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NDA 2020 Maths · Mathematics · Complex Numbers and Quadratic Equations · hard

If z is a complex number satisfying |z-4/z|=2, then the greatest possible value of |z| is

Single correct option

If zz is a complex number satisfying ∣z−4z∣=2\left|z-\dfrac4z\right|=2, then the greatest possible value of ∣z∣|z| is
  1. A. 5−1\sqrt5-1
  2. B. 5+1\sqrt5+1
  3. C. 5+2\sqrt5+2
  4. D. 33
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Answer: (B) 5+1\sqrt5+1

Using ∣a−b∣≥∣∣a∣−∣b∣∣|a-b|\ge\big||a|-|b|\big|: 2=∣z−4z∣≥∣z∣−4∣z∣2=\left|z-\frac4z\right|\ge |z|-\frac4{|z|}. With r=∣z∣r=|z| this gives r2−2r−4≤0r^2-2r-4\le0, so r≤1+5r\le1+\sqrt5. Equality is attained for z=rz=r real positive with r−4/r=2r-4/r=2. The smaller root 1−51-\sqrt5 is negative and is rejected.
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Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake