$Consider the equation$${\sin }^{-1}\left(x^2-6x+\frac{17}{2}\right)+{\cos }^{-1}k=\frac{\pi }{2}$ $then$

$The largest value of k for which the equation has 2 distinct solutions is 1$

$The eqution must have real roots if k\in \left(-\frac{1}{2},1\right)$

$the equation must have real roots if k\in \left(-1,\frac{1}{2}\right)$

$The equation has unique solution if k=-\frac{1}{2}$