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

Question

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

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`

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - 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.