Question

If all values of $a$ for which the equation ${2x}^2-2(2a+1)x+a(a-1)=0$ has roots $\alpha$ and $\beta$ satisfying the condition $\alpha <a<\beta$ belong to the set A, then the number of integers in the intersection of A and $[-3,0]$ is

Answer 1

Number $a$ lies between the roots of the given equation, then
$2f\left(a\right)<0\Rightarrow f\left(a\right)<0{\therefore 2a}^2-2(2a+1)a+a(a-1)<0\Rightarrow -a^2-3a<0\Rightarrow a(a+3)>0$
$\therefore a\in (-\infty ,-3)\cup (0,\infty )$.