If two equal chords are drawn parallelly to a circle such that they divide the circle into three parts whose areas are equal, then how many of the following can be the angle subtended by any of the chords at the centre of the circle? i. 75° ii. 90° iii. 105° iv. 108° v. 120°

For two parallel chords to divide the circle into three equal areas, the areas between the chords and the circumference must each be one-sixth of the total circle area. If a chord subtends an angle theta at the center, the area of its corresponding segment is 0.5 R squared times theta minus sin(theta), which must equal one-sixth of pi R squared. This simplifies to the equation theta minus sin(theta) equals pi over 3. Using iterative approximation for this equation yields a required angle of roughly 104 degrees for the segments, meaning an angle of approximately 152 degrees for the chord sectors; none of the provided options satisfy this condition.

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