| Direction: If \[\cos \,\frac{\pi }{7},\,\,\cos \,\frac{3\pi }{7}\]and \[\cos \,\frac{5\pi }{7}\] are the roots of the equations \[8{{x}^{3}}-4{{x}^{2}}-4x+1=0\] then, |
A) 2
B) 4
C) 8
D) None of these
Correct Answer: B
Solution :
Given, \[\cos \,\frac{\pi }{7},\,\,\cos \,\frac{3\pi }{7},\,\,\cos \,\frac{5\pi }{7}\] are the roots of the equation \[8{{x}^{3}}-4{{x}^{2}}-4x+1=0\] ?(i) Replacing \[x\] by \[\frac{1}{x}\] in Eq. (i), we get \[{{x}^{3}}-4{{x}^{2}}-4x+8=0\] ?(ii) \[\Rightarrow \,\sec d\,\frac{\pi }{7},\,\sec \,\frac{3\pi }{7},\,\sec \,\frac{5\pi }{7}\] are the roots of Eq. (ii). \[\therefore \] \[\sec \frac{\pi }{7}+\sec \,\frac{3\pi }{7}+\sec \,\frac{5\pi }{7}=4\]
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