Q69 / 200 · Manipal (MET) 2010 · Mathematics

The number of terms in the expansion of $(\sqrt{5}+\sqrt[4]{11})^{124}$ which are integers, is equal to

  • A
    $0$
  • B
    $30$
  • C
    $31$
  • Correct answer
    D
    $32$

Solution

The general term in the expansion of $(\sqrt{5}+4 \sqrt{11})^{124}$ is $\begin{gathered}T_{r+1}={ }^{124} C_r(\sqrt{5})^{124-r}(\sqrt[4]{11})^r \\ \quad={ }^{124} C_r \cdot 5^{(124-r) / 2} \cdot 11^{r / 4}\end{gathered}$ For integer terms $r=0,4,8,12, \ldots, 124$ $\therefore$ Number of terms which are integers $=32$

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

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