The coefficient of term independent of x in the expression of `((3x^2)/2-1/3x)^9` is `lamda` then `18lamda` is

Options

A. 9
B. 7
C. 6
D. 4

Answer

To find the coefficient of the term independent of \( x \) in the expression \(\left(\frac{3x^2}{2} - \frac{1}{3}x\right)^9\), we can follow these steps:

### Step 1: Identify the binomial expansion
The expression can be written in the form of a binomial expansion:
\[ \left(a + b\right)^n \]
where \( a = \frac{3x^2}{2} \) and \( b = -\frac{1}{3}x \) and \( n = 9 \).

### Step 2: Write the general term of the expansion
The general term \( T_r \) in the expansion of \((a + b)^n\) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
Substituting \( a \) and \( b \):
\[ T_r = \binom{9}{r} \left(\frac{3x^2}{2}\right)^{9-r} \left(-\frac{1}{3}x\right)^r \]

### Step 3: Simplify the general term
Now, simplifying \( T_r \):
\[ T_r = \binom{9}{r} \left(\frac{3^{9-r}}{2^{9-r}} x^{2(9-r)}\right) \left(-\frac{1}{3}\right)^r x^r \]
\[ = \binom{9}{r} \frac{3^{9-r}}{2^{9-r}} (-1)^r \frac{1}{3^r} x^{2(9-r) + r} \]
\[ = \binom{9}{r} \frac{3^{9-r}}{2^{9-r}} (-1)^r \frac{1}{3^r} x^{18 - r} \]
\[ = \binom{9}{r} \frac{3^{9-r}}{2^{9-r}} (-1)^r \frac{1}{3^r} x^{18 - r} \]
\[ = \binom{9}{r} (-1)^r \frac{3^{9-r}}{2^{9-r} \cdot 3^r} x^{18 - r} \]
\[ = \binom{9}{r} (-1)^r \frac{3^{9 - 2r}}{2^{9-r}} x^{18 - r} \]

### Step 4: Find the term independent of \( x \)
For the term to be independent of \( x \), we need:
\[ 18 - r = 0 \implies r = 18 \]
However, since \( r \) cannot exceed \( n = 9 \), we need to find when \( 18 - r = 0 \) is not possible. Instead, we need \( 2(9 - r) + r = 0 \):
\[ 18 - 3r = 0 \implies r = 6 \]

### Step 5: Calculate the coefficient for \( r = 6 \)
Now substituting \( r = 6 \):
\[ T_6 = \binom{9}{6} (-1)^6 \frac{3^{9 - 2 \cdot 6}}{2^{9 - 6}} = \binom{9}{6} \frac{3^{9 - 12}}{2^{3}} = \binom{9}{6} \frac{3^{-3}}{2^{3}} = \binom{9}{6} \frac{1}{27 \cdot 8} \]
\[ = \binom{9}{3} \frac{1}{216} = 84 \cdot \frac{1}{216} = \frac{84}{216} = \frac{7}{18} \]

### Step 6: Find \( 18\lambda \)
Now, we have \( \lambda = \frac{7}{18} \). Therefore:
\[ 18\lambda = 18 \cdot \frac{7}{18} = 7 \]

### Final Answer
Thus, the value of \( 18\lambda \) is \( 7 \).

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