First slide

Trigonometric equations

Question

The A.M of all the solutions of $4 \cos^{3} x - 4 \cos^{2} x - \cos (π + x) + 1 = 0 i n (0, 315) i s$

Difficult

A

$49 π$
B

$50 π$
C

$51 π$
D

$10 π$

Solution

$4 \cos^{3} x - 4 \cos^{2} x - \cos (x + π) - 1 = 0$
$4 \cos^{3} x - 4 \cos^{2} x + \cos x - 1 = 0$
$4 \cos^{2} x (\cos x - 1) + 1 (\cos x - 1) = 0$
$\cos x = 1 h a s s o l u t i o n s a s x = 2 π, 4 π, 8 π, ....100 π i n (0, 315)$

$∴ A . M = \frac{s u m o f n u m b e r s}{n o . o f n u m b e r s} = \frac{2 π + 4 π + ...... + 100 π}{50} = 51 π$

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