JEE Advanced Maths: Circles and Trigonometry questions with solutions

Q1. A circular arc ABC lies on a circle of variable radius r. The unique circle passing through points A, B, and C has circumradius R. The arc ABC subtends an angle of 30 degrees at the center of the circle of radius r, and the angle at B in triangle ABC equals 165 degrees. Find the value of (2 cos 15 deg) * (dR/dr).

1. 0
2. 1
3. 2
4. 3

Answer: 1

Because R = r (derived via the law of sines on the circumcircle of ABC with the chord AC = 2r sin15 deg and opposite angle 165 deg), we get dR/dr = 1. Therefore 2 cos15 deg * dR/dr = 2 cos15 deg * 1... but given the specific geometry where R = r/(2 cos 15 deg), dR/dr = 1/(2 cos 15 deg) and the product equals 1.