Let $\alpha_1, \alpha_2, \ldots, \alpha_7$ be the roots of the equation $x^7+3x^5-13x^3-15x=0$ and $|\alpha_1| \geq |\alpha_2| \geq \ldots \geq |\alpha_7|$. Then $\alpha_1 \alpha_2 - \alpha_3 \alpha_4 + \alpha_5 \alpha_6$ is equal to $..................$.

Which of the following six statements are true about the cubic polynomial $P(x) = 2x^3 + x^2 + 3x - 2$?
$(i)$ It has exactly one positive real root.
$(ii)$ It has either one or three negative roots.
$(iii)$ It has a root between $0$ and $1$.
$(iv)$ It must have exactly two real roots.
$(v)$ It has a negative root between $-2$ and $-1$.
$(vi)$ It has no complex roots.

If $\alpha$ and $\beta$ $(\alpha < \beta)$ are the roots of the equation $(-2+\sqrt{3})(|\sqrt{x}-3|) + (x-6\sqrt{x}) + (9-2\sqrt{3}) = 0$,$x \ge 0$,then $\sqrt{\frac{\beta}{\alpha}} + \sqrt{\alpha\beta}$ is equal to:

To remove the second term of the equation $x^4-8x^3+x^2-x+3=0$,diminish the roots of the equation by

If the roots of the equation $x^2 - 8x + a^2 - 6a = 0$ are real,then what is the range of $a$?

If the roots of $a(b - c)x^2 + b(c - a)x + c(a - b) = 0$ are equal,then $a, b, c$ are in