Multiple choice

If the equation $\displaystyle \lambda x^{2}-2x+3= 0$ has positive roots for some real $\displaystyle \lambda $, then

  1. $\displaystyle \lambda > 0$
  2. $\displaystyle \lambda > \frac{1}{3}$
  3. $\displaystyle 0 < \lambda \leq \frac{1}{3}$
  4. $\displaystyle \lambda \leq \frac{1}{3}$
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C Correct answer
Explanation

For roots to be positive, sum of roots = 2/lambda > 0 (so lambda > 0) and product of roots = 3/lambda > 0 (so lambda > 0). Discriminant D = 4 - 12*lambda >= 0. 4 >= 12*lambda, so lambda <= 1/3. Combined: 0 < lambda <= 1/3.

AI explanation

For the quadratic to have positive roots, lambda must be positive. The discriminant condition requires 4 - 12(lambda) >= 0, which gives lambda <= 1/3. Combining these conditions, lambda must be greater than 0 and less than or equal to 1/3. The result is 0 < lambda <= 1/3.

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