First slide

Trigonometric equations

Question

The sum of the roots of the equation $4 \cos^{3} x - 4 \cos^{2} x - \cos (π + x) - 1 = 0 in the in terval [0, 315] is p π$ $where p is equal to$

Moderate

A

2500
B

2550
C

2600
D

2651

Solution

$\begin{array}{l} 4 \cos^{3} x - 4 \cos^{2} x + \cos x - 1 = 0 \\ \Rightarrow (4 \cos^{2} x + 1) (\cos x - 1) = 0 \\ \Rightarrow \cos x = 1 \Rightarrow x = 2 n π \\ 100 π < 315 < 101 π \\ Required sum = 2 π + 4 π + \dots 100 π \\ = 2 (1 + 2 + \dots 50) π \\ = \frac{2 \times 50 \times 51}{2} π = 2550 π \end{array}$

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