cmi-entrance 2022 QA9

cmi-entrance · India · ugmath_22may 4 marks Complex Numbers Arithmetic True/False or Property Verification Statements

In this question $z$ denotes a non-real complex number, i.e., a number of the form $a + ib$ (with $a, b$ real) whose imaginary part $b$ is nonzero. Let $f(z) = z^{222} + \frac{1}{z^{222}}$.
Statements
(33) If $|z| = 1$, then $f(z)$ must be real. (34) If $z + \frac{1}{z} = 1$, then $f(z) = 2$. (35) If $z + \frac{1}{z}$ is real, then $|f(z)| \leq 2$. (36) If $f(z)$ is a real number, then $f(z)$ must be positive.

In this question $z$ denotes a non-real complex number, i.e., a number of the form $a + ib$ (with $a, b$ real) whose imaginary part $b$ is nonzero. Let $f(z) = z^{222} + \frac{1}{z^{222}}$.

Statements

(33) If $|z| = 1$, then $f(z)$ must be real.\\
(34) If $z + \frac{1}{z} = 1$, then $f(z) = 2$.\\
(35) If $z + \frac{1}{z}$ is real, then $|f(z)| \leq 2$.\\
(36) If $f(z)$ is a real number, then $f(z)$ must be positive.

Paper Questions

QA1 QA10 QA2 QA3 QA4 QA5 QA6 QA7 QA8 QA9 QB1 QB2 QB3 QB4 QB5 QB6