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Question
If the term free from x in the expansion of \[\left( \sqrt{x} - \frac{k}{x^2} \right)^{10}\] is 405, find the value of k.
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Solution
Let (r + 1)th term, in the expansion of \[\left( \sqrt{x} - \frac{k}{x^2} \right)^{10}\] , be free from x and be equal to Tr + 1. Then, \[T_{r + 1} =^{10} C_r \left( \sqrt{x} \right)^{10 - r} \left( \frac{- k}{x^2} \right)^r =^{10} C_r x^{5 - \frac{5r}{2}} \left( - k \right)^r . . . . (1)\] If Tr + 1 is independent of x, then \[5 - \frac{5r}{2} = 0 \Rightarrow r = 2\] Putting r = 2 in (1), we obtain \[T_3 =^{10} C_2 \left( - k \right)^2 = 45 k^2\] But it is given that the value of the term free from x is 405. \[\therefore 45 k^2 = 405 \Rightarrow k^2 = 9 \Rightarrow k = \pm 3\] Hence, the value of k is \[\pm 3\]
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