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Question

The number of terms which are free from radical signs in the expansion of $$(y^{\frac 15} + x^{\frac 1{10}})^{55}$$ are


A
5
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B
6
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C
7
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D
none of these
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Solution

The correct option is B 6
In the expansion of $$(y^{1/5} + x^{1/10})^{55}$$,the general term is
$$T_{r+1} = ^{55}C_r (y^{1/5})^{55-r} (x^{1/10})^r = ^{55}C_r.y^{11-r/5}x^{r/10}$$
This $$T_{r+1}$$ will be independent of radicals if the exponents r/5 and r/10 are integers, for $$0\leq r \leq 55$$ which is possible only when $$r = 0, 10, 20, 30, 40, 50$$.
$$\therefore$$ There are six terms viz. $$T_1,T_{11},T_{21},T_{31},T_{41},T_{51}$$, which are independent of radicals.

Mathematics

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