For what values of $$m,$$ does the system of equations $$$\matrix{ {3x + my = m} \cr {2x - 5y = 20} \cr } $$$

has solution satisfying the conditions $$x > 0,\,y > 0.$$

The least value of the expression $$2\,\,{\log _{10}}\,x\, - \,{\log _x}(0.01)$$ for x > 1, is

Given $${n^4} < {10^n}$$ for a fixed positive integer $$n \ge 2,$$ prove that $${\left( {n + 1} \right)^4} < {10^{n + 1}}.$$

Both the roots of the equation (x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0 are always