Multiple choice

If $\alpha$, $\beta$ are the roots of the equation $3x^2-6x+4=0$, the value of $\displaystyle\left(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\right)+2\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)+3\alpha\beta$ will be

  1. $6$
  2. $\displaystyle\frac{4}{3}$
  3. $2$
  4. $8$
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D Correct answer
Explanation

alpha+beta = 2, alpha*beta = 4/3. Expression = (alpha^2+beta^2)/(alpha*beta) + 2(alpha+beta)/(alpha*beta) + 3(alpha*beta). = ((alpha+beta)^2 - 2*alpha*beta)/(alpha*beta) + 2(2)/(4/3) + 3(4/3). = (4 - 8/3)/(4/3) + 3 + 4 = (4/3)/(4/3) + 7 = 1 + 7 = 8.

AI explanation

From the equation 3x^2 - 6x + 4 = 0, the sum of the roots alpha + beta is -(-6)/3 = 2 and the product alpha * beta is 4/3. The expression simplifies as (alpha^2 + beta^2)/(alpha * beta) + 2(alpha + beta)/(alpha * beta) + 3(alpha * beta). Substituting the values gives ((2)^2 - 2(4/3))/(4/3) + 2(2)/(4/3) + 3(4/3), which is (4/3)/(4/3) + 3 + 4. Adding these results gives 1 + 3 + 4 = 8.

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