Which of the following is not a dimensionless quantity?

We begin by recalling that a dimensionless quantity is one whose numerical value does not change when we change the fundamental units of measurement. In other words, if we write its dimensional formula in terms of the basic dimensions $$[M]$$ (mass), $$[L]$$ (length), $$[T]$$ (time), $$[I]$$ (electric current), and so on, every exponent must be zero, giving the overall dimensional formula $$[M^0L^0T^0I^0]$$, which we usually denote simply by $$1$$.

Now we examine each option one by one.

Option A : Power factor
The power factor is defined as $$\cos\phi$$, where $$\phi$$ is the phase angle between the alternating voltage and the alternating current. Since cosine is the ratio of two sides of a right-angled triangle in phasor representation, or more simply, the cosine of an angle, it is a pure number. Hence its dimensional formula is $$[M^0L^0T^0I^0]$$. Therefore, the power factor is dimensionless.

Option B : Quality factor
For an LCR resonant circuit, the quality factor $$Q$$ is defined by the relation $$ Q = \dfrac{\omega_0 L}{R} = \dfrac{1}{\omega_0 C R}, $$ where $$\omega_0$$ is the resonant angular frequency, $$L$$ is inductance, $$C$$ is capacitance, and $$R$$ is resistance. In every equivalent formula, the units in the numerator and denominator cancel out completely. This leaves the dimensional formula $$[M^0L^0T^0I^0]$$. Thus the quality factor is also dimensionless.

Option C : Permeability of free space $$\mu_0$$
The magnetic permeability of free space is defined from the relation $$ F = \dfrac{\mu_0 I_1 I_2 l}{2\pi r}, $$ or more commonly appears in the force law between two current-carrying conductors, in Ampère’s law, and in many electromagnetic formulas. From any such formula we can extract its dimensional formula. One very convenient starting point is the expression for the speed of light in vacuum: $$ c = \dfrac{1}{\sqrt{\mu_0 \varepsilon_0}}, $$ which gives $$ \mu_0 = \dfrac{1}{c^2 \varepsilon_0}. $$ Because $$\varepsilon_0$$ (permittivity of free space) has the dimensions $$ [\varepsilon_0] = [M^{-1}L^{-3}T^{4}I^{2}], $$ and the speed of light has dimensions $$ [c] = [LT^{-1}], $$ we substitute these into the above relation. We have $$ [\mu_0] = \dfrac{1}{[c]^2 \,[\varepsilon_0]} = \dfrac{1}{\bigl([L T^{-1}]\bigr)^2 \,[M^{-1}L^{-3}T^{4}I^{2}]} $$ $$\;\;\;= \dfrac{1}{[L^{2}T^{-2}]\,[M^{-1}L^{-3}T^{4}I^{2}]} $$ $$\;\;\;= [M^{1} L^{1} T^{-2} I^{-2}]. $$ The exponents here are not all zero, so $$\mu_0$$ is not dimensionless. Indeed, in SI units its unit is henry per metre (H m−1).

Option D : Relative magnetic permeability $$\mu_r$$
Relative permeability is defined as the ratio $$ \mu_r = \dfrac{\mu}{\mu_0}, $$ where $$\mu$$ is the absolute permeability of the medium. Because both $$\mu$$ and $$\mu_0$$ carry exactly the same dimensions, the ratio cancels those dimensions out completely. Therefore $$ [\mu_r] = \dfrac{[\mu]}{[\mu_0]} = 1, $$ so $$\mu_r$$ is dimensionless.

Gathering the conclusions:

Option A: dimensionless.
Option B: dimensionless.
Option C: has dimensions.
Option D: dimensionless.

Hence, the only quantity that is not dimensionless is the permeability of free space $$\mu_0$$, which corresponds to Option C (Option 3).

Hence, the correct answer is Option 3.