First slide

Trigonometric equations

Question

The value of k if the equation 2 cos x + cos 2kx = 3 has only one solution is

Moderate

A

0
B

2
C

$\sqrt{2}$
D

$\frac{1}{2}$

Solution

$\begin{array}{l} 2 \cos x + \cos 2 kx = 3 \\ \Rightarrow 2 (1 - 2 \sin^{2} \frac{x}{2}) + 1 - 2 \sin^{2} kx = 3 \\ \Rightarrow 2 \sin^{2} \frac{x}{2} + \sin^{2} kx = 0 - - - (1) \end{array}$
Since $\sin^{2} \frac{x}{2}$ and $\sin^{2} kx$ are both non-negative, Eq.(1)
is possible only if $\sin^{2} \frac{x}{2} = 0$ and $\sin^{2} kx = 0$
$x = 0, \pm 2 π, \pm 4 π, \dots$
But for Eq. (1) to have unique solution, the possible value of k is irrational.
$∴ k = \sqrt{2}$ is the possible option.

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