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. 0
B. 2
C. 1
D. 3

Answer

C

Explanation

We have, `|(sin x,cos x,cos x),(cos x,sin x,cos x),(cos x,cos x,sin x)| = 0`
`rArr |(sin x + 2 cos x,sin x + 2 cos x,sin x + 2 cos x),(cos x,sin x,cos x),(cos x,cos x,sin x)| =0 " " ["Applying " R_(1) rarr R_(1) + R_(2) + R_(3)]`
`rArr (sinx + 2 cos x) |(1,1,1),(cos x,sin x,cos x),(cos x,cos x,sin x)| = 0`
`rArr (sinx + 2 cos x) |(1,0,0),(cos x,sin x - cos x,0),(cos x,0,sin x - cos x)| = 0 " " ["Applying " C_(3) rarr C_(3) - C_(1)"," C_(2) rarr C_(2) - C_(1)]`
`rArr (sin x + 2 cos x) (sin x - cos x)^(2) = 0`
`rArr tan x = -2 or, tan x = 1`
`rArr x = (pi)/(4) " " [ :. x in [pi//4, pi//4] rArr -1 le tan x le 1]`