List of top Mathematics Questions on sequences asked in JEE Advanced

Let $a, b, c$ be positive integers in arithmetic progression such that the equation \[ ax^2 + bx + c = 0 \] has only integer solutions. Then which of the following statements is (are) TRUE?

Let \[ \alpha= \left(1-2\cos\frac{\pi}{11}\right) \left(1-2\cos\frac{3\pi}{11}\right) \left(1-2\cos\frac{9\pi}{11}\right) \left(1-2\cos\frac{27\pi}{11}\right) \left(1-2\cos\frac{81\pi}{11}\right) \] Then the value of \[ 5-\alpha^2 \] is ________.