Let S be the set of all complex numbers z satisfying | z 2 + z + 1 | = 1 . Then which of the following statements is/are TRUE? [2020]

Question

Let $S$ be the set of all complex numbers $z$ satisfying $| z^{2} + z + 1 |$ $= 1 .$ Then which of the following statements is/are TRUE? [2020]

Smart Achievers · Accepted Answer

(2, 3)

$| z^{2} + z + 1 | = 1$

$\Rightarrow$ $| {(z + \frac{1}{2})}^{2} + \frac{3}{4} |$ $\geq 1$ $\Rightarrow$ $| (z + \frac{1}{2}) |^{2}$ $\geq \frac{1}{4}$ $\Rightarrow$ $| z + \frac{1}{2} |$ $\geq \frac{1}{2}$

$also | (z^{2} + z) + 1 | = 1$

$\Rightarrow | z^{2} + z | - 1 \leq 1 \Rightarrow | z^{2} + z | \leq 2$

$\Rightarrow$ $| | z^{2} | - | z | | \leq | z^{2} + z | \leq 2$ $\Rightarrow | r^{2} - r | \leq 2 \Rightarrow r = | z | \leq 2; \forall z \in S$

$Hence, set S is infinite$

Solution

Q.
Let $S$ be the set of all complex numbers $z$ satisfying $| z^{2} + z + 1 |$ $= 1 .$ Then which of the following statements is/are TRUE [2020]

1 $| z + \frac{1}{2} |$ $\leq \frac{1}{2}$ for all $z \in S$

2 $| z |$ $\leq 2$ for all $z \in S$

3 $| z + \frac{1}{2} |$ $\geq \frac{1}{2}$ for all $z \in S$

4 The set $S$ has exactly four elements

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let S be the set of all complex numbers z satisfying |z2+z+1|=1. Then which of the following statements is/are TRUE [2020]

1 |z+12|≤12 for all z∈S

2 |z|≤2 for all z∈S

3 |z+12|≥12 for all z∈S

4 The set S has exactly four elements