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 :

For 10 observations $x_1, x_2, \ldots, x_{10}$, if $\sum\limits_{i=1}^{10}\left(x_i+2\right)^2=180$ and $\sum\limits_{i=1}^{10}\left(x_i-1\right)^2=90$, then their standard deviation is :

In the expansion of $\left(9 x-\frac{1}{3 \sqrt{x}}\right)^{18}, x>0$, if the term independent of $x$ is (221)k, then k is equal to: