Let $$a, b, c$$ be any real numbers. Suppose that there are real numbers $$x, y, z$$ not all zero such that $$x=cy+bz,$$ $$y=az+cx,$$ and $$z=bx+ay.$$ Then $${a^2} + {b^2} + {c^2} + 2abc$$ is equal to :

Let $$A$$ be $$a\,2 \times 2$$ matrix with real entries. Let $$I$$ be the $$2 \times 2$$ identity matrix. Denote by tr$$(A)$$, the sum of diagonal entries of $$a$$. Assume that $${a^2} = I.$$
Statement-1 : If $$A \ne I$$ and $$A \ne - I$$, then det$$(A)=-1$$
Statement- 2 : If $$A \ne I$$ and $$A \ne - I$$, then tr $$(A)$$ $$ \ne 0$$.

Suppose the cubic $${x^3} - px + q$$ has three distinct real roots
where $$p>0$$ and $$q>0$$. Then which one of the following holds?

How many real solutions does the equation
$${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$$ have?