Question

Coefficient independent of x in the expansion of \([\frac{3x^{2}-1}{2x^{5}}]^{7}\) is?

• A. \(\frac{5103}{4}\)
• B. \(\frac{5293}{6}\)
• C. \(\frac{6715}{3}\)
• D. \(\frac{7193}{4}\)

Hint:

To find the coefficient independent of \(x\) in the binomial expansion, set the power of \(x\) in the general term equal to zero. This will help you identify the value of \(k\) that leads to the term that does not contain \(x\). Then calculate the corresponding coefficient.

Answer 1

The Correct Option is A

The given expression is:

\[ \left( \frac{3x^2 - 1}{2x^5} \right)^7 = \sum_{k=0}^{7} \binom{7}{k} \left( \frac{3x^2}{2x^5} \right)^k \left( -\frac{1}{2x^5} \right)^{7-k} \]

Now, for the coefficient independent of \(x\), we need the powers of \(x\) to cancel out. The general term of the expansion is:

\[ T_k = \binom{7}{k} \left( \frac{3^k}{2^k} \right) x^{2k-5k} (-1)^{7-k} \frac{1}{2^{7-k} x^{5(7-k)}} \]

We want the power of \(x\) to be 0, which gives:

\[ 2k - 5k - 5(7-k) = 0 \]

Solving this equation for \(k\) yields \(k = 4\).

The term for \(k = 4\) is:

\[ T_4 = \binom{7}{4} \left( \frac{3^4}{2^4} \right) (-1)^3 \frac{1}{2^3} \]

Now calculate this term:

\[ T_4 = \binom{7}{4} \cdot \frac{81}{16} \cdot \frac{-1}{8} \]

Finally, the coefficient independent of \(x\) is:

\[ \frac{5103}{4} \]

Final Answer: \(\frac{5103}{4}\).