Question

If the constant term in the binomial expansion of \(\left(\sqrt{x}-\frac{k}{x^2}\right)^{10}\) is 405, then \(|k|\) equals :

• A. \(3\)
• B. \(9\)
• C. \(1\)
• D. \(2\)

Hint:

Write the general term and equate the power of x to zero to find r.

Answer 1

The Correct Option is A

General term: \(T_{r+1}=\binom{10}{r}(\sqrt{x})^{10-r}\left(-\frac{k}{x^2}\right)^r=\binom{10}{r}(-k)^r\,x^{\frac{10-r}{2}-2r}\).
Constant term: \(\frac{10-r}{2}-2r=0 \Rightarrow 10-5r=0 \Rightarrow r=2\).
Constant term \(=\binom{10}{2}k^2=45k^2=405 \Rightarrow k^2=9\).
So \(|k|=3\).
Answer: (1) 3