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Question
If the term free from x in the expansion of `(sqrt(x) - k/x^2)^10` is 405, find the value of k.
Solution
The given expression is `(sqrt(x) - k/x^2)^10`
General term `"T"_(r + 1) = ""^n"C"_r x^(n - r) y^r`
= `""^10"C"_r (sqrt(x))^(10 - r) ((-k)/x^2)^r`
= `""^10"C"_r (x)^((10 - r)/2) (-k)^r (1/x^(2r))`
= `""^10"C"_r (x)^((10 - r)/2 - 2r) (-k)^r`
= `""^10"C"_r (x)^((10 - r - 4r)/2) (- k)^r`
= `""^10"C"_r (x)^((10 - 5r)/2) (- k)^r`
For getting term free from x
`(10 - 5r)/2` = 0
⇒ r = 2
On putting the value of r in the above expression
We get `""^10"C"_2 (-k)^2`
According to the condition of the question, we have
`""^10"C"_2 k^2` = 405
⇒ `(10*9)/(2*1) k^2` = 405
⇒ 45k2 = 405
⇒ k2 = `405/45` = 9
∴ k = `+- 3`
Hence, the value of k = ±3
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