Find the truth 7

Algebra Level 5

Read the following statements.

I) It is known that $x$ is a member of the set of complex numbers ( which includes all real numbers also ), then $|x|$ is always a positive real.

II) If $f\left( g\left( x \right) \right) =x$ for all $x$ , then $g\left( f\left( x \right) \right) =x$ for all $x$ .

Which of the above statements are true?

This is a part of set Find the Truth

Only II and III None All Only I and III Only I Only I and II Only II Only III

1 solution

Archit Boobna
May 11, 2015

I) Contradicting Example- If $x=0$ , then $|x|$ is also equal to zero, which is not positive. Hence, I is false .

II) Contradicting Example- It can happen that this is not true, let $g\left( x \right)=\sqrt { x }$ and $f\left( x \right)={ x }^{ 2 }$ . Hence, II is false .

III) When we plot $z$ , $z+1$ and $z-1$ on the complex plot, we get 3 collinear points as their imaginary part is same.

Let's assume their absolute values are equal, they are equidistant from the origin. This means that we can make a circle centered at the origin, with the points $z$ , $z+1$ and $z-1$ lying on it.

By simple geometry we can show that 3 collinear points can't lie on the same circle. So are assumption was false and their absolute values can't be the same. Hence, III is true

Nice Question ! But way too overrated.

Rajdeep Dhingra - 6 years, 1 month ago

If you liked the question, you probably liked the solution too. So it deserves an upvote, right?

Lol, thanks, by the way!

Archit Boobna - 6 years, 1 month ago

Here is an upvote!nice solution,genius

Kaustubh Miglani - 5 years, 4 months ago

Find the truth 7

1 solution

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