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

$\alpha$, $\beta$ ,$\gamma$  are roots of equatiuon $x^3 -x -1 = 0$ then equation whose roots are $\frac{1}{{\beta  + \gamma }},\frac{1}{{\gamma  + \alpha }},\frac{1}{{\alpha  + \beta }}$ is

Let $y = \sqrt {\frac{{(x + 1)(x - 3)}}{{(x - 2)}}} $, then all real values of $x$ for which $y$ takes real values, are

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

If the sum of the two roots of the equation $4{x^3} + 16{x^2} - 9x - 36 = 0$ is zero, then the roots are

Suppose that $x$ and $y$ are positive number with $xy = \frac{1}{9};\,x\left( {y + 1} \right) = \frac{7}{9};\,y\left( {x + 1} \right) = \frac{5}{{18}}$ . The value of $(x + 1) (y + 1)$ equals