If $\alpha=1$ and $\beta=1+i \sqrt{2}$, where $i=\sqrt{-1}$ are two roots of the equation

$x^3+a x^2+b x+c=0, a, b, \in \mathbb{R}$, then $\int_{-1}^1\left(x^3+a x^2+b x+c\right) d x$ is equal to:

If the quadratic equation $(\lambda+2) x^2-3 \lambda x+4 \lambda=0, \lambda \neq-2$, has two positive roots, then the number of possible integral values of $\lambda$ is :

Let $A=\left[\begin{array}{ccc}1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7\end{array}\right]$ and $\operatorname{det}(A-\alpha I)=0$, where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$, then the circle $(x-p)^2+(y-2 p)^2=320$, intersects the co-ordinate axes at

Let $\alpha=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots \infty$ and

$\beta=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\ldots \infty$. Then the value of

$(0.2)^{\log _{\sqrt{5}}(\alpha)}+(0.04)^{\log _5(\beta)}$ is equal to :