IIM-K (BMSAT) 2025 (QA) PYQs

If $a_1, a_2, ..., a_8$ are the roots of the equation $x^8 + x^7 + ... + x + 1 = 0$, then the value of $a_1^{2025} + a_2^{2025} + ... + a_8^{2025}$ is

The zeroes of $ax² + bx + c = 0$ are in the ratio $1:2$ (assume the coefficients to be non-zero). Which of these is one of the zeroes?

What must be added to $x^4+2x^3-2x^2+x-1$ so that the result is exactly divisible by $x^2+2x-3$?