If $a$ and $b$ are two different positive real numbers, then which of the following relations is true

If the minimum value of $f(x)=\frac{5 x^{2}}{2}+\frac{\alpha}{x^{5}}, x>0$, is 14 , then the value of $\alpha$ is equal to.

Let $3, a, b, c$ be in $A.P.$ and $3, a-1, b+1, c+9$ be in $G.P.$ Then, the arithmetic mean of $a, b$ and $c$ is :

$x + y + z = 15$ if $9,\;x,\;y,\;z,\;a$ are in $A.P.$; while $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{5}{3}$ if $9,\;x,\;y,\;z,\;a$ are in $H.P.$, then the value of $a$ will be

Let $a_{1}, a_{2}, \ldots, a_{10}$ be an $AP$ with common difference $-3$ and $\mathrm{b}_{1}, \mathrm{~b}_{2}, \ldots, \mathrm{b}_{10}$ be a $GP$ with common ratio $2.$ Let $c_{k}=a_{k}+b_{k}, k=1,2, \ldots, 10 .$ If $c_{2}=12$ and $\mathrm{c}_{3}=13$, then $\sum_{\mathrm{k}=1}^{10} \mathrm{c}_{\mathrm{k}}$ is equal to ..... .

If $A.M.$ and $G.M.$ of roots of a quadratic equation are $8$ and $5,$ respectively, then obtain the quadratic equation.