JEE Mathematics 2023 Question with Solution

Given: The binomial expansion is $\left(\frac{4x}{5} - \frac{5}{2x}\right)^{2022}$. The $1011^{\text{th}}$ term from the end is $1024$ times the $1011^{\text{th}}$ term from the beginning.

Find: The value of $|x|$.

From the solution, the conclusion shown is that the correct option is A and the final answer stated is $|x| = 10$. However, this conflicts with the given options, and the working in the solution also uses a different expression, namely $\left(4x^5 - \frac{5}{2x}\right)^{2022}$, which does not match the question text.

Using the term positions shown in the solution:

and the $1011^{\text{th}}$ term from the end is taken as the $1012^{\text{th}}$ term from the beginning:

According to the solution,

The working then simplifies this to

so that

and then states

This final numerical statement is inconsistent with the preceding equation, and it is also not present in the options. Since the solution explicitly marks Option A as correct, the answer is taken as A as required by the source-priority rule, while noting the discrepancy.

Therefore, the correct option is A.

The extracted source contains multiple inconsistencies:

1. The question uses $\left(\frac{4x}{5} - \frac{5}{2x}\right)^{2022}$.
2. The solution working uses $\left(4x^5 - \frac{5}{2x}\right)^{2022}$.
3. The solution heading says the correct option is A.
4. The final line of the solution says $|x| = 10$, which is not among the options.
5. The answer key says (3), that is C.

Because the instruction says the solution is the primary source, the extracted answer is resolved from the solution as A, with the inconsistency explicitly recorded.