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 (pi)/4` is

Options

A. 0
B. 2
C. 1
D. 3

Answer

C

Explanation

We have
`|(sinx,cosx,cosx),(cosx,sinx,cosx),(cosx,cosx,sinx)|=0`
Applying `C_(1)toC_(1)+C_(2)+C_(3)`
`|(2cosx+sinx, cosx,cosx),(2cosx+sinx,sinx,cosx),(2cosx+sinx,cosx,sinx)|=0`
On taking `(2cosx+sinx)` common from `C_(1)` we get
`implies (2cosx+sinx)|(1,cosx,cosx),(1,sinx,cosx),(1,cosx,sinx)|=0`
`implies (2cosx+sinx)|(1,cosx,cosx),(0,sinx-cosx,0),(0,0,(sinx-cosx))|=0`
`[ :' R_(2)toR_(2)-R_(1)` and `R_(3)toR_(3)-R_(1)]`
Expanding along `C_(1)`
`(2cosxsinx)[1.(sinx-cosx)^(2)]=0`
`implies (2cosx+sinx)(sinx-cosx^(2))=0`
Either `2cosx=-sinx`
`implies cosx=-1/2sinx`
`implies tanx=-2`..............(i)
But here for `-(pi)/4lexle(pi)/4` we get `-1letanxle1` so no solution possible
and for `(sinx-cosx)^(2)=0,sinx=cosx`
`impliestanx=1="tan"(pi)/4`
`:.x=(pi)/4`
So, only one distinct real root exist.