If m is the area of the circle S and n is the 2r2 tan 9area of semi circle S1 then which one of...........

Quantitative Aptitude (CDS) Geometry

If m is the area of the circle S and n is the 2r2 tan 9area of semi-circle S₁, then which one of the following is correct ?

This questions was previously asked in

2023 CDS -2 MATH

9m = 2n

9m = 4n

3m = 2n

7m = 3n

Let’s break it down:

- Area of circle S (m): m = pr².

- Area of semicircle S1 (n): Diameter is 3r, so radius is (3r/2). Area = ½ × p × (3r/2)² = ½ × p × (9r²/4) = (9/8)pr² ? n = (9/8)pr².

- Now, let’s compare m and n.

- 9m = 9 × pr²

- 2n = 2 × (9/8)pr² = (9/4)pr²

Which option matches?

- Option 1: 9m = 2n ? 9pr² = (9/4)pr²? No.

- Option 2: 9m = 4n ? 9pr² = 4 × (9/8)pr² = (9/2)pr²? No.

- Option 3: 3m = 2n ? 3pr² = 2 × (9/8)pr² = (9/4)pr²? No.

- Option 4: 7m = 3n ? 7pr² = 3 × (9/8)pr² = (27/8)pr²?

Let’s check numerically:

- 7/1 vs 27/8 ? 7 = 3.5 × 2 = 7 = 27/8 ˜ 3.375. So, not equal.

Actually, none match exactly. But based on area comparison, closest is 8m = 9n (from rearranging n = (9/8)m).

But if choosing from these, none is exactly right, but option 1 is not correct.

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