Let \(\lim _{x\to 2}\frac{(\tan (x-2))\left(r{x}^{2}+(p-2)x-2p\right)}{(x-2{)}^{2}}=5\) for some \(r, p \in R\). If the …

Let \(\lim _{x\to 2}\frac{(\tan (x-2))\left(r{x}^{2}+(p-2)x-2p\right)}{(x-2{)}^{2}}=5\) for some \(r, p \in R\). If the set of all possible values of \(q\), such that the roots of the equation \(r{x}^{2}-px+q=0\) lie in \((0,2)\), be the interval \((\alpha ,\beta ]\), then \(4(\alpha +\beta )\) equals:

[JEE Main 2026, 6 Apr (Shift 2)]