11th Class Mathematics Complex Numbers and Quadratic Equations Question Bank Critical Thinking

  • question_answer Let \[{{z}_{1}}\] and \[{{z}_{2}}\] be nth roots of unity which are ends of a line segment that subtend a right angle at the origin.  Then n must be of the form  [IIT Screening 2001; Karnataka 2002]

    A) 4k + 1

    B) 4k + 2

    C) 4k + 3

    D) 4k

    Correct Answer: D

    Solution :

    \[{{1}^{1/n}}=\cos \frac{2r\pi }{n}+i\sin \frac{2r\pi }{n}\] Let\[{{z}_{1}}=\cos \frac{2{{r}_{1}}\pi }{n}+i\sin \frac{2{{r}_{1}}\pi }{n}\] and \[{{z}_{2}}=\cos \frac{2{{r}_{2}}\pi }{n}+i\sin \frac{2{{r}_{2}}\pi }{n}\]. Then \[\angle \,{{Z}_{1}}O{{Z}_{2}}=amp\,\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)=amp\,({{z}_{1}})-amp\,({{z}_{2}})\] \[=\frac{2({{r}_{1}}-{{r}_{2}})\pi }{n}=\frac{\pi }{2}\] (Given) \\[n=4({{r}_{1}}-{{r}_{2}})\]=4 × integer, so n is of the form 4 k.

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