1.

If `alpha,beta(alpha < beta)` are the roots of equation `6x^2+11="" x+3="0` , then which following real? (a) `cos^(-1)alpha` (b) `sin^(-1)beta` (c) `cosec^(-1)alpha` (d) both `cot^(-1)alpha` and `cot^(-1)beta`

Answer» `6x^2+11x+3 = 0`
`=>6x^2+9x+2x+3 = 0`
`=>3x(2x+3)+1(2x+3) = 0`
`=>(2x+3)(3x+1) = 0`
`=> x = -3/2 or x = -1/3`
As `alpha` and `beta` ar the roots of the given equation,
`:. alpha = -3/2 or beta = -1/3`
Now, we will check the options.
(a) `cos^-1(alpha) = cos^-1(-3/2)`
As, domain of `cos^-1` is from `-1` to `1`, so `cos^-1(-3/2)` is not real.
(b) `sin^-1(beta) = sin^-1(-1/3) `
As, domain of `sin^-1` is from `-1` to `1`, so `sin^-1(-1/3)` is real.
(c) `cosec^-1(alpha) = cosec^-1(-3/2)`
As, domain of `cosec^-1` is from `-oo` to `-1`, so `cosec^-1(-3/2)` is real.
(d) `cot^-1(alpha) and cot^-1(beta)`
As, domain of `cot^-1` is from `-oo` to `oo`, so `cot^-1(-3/2) and cot^-1(-1/3)` is real.


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