# JEE Advanced 2016 (6)

**Algebra**Level 5

Say a real number \(r\) is repetitive if there exist two distinct complex numbers \(z_{1},z_{2}\) with \(|z_{1}| = |z_{2}| = 1\) and \(z_{1},z_{2}\) is not equal to \({−i,i}\) such that \[z_{1}(z_{1}^{3} + z_{1}^{2} + rz_{1} + 1) = z_{2}(z_{2}^{3} + z_{2}^{2} + rz_{2} +1)\] There exist real numbers \(a,b\) such that a real number \(r\) is repetitive if and only if \(a < r ≤ b\). If the value of \(|a| + |b|\) can be expressed in the form \(\frac{p}{q}\) for relatively prime positive integers \(p,q\). Find \(100p +q\).

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*Source: OMO Spring 2016*