If `0 lt alpha lt beta lt gamma lt pi//2`, then the equation
`(x-sinbeta)(x-singamma)+(x-sinalpha)(x-singamma)+(x-sinalpha)(x-sinbeta)=0` has

Options

A. real and unequal roots
B. non-real roots
C. real and equal roots
D. real and unequal roots greater than `2`

Answer

A

Explanation

`(a)` Let `f(x)=(x-sinbeta)(x-singamma)+(x-sinalpha)(x-singamma)+(x-sinalpha)(x-sinbeta)`
Now, `f(sin alpha)=(sinalpha-sinbeta)(sinalpha-singamma)`
`=(-)(-)=positive


`f(sinbeta)=(sinbeta-sinalpha)(sinbeta-sinalpha)=(+)(-)=negative
`f(sin gamma)=(sin gamma-sinalpha)(singamma-sinbeta)`=(+)(+)=`positive
`implies "Roots of " f(x)=0` are real and distinct.