The term independent of $ x $ in the expansion of $ \left[ \sqrt{\frac{x}{3}} + \sqrt{\frac{3}{2}} x^2 \right]^{10} $ is ________.

Question

If $ A $ and $ B $ are the two real values of $ k $ for which the system of equations $ x + 2y + z = 1 $, $ x + 3y + 4z = k $, $ x + 5y + 10z = k^2 $ is consistent, then $ A + B = $: (a) 3

Answer 1

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.
Hence, the Answer is: (D)

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 Correct Option is D

Solution and Explanation

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