If the constant terms in the expansion of `(sqrt(x)-k/x^2)^(10)` is 405 then `|k|=`

Options

A. 9
B. `9/2`
C. 3
D. 6

Answer

To find the value of \(|k|\) given that the constant term in the expansion of \((\sqrt{x} - \frac{k}{x^2})^{10}\) is 405, we can follow these steps:

### Step 1: Write the General Term
The general term \(T_{r+1}\) in the binomial expansion of \((a + b)^n\) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
For our expression, we have \(a = \sqrt{x}\), \(b = -\frac{k}{x^2}\), and \(n = 10\). Thus, the general term is:
\[ T_{r+1} = \binom{10}{r} (\sqrt{x})^{10-r} \left(-\frac{k}{x^2}\right)^r \]

### Step 2: Simplify the General Term
Now, simplify \(T_{r+1}\):
\[ T_{r+1} = \binom{10}{r} (-1)^r k^r x^{\frac{10-r}{2} - 2r} \]
This can be rewritten as:
\[ T_{r+1} = \binom{10}{r} (-1)^r k^r x^{\frac{10 - 5r}{2}} \]

### Step 3: Find the Constant Term
The constant term occurs when the exponent of \(x\) is zero:
\[ \frac{10 - 5r}{2} = 0 \]
Multiplying through by 2 gives:
\[ 10 - 5r = 0 \implies 5r = 10 \implies r = 2 \]

### Step 4: Substitute \(r\) into the General Term
Now substitute \(r = 2\) into the general term:
\[ T_{3} = \binom{10}{2} (-1)^2 k^2 x^{\frac{10 - 5(2)}{2}} = \binom{10}{2} k^2 \]
Calculating \(\binom{10}{2}\):
\[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]
Thus, the constant term becomes:
\[ T_{3} = 45 k^2 \]

### Step 5: Set the Constant Term Equal to 405
We know from the problem statement that this constant term equals 405:
\[ 45 k^2 = 405 \]

### Step 6: Solve for \(k^2\)
Dividing both sides by 45:
\[ k^2 = \frac{405}{45} = 9 \]

### Step 7: Find \(|k|\)
Taking the square root gives:
\[ |k| = \sqrt{9} = 3 \]

### Final Answer
Thus, the value of \(|k|\) is:
\[ \boxed{3} \]