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

If $a,b,c$ are real and ${x^3} – 3{b^2}x + 2{c^3}$ is divisible by $x – a$ and$x – b$, then

Number of rational roots of equation $x^{2016} -x^{2015} + x^{1008} + x^{1003} + 1 = 0,$ is equal to

For what value of $\lambda$ the sum of the squares of the roots of ${x^2} + (2 + \lambda )\,x – \frac{1}{2}(1 + \lambda ) = 0$ is minimum

The number of solutions of $\sin ^2 \mathrm{x}+\left(2+2 \mathrm{x}-\mathrm{x}^2\right) \sin \mathrm{x}-3(\mathrm{x}-1)^2=0$, where $-\pi \leq \mathrm{x} \leq \pi$, is………………..