Question: What is the area of the circle C? Statement I: An arc of length 7 cm subtends an angle 30° at the centre of C. Statement II: A chord of length 10 cm subtends an angle 90° at the centre of C.
The Question can be answered by one of the Statements alone but not by the other.
The Question can be answered by either Statement alone.
The Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
The Question cannot be answered even by using both Statements together.
The area of a circle is pi * r^2. Statement I gives arc length L = r * theta (in radians). With L = 7 and theta = 30 degrees (pi/6 radians), r = 7 / (pi/6) = 42/pi. This allows calculating the area. Statement II gives a chord length c = 2 * r * sin(theta/2). With c = 10 and theta = 90 degrees, 10 = 2 * r * sin(45), so 10 = 2 * r * (sqrt(2)/2), which means r = 10/sqrt(2). This also allows calculating the area. Since both statements independently provide enough information to find the radius, the answer is B.
Using Statement I, apply the arc length formula l = rθ to get 7 = r × (30π/180), which solves to r = 42/π and yields the circle's area. Using Statement II, the chord length formula c = 2r sin(θ/2) gives 10 = 2r sin(45°), so r = 5√2 and the area can be found. Since the radius and area can be determined independently from either piece of information, the data is sufficient in both statements alone. The question can be answered by either Statement alone.