A) \[\cos \alpha +i\sin \alpha \]
B) \[\cos (\alpha /2)-i\,\,sin(\alpha /2)\]
C) \[{{e}^{i\alpha /2}}\]
D) \[\sqrt[3]{{{e}^{i\alpha }}}\]
Correct Answer: C
Solution :
[c] \[{{z}_{r}}=\cos \frac{ra}{{{n}^{2}}}+i\sin \frac{ra}{{{n}^{2}}};\] \[{{z}_{1}}=\cos \frac{\alpha }{{{n}^{2}}}+i\sin \frac{\alpha }{{{n}^{2}}};\] \[{{z}_{2}}=\cos \frac{2\alpha }{{{n}^{2}}}+i\sin \frac{2\alpha }{{{n}^{2}}};...,\] \[{{z}_{n}}=\cos \frac{n\alpha }{{{n}^{2}}}+i\sin \frac{n\alpha }{{{n}^{2}}}\] Consider \[\underset{n\to \infty }{\mathop{\lim }}\,({{z}_{1}}{{z}_{2}}{{z}_{3}}...{{z}_{n}})\] \[=\underset{n\to \infty }{\mathop{\lim }}\,\left[ \cos \left\{ \frac{\alpha }{{{n}^{2}}}(1+2+3+...+n) \right\}+i\sin \left\{ \frac{\alpha }{{{n}^{2}}}(1+2+3+...+n) \right\} \right]\]\[=\underset{n\to \infty }{\mathop{\lim }}\,\left[ \cos \frac{an(n+1)}{2{{n}^{2}}}+i\sin \frac{an(n+1)}{2{{n}^{2}}} \right]\] \[=\underset{x\to \infty }{\mathop{\lim }}\,\frac{\cos a\left( 1+1/n \right)}{2}+\frac{i\sin a\left( 1+1/n \right)}{2}\] \[=\cos \frac{\alpha }{2}+i\sin \frac{\alpha }{2}={{e}^{\frac{ia}{2}}}\]
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