If A, B and C are the angles of a triangle and
`|(1,1,1),(1 + sin A,1 + sin B,1 + sin C),(sin A + sin^(2) A,sin B + sin^(2)B,sin C + sin^(2) C)|= 0`, then the triangle ABC is

Options

A. The triangle ABC is isosceles
B. The triangle ABC is equailateral
C. The triangle ABC is scalene
D. No conclusion can be drawn with regard to the nature of the triangle

Answer

A

Explanation

`|{:(" "1," "1," "1),(1+sinA,1+sinB,1+sinC),(sinA+sin^(2)A,sinB+sin^(2)B,sin^(2)A-sin^(2)C):}|`
`|{:(" "1," "0," "0),(1+sinA,sinA-sinB,sinA-sinC),(sinA+sin^(2)A,sinA-sin^(2)B,sin^(2)A-sin^(2)C):}|+0=0`
{Applying `C_(2) rarr C_(1) - C_(2) and C_(3) rarr C_(1) - C_(3)`}
`=(sin A - sin B) xx (sin A - sin C)`
`|{:(" "1," "0," "0),(1+sinA," "1," "1),(sinA+sin^(2)A,(sin A+sin B),(sin A+sin C)):}|=0`
`(sin A - sin B)(sin A - sin C)(sin B - sin C)=0`
`therefore sin A = sin B or sin A = sin C or sin B = "sin C either A"= B or B = C or A = C`.