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  • The natural number m, for which the coefficient of y in the  binomial expansion of    <img alt="\small \left(y^m+\frac{1}{y^2}\right)^{22}" src="https://entrancecorner.oncodec

The natural number m, for which the coefficient of y in the 
binomial expansion of    \small \left(y^m+\frac{1}{y^2}\right)^{22} is 1540, is  

Option: 1

11


Option: 2

12


Option: 3

13


Option: 4

None of these


Answers (1)

best_answer

We have, \left(y^m+\frac{1}{y^2}\right)^{22}

T_{r+1}={ }^{22} C_r\left(y^m\right)^{22-r}\left(\frac{1}{y^2}\right)^r={ }^{22} C_r y^{22 m-m r-2 r}

Since, { }^{22} C_3={ }^{22} C_{19}=1540\ \therefore \quad r=3 or 19

Now, for coefficient of y, 22m – mr – 2r = 1 ⇒ m=\frac{2 r+1}{22-r}

At\ r=3, m=\frac{7}{19} \notin N At\ r=19, m=\frac{39}{3}=13 \in N \\ \therefore \quad m=13

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HARSH KANKARIA

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