If $\alpha, \beta$ are the roots of the equation $x^2 - px + r = 0$ and $\alpha/2, 2\beta$ are the roots of the equation $x^2 - qx + r = 0$,then what is the value of $r$?

The number of integers $k$ for which the equation $x^3-27x+k=0$ has at least two distinct integer roots is

Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0$,with $\alpha>\beta$. For all positive integers $n$,define $a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}, n \geq 1$ and $b_1=1$ and $b_n=a_{n-1}+a_{n+1}, n \geq 2$. Then which of the following options is/are correct?
$(1)$ $\sum_{i=1}^{n} a_i = a_{n+2}-1$ for all $n \geq 1$
$(2)$ $\sum_{n=1}^{\infty} \frac{a_n}{10^n} = \frac{10}{89}$
$(3)$ $\sum_{n=1}^{\infty} \frac{b_n}{10^n} = \frac{8}{89}$
$(4)$ $b_n = \alpha^n+\beta^n$ for all $n \geq 1$

If $\alpha, \beta, \gamma$ are the roots of the equation $2x^3+x^2-13x+6=0$,then $\alpha^3+\beta^3+\gamma^3=$

Let $a, b, c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a, c), (2, b)$ and $(a, b)$ be $\left(\frac{10}{3}, \frac{7}{3}\right)$. If $\alpha, \beta$ are the roots of the equation $ax^{2} + bx + 1 = 0$,then the value of $\alpha^{2} + \beta^{2} - \alpha\beta$ is ....... .

Find the sum of all possible values of $k$ for which the roots of the equation $x^2 + (k + 1)x + \lambda = 0$ are the square of each other.