The number of distinct real roots of `|(sinx, cosx, cosx),(cos x,sin x,cos x),(cos x,cos x,sin x)|=0` in the interval `-(pi)/4 le x le t (pi)/4` is

Options

A. 4
B. 1
C. 2
D. 3

Answer

C

Explanation

We have,
`|{:("cos" x, "sin" x, "sin" x), ("sin" x, "cos" x, "sin" x), ("sin" x, "sin" x,"cos"x):}|=0`
Applying `R_(1) to R_(1) + R_(2) + R_(3)`, we obtain
`|{:("cos" x + "sin" x,"cos"x +2"sin" x, "cos"x +2"sin" x), (" sin" x, "cos" x, "sin" x), (" sin" x, "sin" x,"cos"x):}|=0`
`rArr ("cos" x+ 2"sin" x ) |{:(1,1, 1), ("sin" x, "cos" x, "sin" x), ("sin" x, "sin" x,"cos"x):}|=0`
Applying `C_(2) to C_(2) -C_(1), C_(3) to C_(3)-C_(1)`, we obtain
`rArr ("cos" x + 2"sin"x)|{:(1," "0, " "0), ("sin" x, "cos" x-"sin"x, " "0), ("sin" x, " "0,"cos"x-"sin"x):}|=0`
`rArr ("cos" x + 2 "sin"x) ("cos" x -"sin"x)^(2) = 0`
`rArr "cos" x + 2"sin" x = 0 "or, cos"x = "sin" x`
`rArr "tan"x = (1)/(2) "or, tan"x = 1`
`rArr x = "tan"^(-1)(-(1)/(2)) "or", x = (pi)/(4)`
So, there are two solutions in `[-(pi)/(4), (pi)/(4)]`