For what value of m the ratio of the roots of the equation 12x2 – mx

1.	For what value of m the ratio of the roots of the equation 12x2 – mx + 5 = 0 is 3 : 2 ?(a) 5√10 (b) 10√5 (c) 25√2 (d) 15√5
Answer» (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\(\sqrt{10}\).

For what value of m the ratio of the roots of the equation 12x2 – mx + 5 = 0 is 3 : 2 ?(a) 5√10 (b) 10√5 (c) 25√2 (d) 15√5

Answer»

(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\(\sqrt{10}\).