Doubt from waves

Example 1 :

Example 2 :

For 41) the answer is C

Draw tangents from the observer to the circle (minimum and maximum apparent frequency occurs at these positions)
When velocity of source is towards observer maximum occurs.
When velocity of source is away then minimum occurs

Reason:Why max/min occurs at tangents ...because at these points the velocity of approach towards observer equals to the actual rotating velocity.(The maximum value it can have )

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Very well explained...thank u so much for ur efforts.

It should be of this form .

Also the function should be finite for finite x and t.
Option a is answer as it cannot be expressed in this form.

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B,C,D are finite in finite values of x and t.
They only become infinite at infinite time and x.
And they are in that function they represent travelling waves

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