If the constant term in the expansion of \({\left( {\sqrt x - \frac{k}{{{x^2}}}} \right)^{10}}\) is 405, then what can be the values of k?

Concept:

In the binomial expansion of (a + b)ⁿ, the term which does not involve any variable is said to be an independent term.

The general term in the binomial expansion of (a + b)ⁿ is given by: \({T_{r + 1}} = {\;^n}{C_r} \times {a^{n - r}} \times {b^r}\)

Calculation:

Given: The constant term in the expansion of \({\left( {\sqrt x - \frac{k}{{{x^2}}}} \right)^{10}}\) is 405

i.e the independent term of \({\left( {\sqrt x - \frac{k}{{{x^2}}}} \right)^{10}}\) is 405.

Let (r + 1)^th term be the independent term.

\(\Rightarrow {T_{r + 1}} = {\;^{10}}{C_r} \times {\left( {\sqrt x } \right)^{10 - r}} \times {\left( {\frac{{ - k}}{{{x^2}}}} \right)^r} = \;{\;^{10}}{C_r} \times {\left( { - k} \right)^r} \times {\left( x \right)^{\frac{{10 - 5r}}{2}}}\;\)

∵ (r + 1)th term is the independent term

\(\Rightarrow \frac{{10 - 5r}}{2} = 0 \Rightarrow r = 2\)

\(\Rightarrow {\;^{10}}{C_2} \times {\left( { - k} \right)^2} = 405 \Rightarrow 45{k^2} = 405\)