The value of `k` if the equation `2cosx+cos2k x=3` has only one solution is 2 (b) 2 (c) `sqrt(2)` (d) `1/2`

Options

A. 0
B. 2
C. `sqrt(2)`
D. `1//2`

Answer

C

Explanation

`2 cos x + cos 2 kx=3`
`rArr 2(1-2 "sin"^(2) x/2)+1-2 sin^(2) kx=3`
`rArr 2 "sin"^(2) x/2+sin^(2) kx=0` ...(1)
Since `"sin"^(2) x/2` and `sin^(2) kx` are both non-negative, Eq. (1) is possible only if `sin^(2) x/2=0` and `sin^(2) kx=0`.
`x=0, pm 2pi, pm 4 pi`, ...
But for Eq. (1) to have unique solution, the possible value of k is irrational.
`:. k=sqrt(2)` is the possible option.