Question

The number of solutions of $\sum _{r=1}^5\cos rx=5$ in the interval of $[0,2\pi ]$ is

• A. $0$
• B. $2$
• C. $5$
• D. $10$

Answer 1

Answer: (B)

$2$

$\cos x+\cos 2x+\cos 3x+\cos 4x+\cos 5x=5$
we know, $\cos x,\cos 2x,\cos 3x,\cos 4x,\cos 5x\le 1$
considering both,
$\cos x,\cos 2x,\cos 3x,\cos 4x,\cos 5x=1$
hence general solutions in the given interval are,
$\cos x=1\Rightarrow x=0,2\pi$
$\cos 2x=1\Rightarrow 2x=0,2\pi ,4\pi \Rightarrow x=0,\pi ,2\pi$
$\cos 3x=1\Rightarrow 3x=0,2\pi ,4\pi ,6\pi \Rightarrow x=0,\frac{2\pi }{3},2\pi$
$\cos 4x=1\Rightarrow 4x=0,2\pi ,4\pi ,6\pi ,8\pi \Rightarrow x=0,\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi$
$\cos 5x=1\Rightarrow 5x=0,2\pi ,4\pi ,6\pi ,8\pi ,10\pi \Rightarrow x=0,\frac{2\pi }{5},\frac{4\pi }{5},\frac{6\pi }{5},\frac{8\pi }{5},2\pi$
thus common solutions are,
$x=0,2\pi$
Hence, option 'B' is correct.