Let A and B be two 3 $$\times$$ 3 real matrices such that (A² $$-$$ B²) is invertible matrix. If A⁵ = B⁵ and A³B² = A²B³, then the value of the determinant of the matrix A³ + B³ is equal to :

Let f : (a, b) $$\to$$ R be twice differentiable function such that $$f(x) = \int_a^x {g(t)dt} $$ for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :

Let $$\overrightarrow a = \widehat i - \alpha \widehat j + \beta \widehat k$$,   $$\overrightarrow b = 3\widehat i + \beta \widehat j - \alpha \widehat k$$ and $$\overrightarrow c = -\alpha \widehat i - 2\widehat j + \widehat k$$, where $$\alpha$$ and $$\beta$$ are integers. If $$\overrightarrow a \,.\,\overrightarrow b = - 1$$ and $$\overrightarrow b \,.\,\overrightarrow c = 10$$, then $$\left( {\overrightarrow a \, \times \overrightarrow b } \right).\,\overrightarrow c $$ is equal to ___________.

The distance of the point P(3, 4, 4) from the point of intersection of the line joining the points. Q(3, $$-$$4, $$-$$5) and R(2, $$-$$3, 1) and the plane 2x + y + z = 7, is equal to ______________.