NEET Test Series from KOTA - 10 Papers In MS WORD
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Complex Numbers and Quadratic Equation
118007
If \(\alpha, \beta\) are the roots of \(x^2-2 x+4=0\), for \(n \in\) \(\mathrm{N}\), what is the value of \(\boldsymbol{\alpha}^{\mathrm{n}}+\boldsymbol{\beta}^{\mathrm{n}}=\)
B Given that, \(\alpha\) and \(\beta\) are the roots of the equation \(\mathrm{x}^2-2 \mathrm{x}+4=0\) \(\therefore \alpha+\beta=2\) \(\alpha \cdot \beta=4\) So, \((\alpha-\beta)=\sqrt{(\alpha+\beta)^2-4 \alpha \beta}\) \((\alpha-\beta)=\sqrt{(2)^2-4 \times 4}\) \(\alpha-\beta=\sqrt{-12}\) \((\alpha-\beta)=2 \sqrt{3} \mathrm{i}\) Adding equation (i) and (ii), we get - \(2 \alpha=2+2 \sqrt{3 \mathrm{i}}\) \(\alpha=2\left[\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}\right]\) \(\alpha=2\left[\cos \left(\frac{\pi}{3}\right)+\mathrm{i} \sin \left(\frac{\pi}{3}\right)\right]\) On putting the value of \(\alpha\) in equation (i) we get - \(\beta=\frac{1}{2}[2-2 \sqrt{3} \mathrm{i}]\) \(\beta=2\left[\frac{1}{2}-\frac{\sqrt{3}}{2} \mathrm{i}\right]\) \(=2\left[\cos \left(\frac{\pi}{3}\right)-\mathrm{i} \sin \left(\frac{\pi}{3}\right)\right]\) \(\alpha^1+\beta^1=2^1\left[2 \cos \left(\frac{1 \pi}{3}\right)\right]\) \(\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}=2^{\mathrm{n}}\left[2 \cos \left(\frac{\mathrm{n} \pi}{3}\right)\right]\) \(\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}=2^{\mathrm{n}+1} \cos \left(\frac{\mathrm{n} \pi}{3}\right)\)
AP EAMCET - 18.09.2020 Shift - II
Complex Numbers and Quadratic Equation
118008
The value of \(1+\sum_{\mathrm{k}=0}^{14}\left\{\cos \frac{(2 \mathrm{k}+1)}{15} \pi+\mathrm{i} \sin \frac{(2 \mathrm{k}+1)}{15} \pi\right\} \mathrm{i}\)
118007
If \(\alpha, \beta\) are the roots of \(x^2-2 x+4=0\), for \(n \in\) \(\mathrm{N}\), what is the value of \(\boldsymbol{\alpha}^{\mathrm{n}}+\boldsymbol{\beta}^{\mathrm{n}}=\)
B Given that, \(\alpha\) and \(\beta\) are the roots of the equation \(\mathrm{x}^2-2 \mathrm{x}+4=0\) \(\therefore \alpha+\beta=2\) \(\alpha \cdot \beta=4\) So, \((\alpha-\beta)=\sqrt{(\alpha+\beta)^2-4 \alpha \beta}\) \((\alpha-\beta)=\sqrt{(2)^2-4 \times 4}\) \(\alpha-\beta=\sqrt{-12}\) \((\alpha-\beta)=2 \sqrt{3} \mathrm{i}\) Adding equation (i) and (ii), we get - \(2 \alpha=2+2 \sqrt{3 \mathrm{i}}\) \(\alpha=2\left[\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}\right]\) \(\alpha=2\left[\cos \left(\frac{\pi}{3}\right)+\mathrm{i} \sin \left(\frac{\pi}{3}\right)\right]\) On putting the value of \(\alpha\) in equation (i) we get - \(\beta=\frac{1}{2}[2-2 \sqrt{3} \mathrm{i}]\) \(\beta=2\left[\frac{1}{2}-\frac{\sqrt{3}}{2} \mathrm{i}\right]\) \(=2\left[\cos \left(\frac{\pi}{3}\right)-\mathrm{i} \sin \left(\frac{\pi}{3}\right)\right]\) \(\alpha^1+\beta^1=2^1\left[2 \cos \left(\frac{1 \pi}{3}\right)\right]\) \(\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}=2^{\mathrm{n}}\left[2 \cos \left(\frac{\mathrm{n} \pi}{3}\right)\right]\) \(\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}=2^{\mathrm{n}+1} \cos \left(\frac{\mathrm{n} \pi}{3}\right)\)
AP EAMCET - 18.09.2020 Shift - II
Complex Numbers and Quadratic Equation
118008
The value of \(1+\sum_{\mathrm{k}=0}^{14}\left\{\cos \frac{(2 \mathrm{k}+1)}{15} \pi+\mathrm{i} \sin \frac{(2 \mathrm{k}+1)}{15} \pi\right\} \mathrm{i}\)
118007
If \(\alpha, \beta\) are the roots of \(x^2-2 x+4=0\), for \(n \in\) \(\mathrm{N}\), what is the value of \(\boldsymbol{\alpha}^{\mathrm{n}}+\boldsymbol{\beta}^{\mathrm{n}}=\)
B Given that, \(\alpha\) and \(\beta\) are the roots of the equation \(\mathrm{x}^2-2 \mathrm{x}+4=0\) \(\therefore \alpha+\beta=2\) \(\alpha \cdot \beta=4\) So, \((\alpha-\beta)=\sqrt{(\alpha+\beta)^2-4 \alpha \beta}\) \((\alpha-\beta)=\sqrt{(2)^2-4 \times 4}\) \(\alpha-\beta=\sqrt{-12}\) \((\alpha-\beta)=2 \sqrt{3} \mathrm{i}\) Adding equation (i) and (ii), we get - \(2 \alpha=2+2 \sqrt{3 \mathrm{i}}\) \(\alpha=2\left[\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}\right]\) \(\alpha=2\left[\cos \left(\frac{\pi}{3}\right)+\mathrm{i} \sin \left(\frac{\pi}{3}\right)\right]\) On putting the value of \(\alpha\) in equation (i) we get - \(\beta=\frac{1}{2}[2-2 \sqrt{3} \mathrm{i}]\) \(\beta=2\left[\frac{1}{2}-\frac{\sqrt{3}}{2} \mathrm{i}\right]\) \(=2\left[\cos \left(\frac{\pi}{3}\right)-\mathrm{i} \sin \left(\frac{\pi}{3}\right)\right]\) \(\alpha^1+\beta^1=2^1\left[2 \cos \left(\frac{1 \pi}{3}\right)\right]\) \(\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}=2^{\mathrm{n}}\left[2 \cos \left(\frac{\mathrm{n} \pi}{3}\right)\right]\) \(\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}=2^{\mathrm{n}+1} \cos \left(\frac{\mathrm{n} \pi}{3}\right)\)
AP EAMCET - 18.09.2020 Shift - II
Complex Numbers and Quadratic Equation
118008
The value of \(1+\sum_{\mathrm{k}=0}^{14}\left\{\cos \frac{(2 \mathrm{k}+1)}{15} \pi+\mathrm{i} \sin \frac{(2 \mathrm{k}+1)}{15} \pi\right\} \mathrm{i}\)
NEET Test Series from KOTA - 10 Papers In MS WORD
WhatsApp Here
Complex Numbers and Quadratic Equation
118007
If \(\alpha, \beta\) are the roots of \(x^2-2 x+4=0\), for \(n \in\) \(\mathrm{N}\), what is the value of \(\boldsymbol{\alpha}^{\mathrm{n}}+\boldsymbol{\beta}^{\mathrm{n}}=\)
B Given that, \(\alpha\) and \(\beta\) are the roots of the equation \(\mathrm{x}^2-2 \mathrm{x}+4=0\) \(\therefore \alpha+\beta=2\) \(\alpha \cdot \beta=4\) So, \((\alpha-\beta)=\sqrt{(\alpha+\beta)^2-4 \alpha \beta}\) \((\alpha-\beta)=\sqrt{(2)^2-4 \times 4}\) \(\alpha-\beta=\sqrt{-12}\) \((\alpha-\beta)=2 \sqrt{3} \mathrm{i}\) Adding equation (i) and (ii), we get - \(2 \alpha=2+2 \sqrt{3 \mathrm{i}}\) \(\alpha=2\left[\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}\right]\) \(\alpha=2\left[\cos \left(\frac{\pi}{3}\right)+\mathrm{i} \sin \left(\frac{\pi}{3}\right)\right]\) On putting the value of \(\alpha\) in equation (i) we get - \(\beta=\frac{1}{2}[2-2 \sqrt{3} \mathrm{i}]\) \(\beta=2\left[\frac{1}{2}-\frac{\sqrt{3}}{2} \mathrm{i}\right]\) \(=2\left[\cos \left(\frac{\pi}{3}\right)-\mathrm{i} \sin \left(\frac{\pi}{3}\right)\right]\) \(\alpha^1+\beta^1=2^1\left[2 \cos \left(\frac{1 \pi}{3}\right)\right]\) \(\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}=2^{\mathrm{n}}\left[2 \cos \left(\frac{\mathrm{n} \pi}{3}\right)\right]\) \(\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}=2^{\mathrm{n}+1} \cos \left(\frac{\mathrm{n} \pi}{3}\right)\)
AP EAMCET - 18.09.2020 Shift - II
Complex Numbers and Quadratic Equation
118008
The value of \(1+\sum_{\mathrm{k}=0}^{14}\left\{\cos \frac{(2 \mathrm{k}+1)}{15} \pi+\mathrm{i} \sin \frac{(2 \mathrm{k}+1)}{15} \pi\right\} \mathrm{i}\)