Question

In the expansion of \( \left( x - \frac{3}{x^2} \right)^9 \), the constant term is:

• A. \( ^9C_2 \)
• B. \( -2268 \)
• C. \( 2268 \)
• D. Does not exist

Hint:

To find the constant term in a binomial expansion, set the power of \(x\) in the general term to 0 and solve for \(r\).

Answer 1

The Correct Option is B

Step 1: General term in the expansion.
The general term in the binomial expansion is: \[ T_{r+1} = \binom{9}{r} \cdot x^{9 - r} \cdot \left( -\frac{3}{x^2} \right)^r = \binom{9}{r} \cdot (-3)^r \cdot x^{9 - r - 2r} = \binom{9}{r} \cdot (-3)^r \cdot x^{9 - 3r} \] Step 2: For constant term, power of \(x\) should be 0.
\[ 9 - 3r = 0 \Rightarrow r = 3 \] Step 3: Substitute \(r = 3\): \[ T_4 = \binom{9}{3} \cdot (-3)^3 = 84 \cdot (-27) = -2268 \]