Ge(Z = 32) in its ground state electronic configuration has x completely filled orbitals with m_l = 0. The value of x is ___________.

A₃B₂ is a sparingly soluble salt of molar mass M (g mol^$$-$$1) and solubility x g L^$$-$$1. The solubility product satisfies $${K_{sp}} = a{\left( {{x \over M}} \right)^5}$$. The value of a is _____________. (Integer answer)

The number of real roots of the equation

$${e^{4x}} + 2{e^{3x}} - {e^x} - 6 = 0$$ is :

Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If $$\int_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int_0^x {f(t)dt} } $$, $$0 \le x \le 1$$ and f(0) = 0, then $$\mathop {\lim }\limits_{x \to 0} {1 \over {{x^2}}}\int_0^x {f(t)dt} $$ :