The remainder when the polynomial $2 x^5-3 x^4+5 x^3-3 x^2+7 x-9$ is divided by $x^2-x-3$ is

If $x$ is real,then the value of $\frac{x^2 + 34x - 71}{x^2 + 2x - 7}$ does not lie between

Let $p, q$ be real numbers. If $\alpha$ is a root of $x^{2}+3 p^{2} x+5 q^{2}=0$,$\beta$ is a root of $x^{2}+9 p^{2} x+15 q^{2}=0$ and $0 < \alpha < \beta$,then the equation $x^{2}+6 p^{2} x+10 q^{2}=0$ has a root $\gamma$ that always satisfies:

The number of pairs of real numbers $(x, y)$ such that $x = x^2 + y^2$ and $y = 2xy$ is:

If $\alpha$ is a root of the equation $x^2-x+1=0$,then $\left(\alpha+\frac{1}{\alpha}\right)^3+\left(\alpha^2+\frac{1}{\alpha^2}\right)^3+\left(\alpha^3+\frac{1}{\alpha^3}\right)^3+\left(\alpha^4+\frac{1}{\alpha^4}\right)^3=$

Evaluate $\sqrt{x + 2\sqrt{x - 1}} + \sqrt{x - 2\sqrt{x - 1}}$ for $x \ge 1$.