Question

The possible value of ‘k’ if the equation $2cosx+cos2kx=3$ has only one solution

• A. $0$
• B. $2$
• C. $\sqrt{2}$
• D. $\frac{1}{2}$

Answer 1

The correct option is C

$\sqrt{2}$

$2cosx+cos2kx=3$

$2(1-2si{n}^2\frac{x}{2})+1-2si{n}^{2}kx=3$

$2si{n}^2\frac{x}{2}+si{n}^{2}kx=$ ...(1)

since $si{n}^2\frac{x}{2}$ and $si{n}^{2}kx$ are both positive

(1) is possible only if $si{n}^2\frac{x}{2}=0$ and $si{n}^{2}kx=0$

$\therefore x=0,\pm 2\pi ,\pm 4\pi \cdots$

But for (1) to have unique solution.

The possible value of k is irrational

$\therefore k=\sqrt{2}$ is the possible option.