If the equation ` 2 cos x + cos 2 lambda x=3 ` has only one solution , then ` lambda ` is

Options

A. equal to 1
B. a rational number
C. an irrational number
D. none of these

Answer

C

Explanation

We have,
`2"cos" x + "cos" 2ax =3`
`rArr 2 "cos" x = 2 "and cos" 2ax = 1`
`rArr "cos" x = 1 " and cos" 2ax = 1`
Now, `"cos" x = 1 rArr x = 2n pi, pi in Z`
Infinitely many of these values of x also satisfy the equation `"cos" 2ax = 1 "if" a in Q`. Thus, there are infinitely many solutions if `a in Q`.
If`a notin Q`, then x = 0 is the only value satisfying the two equations.
Hence, a is an irrational number.