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**Hint:**Frequency is defined as the number of waves that pass through a fixed point in the unit of time. It is also defined as the number of oscillations per unit of time. The unit of frequency is Hertz.

**Complete step by step solution:**

Given data:

Speed of a police car, ${v_s} = 22m/s$

Frequency of the sound horn, ${n_{car}} = 176Hz$

Frequency of the siren, ${n_{siren}} = 165Hz$

Speed of the sound, v = 330 m/s

Speed of the motorcyclist, ${v_m}$ =?

It is given that in the first case the police car which is a source of sound is moving at a speed ${v_s}$and is approaching a motorcycle (observer) which in turn is moving away from the police car with a speed of ${v_m}$.

Thus the apparent frequency of the sound heard by the motorcyclist is given by,

$\Rightarrow n' = {n_{car}}\left( {\dfrac{{v - {v_m}}}{{v - {v_s}}}} \right)\_\_\_\_\_\_\_\left( 1 \right)$

Again in the second case the motorcyclist, an observer is approaching a stationary siren, source at a speed of ${v_m}$

Thus the apparent frequency of the sound heard by the motorcyclist is given by,

$\Rightarrow n'' = {n_{siren}}\left( {\dfrac{{v + {v_m}}}{v}} \right)\_\_\_\_\_\_\_\left( 2 \right)$

It is given that the motorcyclist does not observe any beats and this is possible only when the difference in the frequencies heard by the motorcyclist is zero.

Thus $n' - n'' = 0$

$ \Rightarrow n' = n''$

Substituting the values of $n'$ and $n''$ from the equations 1 and 2, we get,

$\Rightarrow {n_{car}}\left( {\dfrac{{v - {v_m}}}{{v - {v_s}}}} \right) = {n_{siren}}\left( {\dfrac{{v + {v_m}}}{v}} \right)$

Thus substituting the values of ${n_{car}},{v_m},{v_s},{n_{siren}},v,$ we get

$\Rightarrow 176\left( {\dfrac{{v - {v_m}}}{{330 - 22}}} \right) = 165\left( {\dfrac{{v + {v_m}}}{{330}}} \right)$

$ \Rightarrow \left( {\dfrac{{v - {v_m}}}{{v + {v_m}}}} \right) = \dfrac{{165}}{{176}} \times \dfrac{{308}}{{330}} = \dfrac{7}{8}$

$ \Rightarrow 8v - 8{v_m} = 7v + 7{v_m}$

$ \Rightarrow 15{v_m} = v$

$ \Rightarrow {v_m} = \dfrac{v}{{15}} = \dfrac{{330}}{{15}} = 22m/s$

Thus the speed of the motorcyclist $ = 22m/s$

**Hence the correct option is B.**

**Note:**The sound source generates the sound waves and creates the vibrations in the surrounding medium. As this continues the vibrations propagate away at the speed of the sound.

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