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If 0 lt alpha lt beta lt gamma lt pi//2...

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

A

real and unequal roots

B

non-real roots

C

real and equal roots

D

real and unequal roots greater than `2`

Text Solution

Verified by Experts

The correct Answer is:
A
`(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.
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