A.M., G.M., H.M. and Relations

Equation $x^4+a{x}^3+b{x}^2+cx+1=0$ has real roots where $a, b, c$ are non-negative then $\min (a+b+c)$ is equal to -

Let $x, y, z>0$ and $9x+3y+z=90$, then maximum value of ${\log }_{10}x+{\log }_{10}y+{\log }_{10}z$ is equal to

Let $x^2-3x+k=0$ has two positive roots $k_1$ and $k_2$ then minimum value of $\frac{4}{k_1}+\frac{1}{k_2}$ is equal to

If $a_1,a_2,\dots \dots \dots a_n\in R^{+}$and $a_1·a_2\dots \dots a_n=1$ then the least value of $\left(1+a_1+a_1^2\right)\left(1+a_2+a_2^2\right)\dots .\left(1+a_n+a_n^2\right)$ is -

Which of the following is true, where $a, b, c>0 ?$

If all roots of the equation $f\left(x\right)=x^6-12x^5+b{x}^4+c{x}^3+d{x}^2+ex+64=0$ are positive, then which has the greatest numerical (absolute) value

A straight line through the vertex $P$ of a triangle $PQR$ intersects the side $QR$ at the point $S$ and the circumcircle of the triangle $PQR$ at the point $T.$ if $S$ is not the centre of the circumcircle, then