The number of solution (s) of the equation
`"sin" x + "cos" x = "min" {sqrt(3), a^(2) - 4a +5}, a in R`, is

Options

A. `1, "if" a lt 2`
B. `2, "if" a gt 2`
C. `0, "if" a = 2`
D. infinitely many for any `a in R`.

Answer

D

Explanation

We have,
`a^(2) - 4a + 5 = (a-2)^(2) + 1 ge 1`
`therefore " min" {a^(2) - 4a + 5, sqrt(3)} = [1, sqrt(3)]`
Thus, we have,
`"sin" x + "cos" x = lambda, "where "lambda in [1, sqrt(3)]`
Also, `-sqrt(2) le "sin" x + "cos" x le sqrt(2) " for all " x in R`
So, the given equation reduces to
`"sin" x + "cos" x = lambda, " where " lambda in [1, sqrt(2)]`
Clearly, it has infinitely many solutions.