1.

In a `Delta ABC " if " |(1,a,b),(1,c,a),(1,b,c)| =0`, then `sin^(2) A + sin^(2) B + sin^(2) C` isA. `(3 sqrt3)/(2)`B. `(9)/(4)`C. `(5)/(4)`D. 2

Answer» Correct Answer - B
We have,
`|(1,a,b),(1,c,a),(1,b,c)| =0`
`rArr |(1,a,b),(0,c -a,a -b),(0,b -a,c -a)| = 0 " " [("Applying" R_(2) rarr R_(2) - R_(1)),(R_(3) rarr R_(3) - R_(1))]`
`rArr (c -a) (c -b) + (a-b)^(2) = 0`
`rArr a^(2) + b^(2) + c^(2) - ab - bc - ca = 0`
`rArr 2a^(2) + 2b^(2) + 2c^(2) - 2ab - 2bc - 2ca = 0`
`rArr 2a^(2) + 2b^(2) + 2c^(2) - 2ab - 2bc - 2ca = 0`
`rArr (a -b)^(2) + (b -c)^(2) + (c -a)^(2) = 0`
`rArr a = b = c`
`rArr Delta ABC` is equilateral
`rArr A = B = C = (pi)/(3)`
`:. sin^(2) A + sin^(2) B + sin^(2) C = 3 sin^(2) (pi)/(3) = (9)/(4)`


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