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Two circles with centres $\text{P}$ and $\text{Q}$ cut each other at two distinct points $\text{A}$ and $\text{B}$. The circles have the same radii and neither $\text{P}$ nor $\text{Q}$ falls within the intersection of the circles. What is the smallest range that includes all possible values of the angle $\text{AQP}$ in degrees?

1. Between $0$ and $90$
2. Between $0$ and $30$
3. Between $0$ and $60$
4. Between $0$ and $75$

Let’s draw the diagram.

DIAGRAM

Minimum value of Angle $\text{AQP}$ is possible when, $\text{PQ = AP = AQ = r.}$ (When $\text{P}$ and $\text{Q}$ are at the intersection)

So, the triangle $\text{APQ}$ is an equilateral triangle.

Therefore each angle will be $60^{\circ}.$

If neither $\text{P}$ nor $\text{Q}$ full within the intersection of the circles.

Then $< \text{AQP}<60^{\circ}.$

If we streach the circles then horizontal distance will be increases.

So, the value of $\text{AQP}$ will be decreased. So, $< \text{AQP}>0^{\circ}.$

$\therefore \boxed{ 0^{\circ} < \text{AQP}<60^{\circ}}$

Correct Answer $:\text{C}$
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