question_answer

If \[\left| \,z-4+3i\, \right|\le 1\,\,and\,\,\alpha \,\,and\,\,\beta \] and a and p be the least and greatest values of \[\left| \,z\, \right|\] and \[k\] be the least value of \[\frac{{{x}^{4}}+{{x}^{2}}+4}{x}\] on the x interval \[(0,\infty )\] then k is equal to-

A) \[\alpha \]

B) \[\beta \]

C) \[\alpha +\beta \]

D) None of these

Correct Answer: B

Solution :

Given,

\[\Rightarrow \left| \,z\, \right|\le 6\]and \[\left| \,z\, \right|\ge 4\]\[\Rightarrow 4\le \left| \,z\, \right|\le 6\]\[\Rightarrow \alpha =4,\beta =6\]

Let \[y=\frac{{{x}^{4}}+{{x}^{2}}+4}{x}={{x}^{3}}+x+\frac{4}{x}\]\[={{x}^{3}}+x+\frac{1}{x}+\frac{1}{x}+\frac{1}{x}+\frac{1}{x}\]

Since \[x\in (0,\infty ),\] therefore \[{{x}^{3}},x,\frac{1}{x}\] are positive.

Sum will be least when \[{{x}^{3}}=x=\frac{1}{x}\]\[\Rightarrow x=1\]

\[\therefore k=6\]

Hence, \[k=\beta \]