In sub - part (i) to (x) choose the correct option and in sub - part (xi) to (xy), answer the questions as intructed .
Find the value (s) of so that the term independent of x in the expansion of `(sqrt(x)-(k)/(x^(2)))^(10)` is 405.

Answer

To find the value(s) of \( k \) such that the term independent of \( x \) in the expansion of \( \left( \sqrt{x} - \frac{k}{x^2} \right)^{10} \) is 405, we can follow these steps:

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

### Step 2: Simplify the general term
Now, simplify \( T_{r+1} \):
\[ T_{r+1} = \binom{10}{r} \left( \sqrt{x} \right)^{10-r} \left( -k \right)^r \left( \frac{1}{x^2} \right)^r \]
This can be rewritten as:
\[ T_{r+1} = \binom{10}{r} (-k)^r x^{\frac{10-r}{2} - 2r} \]
\[ = \binom{10}{r} (-k)^r x^{\frac{10 - r - 4r}{2}} = \binom{10}{r} (-k)^r x^{\frac{10 - 5r}{2}} \]

### Step 3: Find the condition for the term to be independent of \( x \)
For the term to be independent of \( x \), the exponent of \( x \) must be zero:
\[ \frac{10 - 5r}{2} = 0 \]
Solving for \( r \):
\[ 10 - 5r = 0 \implies 5r = 10 \implies r = 2 \]

### Step 4: Substitute \( r \) back into the general term
Now substitute \( r = 2 \) into the general term:
\[ T_{3} = \binom{10}{2} (-k)^2 x^{\frac{10 - 5 \cdot 2}{2}} = \binom{10}{2} (-k)^2 x^{0} \]
This simplifies to:
\[ T_{3} = \binom{10}{2} k^2 \]

### Step 5: Calculate \( \binom{10}{2} \)
Calculate \( \binom{10}{2} \):
\[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]
Thus, we have:
\[ T_{3} = 45 k^2 \]

### Step 6: Set the equation equal to 405
We know that this term is equal to 405:
\[ 45 k^2 = 405 \]

### Step 7: Solve for \( k^2 \)
Dividing both sides by 45:
\[ k^2 = \frac{405}{45} = 9 \]

### Step 8: Find the values of \( k \)
Taking the square root of both sides:
\[ k = \pm 3 \]

### Final Answer
Thus, the values of \( k \) are:
\[ k = 3 \quad \text{or} \quad k = -3 \]