(a) 5\(\sqrt{10}\).
Given, the roots of the given equation 12x2 – mx + 5 = 0 are in the ratio 3 : 2. Let the roots of the given equation be 3α and 2α. Then,
Sum of roots = 3α + 2α = \(\frac{m}{12}\) ⇒ 5α = \(\frac{m}{12}\) ........(i)
and (3α)(2α) = \(\frac{5}{12}\) ⇒ 6α2 = \(\frac{5}{12}\) ⇒ α2 = \(\frac{5}{72}\)
⇒ α = \(\sqrt{\frac{5}{12}}\)
∴ From (i) and (ii)
5. \(\sqrt{\frac{5}{12}}\) = \(\frac{m}{12}\) ⇒ m = 60\(\sqrt{\frac{5}{12}}\) = 60.\(\frac{\sqrt5}{6\sqrt2}\) = 10\(\sqrt{\frac{5}{2}}\)
= 10.\(\frac{\sqrt5}{\sqrt2}\).\(\frac{\sqrt2}{\sqrt2}\) = \(\frac{10}{2}\). \(\sqrt{10}\) = 5\(\sqrt{10}\).