Multiple choice

Let $\alpha,\,\beta$ be the roots of the equation ${x}^{2}-5x+r=0$ and $\displaystyle \frac{\alpha}{3},3\beta$ be the roots of the equation ${x}^{2}-3x+r=0$, then the value of $r$ is -

  1. $\displaystyle \frac{9}{4}$
  2. $\displaystyle \frac{4}{9}$
  3. $\displaystyle \frac{5}{4}$
  4. $\displaystyle \frac{4}{5}$
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A Correct answer
Explanation

From the first equation, alpha + beta = 5 and alpha * beta = r. From the second, (alpha/3) + 3beta = 3 and (alpha/3) * 3beta = r, so alpha * beta = r. Substituting alpha = 3(3 - 3beta) into alpha + beta = 5: 9 - 9beta + beta = 5 => 8beta = 4 => beta = 0.5. Then alpha = 4.5. r = alpha * beta = 4.5 * 0.5 = 2.25 = 9/4.

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