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NDA 2021 Maths · Mathematics · Complex Numbers and Quadratic Equations · hard

If z=(√3+i)/(2), then z^101 equals

Single correct option

If z=3+i2z=\dfrac{\sqrt3+i}{2}, then z101z^{101} equals
  1. A. 3+i2\dfrac{\sqrt3+i}{2}
  2. B. −3+i2\dfrac{-\sqrt3+i}{2}
  3. C. −3−i2\dfrac{-\sqrt3-i}{2}
  4. D. 3−i2\dfrac{\sqrt3-i}{2}
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Answer: (B) −3+i2\dfrac{-\sqrt3+i}{2}

z=cos⁡30∘+isin⁡30∘z=\cos30^\circ+i\sin30^\circ, so z12=1z^{12}=1 by de Moivre. Since 101=12×8+5101=12\times8+5, z101=z5=cos⁡150∘+isin⁡150∘=−3+i2z^{101}=z^5=\cos150^\circ+i\sin150^\circ=\dfrac{-\sqrt3+i}{2}. Using the exponent 101 directly as an angle is the usual slip.
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Original question written in the NDA 2021 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