cmi-entrance 2019 QB2

cmi-entrance · India · ugmath 10 marks Complex numbers 2 Roots of Unity and Cyclotomic Properties
(a) Count the number of roots $w$ of the equation $z^{2019} - 1 = 0$ over complex numbers that satisfy $|w + 1| \geq \sqrt{2 + \sqrt{2}}$.
(b) Find all real numbers $x$ that satisfy the following equation: $$\frac{8^{x} + 27^{x}}{12^{x} + 18^{x}} = \frac{7}{6}$$ 
(a) Count the number of roots $w$ of the equation $z^{2019} - 1 = 0$ over complex numbers that satisfy $|w + 1| \geq \sqrt{2 + \sqrt{2}}$.

(b) Find all real numbers $x$ that satisfy the following equation:
$$\frac{8^{x} + 27^{x}}{12^{x} + 18^{x}} = \frac{7}{6}$$
Paper Questions
QA1 QA10 QA2 QA3 QA4 QA5 QA6 QA7 QA8 QA9 QB1 QB2 QB3 QB4 QB5 QB6