Yes. If X and Y lie on the same side of AB on the circle, then ∠AXB and ∠AYB subtend the same chord AB. Hence, they are equal, so they cannot be different.
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Since OA and OB are radii, triangle AOB is isosceles. Given ∠AOB = 60°, the other two angles are also 60°. Thus, triangle AOB is equilateral, so AB = OA = 12 cm.
No, we cannot conclude that CD = 2AB. According to the chapter, the longer chord is closer to the centre. Chord length depends on radius and distance from the centre, not directly on distance alone.
The perpendicular from the centre bisects the chord, forming two right triangles. Applying the Baudhāyana–Pythagoras Theorem gives the half-chord as √(r² − d²). Hence, the chord length is 2√(r² − d²).
Using the chord-length formula or the Baudhāyana–Pythagoras Theorem, the half-chord measures √13 cm. Therefore, the complete chord length is 2√13 cm.