Multiple choice

The condition that the roots of the equation $lx^{2} + mx + n = 0$ may be in the ratio 3 : 4 is

  1. $14n^{2} = 49 ml$
  2. $m^{2} = 9 nl$
  3. $12 m^{2} = 49 nl$
  4. $4l^{2} = 49 ml$
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C Correct answer
Explanation

Roots are 3k and 4k. Sum = 7k = -m/l. Product = 12k^2 = n/l. k = -m/(7l). 12 * (-m/(7l))^2 = n/l. 12 * m^2 / (49l^2) = n/l. 12m^2 = 49nl.

AI explanation

Let the roots of the quadratic equation be 3 alpha and 4 alpha. The sum of the roots is 7 alpha, which equals negative m divided by l, and the product of the roots is 12 alpha squared, which equals n divided by l. From the first relation, alpha equals negative m divided by 7 l, and substituting this into the second relation gives 12 times m squared divided by 49 l squared equals n divided by l. Simplifying this yields the required condition 12 m squared equals 49 n l.

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