Question

If \[lo{g_{\sqrt 3 }}\left( {\dfrac{{{{\left| z \right|}^2} - \left| z \right| + 1}}{{2 + |z|}}} \right) < 2\], then the locus of z is
A) \[\left| z \right| = 5\]
B) \[\left| z \right| < 5\]
C) \[\left| z \right| > 5\]
D) None of these

Answer 1

Hint: in this question we have to find the locus of complex number z. Just simplify the given equation and use the factorization method to find the roots of the equation. Then use select the root which satisfy the condition of mod function



Formula Used:Factorization is used to find root of complex number
\[(a + b)(c + d) = ac + ad + bc + bd\]
Where
A,b,c,d are any numbers or variable



Complete step by step solution:Given: A complex number equation
Now we have complex number \[lo{g_{\sqrt 3 }}\left( {\dfrac{{{{\left| z \right|}^2} - \left| z \right| + 1}}{{2 + |z|}}} \right) < 2\]
\[\left( {\dfrac{{{{\left| z \right|}^2} - \left| z \right| + 1}}{{2 + |z|}}} \right) < {(\sqrt 3 )^2}\]
\[|z{|^2} - 4|z| - 5 < 0\]
\[|z{|^2} - 5|z| + |z| - 5 < 0\]
\[|z|(|z| - 5) + (|z| - 5) < 0\]
\[(|z| - 5)(|z| + 1) < 0\]
\[ - 1 < |z| < 5\]
\[|z| < 5\] (Because\[|z| > 0\])



Option ‘B’ is correct



Note: Complex number is a number which is a combination of real and imaginary numbers. So in complex number questions, we have to represent the number as a combination of real and its imaginary part. Imaginary part is known as iota. Square of iota is equal to the negative one.
Don’t try to put the value of z in a given equation first try to simplify it after that value of z would be put in the equation.