Question

which of the following statements is/are TRUE?

Answer 1

Answer:

B , C

Step-by-step explanation:

Solution is in refer to attachment .

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Answer 2

Let S be the set of all complex numbers z satisfying |z² + z + 1| = 1. then which of the following statements is/are TRUE ?

solution : |z² + z + 1| = 1

⇒|z² + 2(1/2).z + (1/2)² - (1/2)² + 1 | = 1

⇒|(z + 1/2)² + 3/4| = 1

we know, |z₁ + z₂| ≤ |z₁| + |z₂|

so, |(z + 1/2)² + 3/4| ≤ |(z + 1/2)|² + 3/4

⇒1 ≤ |(z + 1/2)|² + 3/4

⇒1/4 ≤ |z + 1/2|²

⇒|z + 1/2| ≥ 1/2

Therefore option (C) is correct for all z belongs to S

also we know, |z₁ + z₂| ≥ ||z₁| - |z₂||

so, |(z² + z) + 1| ≥ ||z² + z| - 1|

⇒1 ≥ |z² + z| - 1

⇒|z² + z| ≤ 2

||z²| - |z|| ≤ |z² + z| ≤ 2

⇒|z² - z| ≤ 2

⇒|z|² - |z| ≤ 2

⇒|z|² - |z| - 2 ≤ 0

⇒(|z| + 1)(|z| - 2) ≤ 0

⇒|z| ≤ 2 for all z belongs to S.

Therefore option (B) is also correct choice.

Therefore the correct options are (B) and (C) choices.