The quadratic equation with rational coefficients,whose one root is $\sqrt{5}$,is:

If $\frac{1}{\sqrt{\alpha}}$ and $\frac{1}{\sqrt{\beta}}$ are the roots of the equation $ax^2 + bx + 1 = 0$ $(a \neq 0, a, b \in R)$,then the equation $x(x + b^3) + (a^3 - 3abx) = 0$ has roots:

If $\alpha$ and $\beta$ are roots of the equation $x^2 - 4\sqrt{2}kx + 2e^{4\ln k} - 1 = 0$ for some $k$, and $\alpha^2 + \beta^2 = 66$, then $\alpha^3 + \beta^3$ is equal to: (in $\sqrt{2}$)

Find the value$(s)$ of $P$ for which the given quadratic equation $3x^{2}-5x+P=0$ has equal roots.

If $\alpha$ and $\beta$ are the roots of $x^2 - 3x + a = 0$,where $a \in R$ and $\alpha < 1 < \beta$,then:

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 6x + a = 0$ and satisfy the relation $3\alpha + 2\beta = 16$,then the value of $a$ is