × time = [MLT⁻²][T] = [MLT⁻¹]
Momentum = mass × velocity = [M][LT⁻¹] = [MLT⁻¹]
Both have the same dimensions [MLT⁻¹].

Correct Answer: B — N m² kg⁻²

From Newton's law of gravitation: F = Gm₁m₂/r²
G = Fr²/(m₁m₂) → [N·m²/kg²] = N m² kg⁻²

Correct Answer: B

In scientific notation (a × 10ᵇ), all zeros in the base number a are always significant — there is no ambiguity about trailing zeros. E.g. 4700 mm = 4.700 × 10³ mm clearly shows 4 sig. figs.

Correct Answer: B — [M L⁻³ T⁰]

Density = mass / volume = [M] / [L³] = [M L⁻³ T⁰]

Correct Answer: B — 663.8 g

For addition, retain the fewest decimal places. The least precise number is 227.2 g (1 decimal place). Sum = 663.821 → rounded to 1 decimal place = 663.8 g.

Correct Answer: B

The principle of homogeneity states only similar physical quantities (same dimensions) can be added/subtracted. You cannot add velocity to force, for example.

Correct Answer: A — 9.46 × 10¹⁵ m

1 light year = speed of light × 1 year = (3×10⁸ m/s) × (3.156×10⁷ s) ≈ 9.47 × 10¹⁵ m. Note: 3.08×10¹⁶ m = 1 parsec, and 1.50×10¹¹ m = 1 AU.

Correct Answer: A — Strain

Strain = change in length / original length = [L]/[L] = dimensionless [M⁰L⁰T⁰]. Refractive index, angle, and relative density are other common dimensionless quantities.

Correct Answer: A

K = m²v³ → dimensions = [M²][LT⁻¹]³ = [M²L³T⁻³]. But kinetic energy has dimensions [ML²T⁻²]. Since dimensions don't match, the formula is dimensionally incorrect and hence wrong.

Correct Answer: B

Order of magnitude = the power b in 10ᵇ after rounding a to 1 (if a ≤ 5) or 10 (if a > 5). E.g., diameter of Earth ≈ 1.28 × 10⁷ m → order of magnitude = 7.

Correct Answer: B

The SI second is defined by the fixed numerical value of the caesium frequency Δν_cs = 9,192,631,770 Hz (hyperfine transition of Cs-133 atom). This is far more precise than astronomical definitions.

Correct Answer: B — Power

Power = Work / time = [ML²T⁻²] / [T] = [ML²T⁻³]. Work = [ML²T⁻²], Pressure = [ML⁻¹T⁻²], Momentum = [MLT⁻¹].

Correct Answer: C — 10⁻¹⁰ m

1 Å = 10⁻¹⁰ m. The angstrom is a convenient unit at the atomic scale. The size of a hydrogen atom is about 0.5 Å = 5 × 10⁻¹¹ m.

Given: Side a = 7.203 m → 4 significant figures

(a) Surface Area:
SA = 6a² = 6 × (7.203)² = 6 × 51.883209 = 311.299 m² Rounded to 4 sig. figs → SA = 311.3 m²

(b) Volume:
V = a³ = (7.203)³ = 373.715 m³ Rounded to 4 sig. figs → V = 373.7 m³

Mass = 5.74 g → 3 significant figures
Volume = 1.2 cm³ → 2 significant figures

For multiplication/division: retain the least number of sig. figs = 2
Density = 5.74 / 1.2 = 4.783... g/cm³ Rounded to 2 sig. figs → ρ = 4.8 g cm⁻³

LHS: ½mv²
[M][LT⁻¹]² = [M][L²T⁻²] = [ML²T⁻²] RHS: mgh
[M][LT⁻²][L] = [M][L²T⁻²] = [ML²T⁻²] LHS dimensions = RHS dimensions = [ML²T⁻²]

✔ Equation is dimensionally correct

1 kg = 10³ g  |  1 m = 10² cm  |  1 s = 1 s

1 kg m² s⁻² = 1 × (10³ g) × (10² cm)² × (1 s)⁻² = 1 × 10³ × 10⁴ g cm² s⁻² = 10⁷ g cm² s⁻² 1 kg m² s⁻² = 10⁷ g cm² s⁻²

Speed of light = 1 (new unit of length / second)
Time = 8 min 20 s = 8 × 60 + 20 = 500 s

Distance = Speed × Time = 1 × 500 = 500 new units Sun–Earth distance = 500 new units of length
(This is equivalent to 500 light-seconds ≈ 1.50 × 10¹¹ m = 1 AU)

Observed width = 3.5 mm, Magnification = 100

Actual thickness = Observed width / Magnification = 3.5 mm / 100 = 0.035 mm 3.5 mm has 2 significant figures → answer has 2 sig. figs.
Thickness = 0.035 mm = 3.5 × 10⁻² mm

Areal magnification = Screen area / Slide area
= 1.55 m² / 1.75 cm² = 1.55 / (1.75 × 10⁻⁴ m²)
Areal magnification = 1.55 / 1.75 × 10⁻⁴ = 8857 Linear magnification = √(Areal magnification)
m = √8857 ≈ 94.1 Linear magnification ≈ 94

l = 16.2 ± 0.1 cm → relative error = 0.1/16.2 = 0.617%
b = 10.1 ± 0.1 cm → relative error = 0.1/10.1 = 0.990% ≈ 1%

Area = l × b = 16.2 × 10.1 = 163.62 cm² Combined relative error = 0.617% + 0.990% ≈ 1.6%
Absolute error = 1.6% of 163.62 = 2.6 cm²

Area = 164 ± 3 cm²

Assume T = k · lˣ · mʸ · gᶻ (k = dimensionless constant)

Dimensions: [T¹] = [L]ˣ [M]ʸ [LT⁻²]ᶻ = [Mʸ · L^(x+z) · T^(-2z)]

Equating powers:
  M: y = 0
  T: -2z = 1 → z = -½
  L: x + z = 0 → x = ½

T = k · l^(½) · g^(-½) = k√(l/g) Actual value: k = 2π
T = 2π √(l/g)

(a) Total mass:
Box = 2300.00 g, Piece 1 = 20.15 g, Piece 2 = 20.17 g
Total = 2300.00 + 20.15 + 20.17 = 2340.32 g 2.30 kg has precision to only 0.01 kg = 10 g → round to nearest 10 g
(a) Total mass = 2.34 kg

(b) Difference:
Δm = 20.17 − 20.15 = 0.02 g Both precise to 2 decimal places → (b) Δm = 0.02 g

l = 4.234 m (4 s.f.), b = 1.005 m (4 s.f.), t = 0.0201 m (3 s.f.)

Area: (one face — l × b) Area = 4.234 × 1.005 = 4.255... m² Minimum sig. figs = 4 → Area = 4.255 m²

Volume: Volume = 4.234 × 1.005 × 0.0201 = 0.08551... m³ Minimum sig. figs = 3 → Volume = 0.0855 m³

Radius r = 0.5 Å / 2 = 0.25 Å = 0.25 × 10⁻¹⁰ m = 2.5 × 10⁻¹¹ m

Volume of one atom: V₁ = (4/3)πr³ = (4/3) × π × (2.5×10⁻¹¹)³ = (4/3) × 3.14159 × 1.5625×10⁻³² ≈ 6.54 × 10⁻³² m³ Total volume for 1 mole: V = N_A × V₁ = 6.022×10²³ × 6.54×10⁻³² ≈ 3.94 × 10⁻⁸ m³ Total atomic volume ≈ 4 × 10⁻⁸ m³

Volume of Sun: V = (4/3)πR³ = (4/3) × 3.14159 × (7.0×10⁸)³ = (4/3) × 3.14159 × 3.43×10²⁶ = 1.437 × 10²⁷ m³ Density: ρ = M/V = 2.0×10³⁰ / 1.437×10²⁷ = 1392 kg/m³ ≈ 1.4 × 10³ kg/m³ ρ ≈ 1.4 × 10³ kg m⁻³

This is comparable to the density of water (10³ kg/m³) and falls in the range of liquids/solids — despite the Sun being a hot plasma gas at extreme temperatures.

Dimensional formula of G: [M⁻¹L³T⁻²]
Conversion factors: 1 kg = 10³ g, 1 m = 10² cm, 1 s = 1 s

G(CGS) = G(SI) × (1 kg)⁻¹ × (1 m)³ × (1 s)⁻² × (1/10³ g)⁻¹ × (1/10² cm)³ = 6.67×10⁻¹¹ × (10³)¹ × (10²)³ × (1)⁻² = 6.67×10⁻¹¹ × 10³ × 10⁶ = 6.67×10⁻¹¹⁺⁹ = 6.67×10⁻² cm³ s⁻² g⁻¹ G = 6.67 × 10⁻⁸ cm³ g⁻¹ s⁻²

Dimensional formula of energy (calorie): [M L² T⁻²]

In SI: 1 calorie = 4.2 kg m² s⁻²

In new system: unit of mass = α kg, unit of length = β m, unit of time = γ s

n₂ = n₁ × [M₁/M₂]¹ × [L₁/L₂]² × [T₁/T₂]⁻² = 4.2 × (1 kg/α kg)¹ × (1 m/β m)² × (1 s/γ s)⁻² = 4.2 × (1/α) × (1/β)² × γ² Magnitude = 4.2 α⁻¹ β⁻² γ² ✓ (proved)