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

If $\alpha$ and $\beta$ are the roots of the equation $x^2 + ax + b = 0$,then the value of $\alpha^3 + \beta^3$ is equal to:

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 + 2(a + b)x + a^2 + b^2 = 0$,then the equation whose roots are $(\alpha + \beta)^2$ and $(\alpha - \beta)^2$ is

The number of integers satisfying the inequality $\sqrt{\log_3(x) - 1} + \frac{\frac{1}{2}\log_3(x^3)}{\log_3(\frac{1}{3})} + 2 > 0$ is

If the graph of $y = ax^3 + bx^2 + cx + d$ is symmetric about the line $x = k$,then:

Let two numbers have an arithmetic mean of $9$ and a geometric mean of $4$. Then these numbers are the roots of the quadratic equation: