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If $\Delta_{r}=\left|\begin{array}{lll}2 r-1 & { }^{m} C_{r} & 1 \\ m^{2}-1 & 2^{m} & m+1 \\ \sin ^{2}\left(\mathrm{~m}^{2}\right) & \sin ^{2}(\mathrm{~m}) & \sin ^{2}(\mathrm{~m}+1)\end{array}\right|$, then the value of $\sum_{r=0}^{m} \Delta_{r}, i s$
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$\Delta_{\mathrm{r}}=\left|\begin{array}{ccc}2 \mathrm{r}-1 & \mathrm{~m}_{\mathrm{r}} & 1 \\ \mathrm{~m}^{2}-1 & 2^{\mathrm{m}} & \mathrm{m}+1 \\ \sin ^{2}\left(\mathrm{~m}^{2}\right) & \sin ^{2}(\mathrm{~m}) & \sin ^{2}(\mathrm{~m}+1)\end{array}\right|$
$\begin{aligned} \therefore \sum_{\mathrm{r}=0}^{\mathrm{m}} \Delta_{\mathrm{r}} &=\left|\begin{array}{ccc}\sum_{\mathrm{r}=0}^{\mathrm{m}}(2 \mathrm{r}-1) & \sum_{\mathrm{r}=0}^{\mathrm{m}} \mathrm{m}_{\mathrm{C}} \mathrm{C}_{\mathrm{r}} & \sum_{\mathrm{r}=0}^{\mathrm{m}} 1 \\ \mathrm{~m}^{2}-1 & 2^{\mathrm{m}} & \mathrm{m}+1 \\ \sin ^{2}\left(\mathrm{~m}^{2}\right) & \sin ^{2}(\mathrm{~m}) & \sin ^{2}(\mathrm{~m}+1)\end{array}\right| \\ &=\left|\begin{array}{ccc}\mathrm{m}^{2}-1 & 2^{\mathrm{m}} & \mathrm{m}+1 \\ \mathrm{~m}^{2}-1 & 2^{\mathrm{m}} & \mathrm{m}+1 \\ \sin ^{2}\left(\mathrm{~m}^{2}\right) & \sin ^{2}(\mathrm{~m}) & \sin ^{2}(\mathrm{~m}+1)\end{array}\right| \\ &=0(\because \mathrm{tworows} \text { are identical }) \end{aligned}$
$\begin{aligned} \therefore \sum_{\mathrm{r}=0}^{\mathrm{m}} \Delta_{\mathrm{r}} &=\left|\begin{array}{ccc}\sum_{\mathrm{r}=0}^{\mathrm{m}}(2 \mathrm{r}-1) & \sum_{\mathrm{r}=0}^{\mathrm{m}} \mathrm{m}_{\mathrm{C}} \mathrm{C}_{\mathrm{r}} & \sum_{\mathrm{r}=0}^{\mathrm{m}} 1 \\ \mathrm{~m}^{2}-1 & 2^{\mathrm{m}} & \mathrm{m}+1 \\ \sin ^{2}\left(\mathrm{~m}^{2}\right) & \sin ^{2}(\mathrm{~m}) & \sin ^{2}(\mathrm{~m}+1)\end{array}\right| \\ &=\left|\begin{array}{ccc}\mathrm{m}^{2}-1 & 2^{\mathrm{m}} & \mathrm{m}+1 \\ \mathrm{~m}^{2}-1 & 2^{\mathrm{m}} & \mathrm{m}+1 \\ \sin ^{2}\left(\mathrm{~m}^{2}\right) & \sin ^{2}(\mathrm{~m}) & \sin ^{2}(\mathrm{~m}+1)\end{array}\right| \\ &=0(\because \mathrm{tworows} \text { are identical }) \end{aligned}$
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