Q64 / 110 · VITEEE 2022 · Mathematics

If the binary operation \(*\) is defined on the set \(\mathrm{Q}^{+}\) of all positive rational numbers by \(\mathrm{a} * \mathrm{~b}=\frac{\mathrm{ab}}{4}\). Then \(3 *\left(\frac{1}{5} * \frac{1}{2}\right)\) is equal to

  • Correct answer
    A
    \(\frac{3}{160}\)
  • B
    \(\frac{5}{160}\)
  • C
    \(\frac{3}{10}\)
  • D
    \(\frac{3}{40}\)

Solution

Given that \(\mathrm{a} * \mathrm{~b}=\frac{\mathrm{ab}}{4} \forall \mathrm{a}, \mathrm{b} \in \mathrm{Q}^{+}\) \(\begin{aligned} \therefore \quad 3 *\left(\frac{1}{5} * \frac{1}{2}\right) & =3 *\left\{\frac{\frac{1}{5} \times \frac{1}{2}}{4}\right\}=3 * \frac{1}{40}=\frac{\left(3 \times \frac{1}{40}\right)}{4} \\ & =\frac{3}{160} \end{aligned}\)

Hence, the correct answer is option: 'A'.

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