Multiple choice

Consider a circle with unit radius. There are 7 adjacent sectors, S1, S2, S3, ..., S7 in the circle such that their total area is (1/8)th of the area of the circle. Further, the area of the jth sector is twice the area of the (j - 1)th sector, for j = 2, …, 7. What is the measure of the angle (in radians) subtended by the arc of S1 at the centre of the circle?

  1. π/508

  2. π/2040

  3. π/1016

  4. π/1524

  5. None of these
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A Correct answer
Explanation

Total area = (1/8) * pi * r^2 = pi/8 (since r=1). Let the area of S1 be A. Then A + 2A + 4A + ... + 64A = pi/8. A(2^7 - 1)/(2-1) = pi/8. 127A = pi/8, so A = pi/1016. The area of a sector is (theta/2) * r^2. Since r=1, A = theta/2. So theta/2 = pi/1016, theta = pi/508.

AI explanation

The combined area of the seven sectors is 1/8 of the circle's area, so their total area is (1/8) * pi * 1^2 = pi/8. Let the area of the first sector be A; the areas form a geometric progression A, 2A, 4A up to 64A, summing to 127A. Equating this to the total area gives 127A = pi/8, meaning the area of S1 is A = pi/1016. The central angle equals the area divided by the radius, which gives the angle as pi/1016 radians.

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