The term independent of x in the expansion of ( x + 1x^3/2 + 1 - x x + 1x - x)^10 for x 1 is:

The term independent of x in the expansion of ( x + 1x^3/2 + 1 -… — Solution | JEE Mathematics

MCQMediumJEE 2025Binomial Expansion

The term independent of $x$ in the expansion of $\left( \frac{x + 1}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{x + 1}{x - \sqrt{x}} \right)^{10}$ for $x > 1$ is:

A
$210$
B
$150$
C
$240$
D
$120$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: Find the term independent of $x$ in the expansion of

\left( \frac{x + 1}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{x + 1}{x - \sqrt{x}} \right)^{10}

for $x > 1$ .

Find: The constant term in the expansion.

Simplify the given expression:

\left( \frac{(x + 1)^2}{x^{3/2} + 1 - \sqrt{x}} \cdot \frac{1}{x - \sqrt{x}} \right)^{10}

\Rightarrow \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^{10}

Now use binomial expansion. The general term is

T_r = {10 \choose r} \cdot x^{\frac{10 - 2r}{2}}

For the term independent of $x$ , set the power of $x$ equal to $0$ :

\frac{10 - 2r}{2} = 0

\Rightarrow r = 5

Hence the required term is

T_5 = {10 \choose 5} = 210

Therefore, the term independent of $x$ is $210$ . The correct option is A.

Common mistakes

A common mistake is to expand the original expression directly without first simplifying it. This is wrong because the constant term becomes much easier to identify after reducing the expression to $\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^{10}$ . Simplify the base expression before applying the binomial theorem.
Another mistake is using the wrong general term exponent for $x$ . This is wrong because the powers come from $\left(\sqrt{x}\right)^{10-r} \left(\frac{1}{\sqrt{x}}\right)^r$ , giving net power $\frac{10-2r}{2}$ . Always combine exponents carefully before setting the power equal to zero.
Some students take the middle term automatically as the constant term without checking the exponent of $x$ . This is wrong because the middle term is not always independent of the variable. Always set the exponent of $x$ to $0$ explicitly.

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Answer

Step-by-step solution

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