Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $3x^2-5x+2=0$, then find the values of $\dfrac{\alpha }{\beta }+\dfrac{\beta }{\alpha }$

  1. $\dfrac{13}{6}$
  2. $\dfrac{-13}{6}$
  3. $\dfrac{12}{6}$

  4. None of these
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A Correct answer
Explanation

For 3x^2 - 5x + 2 = 0, alpha + beta = 5/3 and alpha * beta = 2/3. alpha/beta + beta/alpha = (alpha^2 + beta^2) / (alpha * beta) = ((alpha+beta)^2 - 2*alpha*beta) / (alpha*beta). = ((25/9) - 4/3) / (2/3) = (13/9) / (2/3) = 13/6.

AI explanation

From the equation 3x^2-5x+2=0, the sum of roots a+b is 5/3 and the product ab is 2/3. We evaluate the expression a/b + b/a by combining fractions to get (a^2+b^2)/ab, which equals ((a+b)^2-2ab)/ab. Substituting the values gives ((5/3)^2-2*(2/3))/(2/3) = (25/9 - 12/9)/(2/3) = 13/9 * 3/2 = 13/6.

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