Question

If the equation `sin ^(2) x - k sin x - 3 = 0` has exactly two distinct real roots in `[0, pi]`, then find the values of k .

Answer 1

Let `f(t) = t^(2) - kt - 3`, where t = siin x.
Since equation has exactly two distinct real roots in `[ 0, pi], f(t) = 0` must have exactly one root in (0, 1).
Now `f(0) = -3` .
So., we must have `f(1) = - k - 2 gt 0`
or k `lt`-2.