The ratio of the coefficient of `x^(3)` to the term independent of x in `(2x+1/(x^(2)))^(12)` is

Options

A. `9:8`
B. `8:9`
C. `8:1`
D. `9:1`

Answer

To solve the problem, we need to find the ratio of the coefficient of \( x^3 \) to the term independent of \( x \) in the expression \( \left( 2x + \frac{1}{x^2} \right)^{12} \).

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
In our case, \( a = 2x \), \( b = \frac{1}{x^2} \), and \( n = 12 \). Thus, the general term becomes:
\[ T_r = \binom{12}{r} (2x)^{12-r} \left(\frac{1}{x^2}\right)^r = \binom{12}{r} 2^{12-r} x^{12-r} x^{-2r} \]
Simplifying this gives:
\[ T_r = \binom{12}{r} 2^{12-r} x^{12 - 3r} \]

2. **Find the Coefficient of \( x^3 \)**:
To find the coefficient of \( x^3 \), we set the exponent of \( x \) equal to 3:
\[ 12 - 3r = 3 \]
Solving for \( r \):
\[ 12 - 3r = 3 \implies 3r = 9 \implies r = 3 \]
Now, substituting \( r = 3 \) into the general term:
\[ T_3 = \binom{12}{3} 2^{12-3} = \binom{12}{3} 2^9 \]

3. **Calculate \( \binom{12}{3} \)**:
\[ \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]
Therefore, the coefficient of \( x^3 \) is:
\[ 220 \times 2^9 = 220 \times 512 = 112640 \]

4. **Find the Coefficient of the Term Independent of \( x \)**:
The term independent of \( x \) occurs when the exponent of \( x \) is 0:
\[ 12 - 3r = 0 \]
Solving for \( r \):
\[ 12 - 3r = 0 \implies 3r = 12 \implies r = 4 \]
Substituting \( r = 4 \) into the general term:
\[ T_4 = \binom{12}{4} 2^{12-4} = \binom{12}{4} 2^8 \]

5. **Calculate \( \binom{12}{4} \)**:
\[ \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]
Therefore, the coefficient of the term independent of \( x \) is:
\[ 495 \times 2^8 = 495 \times 256 = 126720 \]

6. **Find the Ratio**:
Now, we find the ratio of the coefficient of \( x^3 \) to the term independent of \( x \):
\[ \text{Ratio} = \frac{112640}{126720} \]
Simplifying this fraction:
\[ \text{Ratio} = \frac{112640 \div 112640}{126720 \div 112640} = \frac{1}{1.125} = \frac{8}{9} \]

### Final Answer:
The ratio of the coefficient of \( x^3 \) to the term independent of \( x \) is \( \frac{8}{9} \).