Question:
The term independent of \( x \) in the expansion of \( \left[ \sqrt{\frac{x}{3}} + \sqrt{\frac{3}{2}} x^2 \right]^{10} \) is ________.
The term independent of \( x \) in the expansion of \( \left[ \sqrt{\frac{x}{3}} + \sqrt{\frac{3}{2}} x^2 \right]^{10} \) is ________.
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The term independent of x in the expansion of $[\sqrtx/3 + \sqrt3/2x]
Updated On: Apr 15, 2026
- 1
- ${}^{10}C_{1}$
- $5/12$
- None of these
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The Correct Option is D
Solution and Explanation
Step 1: General Term
$T_{r+1} = {}^{10}C_r (\frac{x}{3})^{\frac{10-r}{2}} (\frac{3}{2x^2})^{\frac{r}{2}}$.
Step 2: Exponent of x
Power of $x = \frac{10-r}{2} - r = \frac{10 - 3r}{2}$.
Step 3: Condition
For term independent of $x$: $10 - 3r = 0 \Rightarrow r = 10/3$.
Step 4: Conclusion
Since $r$ must be an integer ($0 \le r \le 10$), no such term exists.
Final Answer: (D)
$T_{r+1} = {}^{10}C_r (\frac{x}{3})^{\frac{10-r}{2}} (\frac{3}{2x^2})^{\frac{r}{2}}$.
Step 2: Exponent of x
Power of $x = \frac{10-r}{2} - r = \frac{10 - 3r}{2}$.
Step 3: Condition
For term independent of $x$: $10 - 3r = 0 \Rightarrow r = 10/3$.
Step 4: Conclusion
Since $r$ must be an integer ($0 \le r \le 10$), no such term exists.
Final Answer: (D)
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