यदि a_{r}=cos frac{2 r pi}{9}+i sin frac{2 r | vedclass

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Question

$\mathrm{a}_{\mathrm{r}}=\mathrm{e}^{\frac{\mathrm{i} 2 \pi \mathrm{r}}{9}}, \mathrm{r}=1,2,3, \ldots \mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3},

Vedclass · Accepted Answer

$a_{1} a_{9}-a_{3} a_{7}$

Vedclass · Answer

$\mathrm{a}_{\mathrm{r}}=\mathrm{e}^{\frac{\mathrm{i} 2 \pi \mathrm{r}}{9}}, \mathrm{r}=1,2,3, \ldots \mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots$ are in $G.P.$

$\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \ a_{n} & a_{5} & a_{6} \ a_{7} & a_{8} & a_{9}\end{array}ight|=\left|\begin{array}{lll}a_{1} & a_{2}^{2} & a_{1}^{3} \ a_{1}^{4} & a_{1}^{5} & a_{1}^{6} \ a_{1}^{7} & a_{1}^{8} & a_{1}^{9}\end{array}ight|=a_{1} \cdot a_{1}^{4} \cdot a_{1}^{7}\left|\begin{array}{lll}1 & a_{1} & a_{1}^{2} \ 1 & a_{1} & a_{1}^{2} \ 1 & a_{1} & a_{1}^{2}\end{array}ight|=0$

Now $a_{1} a_{9}-a_{3} a_{7}=a_{1}^{10}-a_{1}^{10}=0$

यदि $a_{r}=\cos \frac{2 r \pi}{9}+i \sin \frac{2 r \pi}{9}, \quad r=1,2,3, \ldots$, $i=\sqrt{-1}$, तो सारणिक $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ बराबर है

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