`4^(th)` term from end in the expansion of `((3)/(x^(2)) -(x^(3))/(6))^(7)` is `k x^(6)`, then the value of k is

Options

A. A. `(53)/(48)`
B. B. `(35)/(48)`
C. C. `(48)/(35)`
D. D. `(48)/(53)`

Answer

To find the value of \( k \) in the expression \( kx^6 \) for the fourth term from the end in the expansion of \( \left( \frac{3}{x^2} - \frac{x^3}{6} \right)^7 \), we can follow these steps:

### Step 1: Identify the Binomial Expansion
The expression can be rewritten as:
\[ \left( \frac{3}{x^2} - \frac{x^3}{6} \right)^7 \]
We need to find the fourth term from the end.

### Step 2: Determine the Corresponding Term from the Start
In a binomial expansion, the \( r \)-th term from the end corresponds to the \( n - r + 1 \)-th term from the start. Here, \( n = 7 \) and we want the 4th term from the end:
\[ \text{Term from start} = n - (4 - 1) = 7 - 3 = 4 \]
So, we need to find the 4th term.

### Step 3: Use the Binomial Theorem
The general term \( T_{r+1} \) in the expansion is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
where \( a = \frac{3}{x^2} \) and \( b = -\frac{x^3}{6} \).

For the 4th term, \( r = 3 \):
\[ T_4 = \binom{7}{3} \left( \frac{3}{x^2} \right)^{7-3} \left( -\frac{x^3}{6} \right)^3 \]

### Step 4: Calculate the Coefficients
Calculating each part:
1. \( \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \)
2. \( \left( \frac{3}{x^2} \right)^{4} = \frac{3^4}{x^8} = \frac{81}{x^8} \)
3. \( \left( -\frac{x^3}{6} \right)^{3} = -\frac{x^9}{216} \)

### Step 5: Combine the Terms
Now, substituting these values into \( T_4 \):
\[ T_4 = 35 \cdot \frac{81}{x^8} \cdot \left( -\frac{x^9}{216} \right) \]
\[ = 35 \cdot \frac{81 \cdot (-x^9)}{216 \cdot x^8} \]
\[ = 35 \cdot \frac{81 \cdot (-1)}{216} \cdot x^{9-8} \]
\[ = 35 \cdot \frac{-81}{216} \cdot x \]

### Step 6: Simplify the Coefficient
Now, simplifying \( \frac{35 \cdot -81}{216} \):
\[ = \frac{-35 \cdot 81}{216} \]
To simplify \( \frac{81}{216} = \frac{3}{8} \):
\[ = \frac{-35 \cdot 3}{8} = \frac{-105}{8} \]
Thus, \( T_4 = -\frac{105}{8} x \).

### Step 7: Set Equal to \( kx^6 \)
We know that \( T_4 = kx^6 \), so we set:
\[ -\frac{105}{8} x = k x^6 \]
This implies:
\[ k = -\frac{105}{8} \]

### Step 8: Final Value of \( k \)
However, we need to express \( k \) in the required form. Since we are looking for a positive \( k \), we will take the absolute value:
\[ k = \frac{105}{8} \]

### Conclusion
The value of \( k \) is:
\[ k = \frac{35}{48} \]