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4-2.Quadratic Equations and Inequations
normal
જો $\alpha , \beta $ એ સમીકરણ $x^2 - 2x + 4 = 0$ ના બીજો હોય તો $\alpha ^n +\beta ^n$ ની કિમત મેળવો
A
${2^n}\cos \left( {\frac{{n\pi }}{3}} \right)$
B
${2^{n + 1}}\cos \left( {\frac{{n\pi }}{3}} \right)$
C
${2^n}\sin \left( {\frac{{n\pi }}{3}} \right)$
D
${2^{n + 1}}\sin \left( {\frac{{n\pi }}{3}} \right)$
Solution
$\alpha+\beta=2$
$\alpha . \beta=4$
by solving,
$\alpha=2[\cos \pi / 3+i \sin \pi / 3]$
$\beta=2[\cos \pi / 3-i \sin \pi / 3]$
$\mathrm{SO},$
$2^{n}\left[\cos \frac{n \pi}{3}+i \sin \frac{n \pi}{3}\right]+2^{n}\left[\cos \frac{n \pi}{3}-i \sin \frac{n \pi}{3}\right]$
$=2^{n}\left[\cos \frac{n \pi}{3}\right]$
$=2^{n+1} \cdot \cos \frac{n \pi}{3}$
Standard 11
Mathematics
