For $$\alpha, \beta, \gamma \neq 0$$, if $$\sin ^{-1} \alpha+\sin ^{-1} \beta+\sin ^{-1} \gamma=\pi$$ and $$(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3 \alpha \beta$$, then $$\gamma$$ equals

Let $$\mathrm{S}$$ be the set of positive integral values of $$a$$ for which $$\frac{a x^2+2(a+1) x+9 a+4}{x^2-8 x+32} < 0, \forall x \in \mathbb{R}$$. Then, the number of elements in $$\mathrm{S}$$ is :

Let $$\alpha, \beta, \gamma, \delta \in \mathbb{Z}$$ and let $$A(\alpha, \beta), B(1,0), C(\gamma, \delta)$$ and $$D(1,2)$$ be the vertices of a parallelogram $$\mathrm{ABCD}$$. If $$A B=\sqrt{10}$$ and the points $$\mathrm{A}$$ and $$\mathrm{C}$$ lie on the line $$3 y=2 x+1$$, then $$2(\alpha+\beta+\gamma+\delta)$$ is equal to

For $$0 < c < b < a$$, let $$(a+b-2 c) x^2+(b+c-2 a) x+(c+a-2 b)=0$$ and $$\alpha \neq 1$$ be one of its root. Then, among the two statements

(I) If $$\alpha \in(-1,0)$$, then $$b$$ cannot be the geometric mean of $a$ and $$c$$

(II) If $$\alpha \in(0,1)$$, then $$b$$ may be the geometric mean of $$a$$ and $$c$$