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Question #40574

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all integers $r$ such that the binomial expansion of $f_r(x) = \left( x^{\frac{1}{\sqrt{r}}} + x^{-\frac{1}{\sqrt{201-r}}} \right)^{201}$ contains a term independent of $x$ . Find the number of elements in $S$ .

Options

A.
0
B.
1
C.
2
D.
4

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Selection vs Arrangement|Term Independent of x|Domain Restrictions

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of binomial expansion, Diophantine equations, and domain restrictions.

The student must first set up the condition for a term independent of x, leading to the equation r = 201(201-k)^2 / (k^2 + (201-k)^2).

Using number theory (specifically that a sum of coprime squares cannot have prime factors of the form 4m+3), the only algebraic solutions are r=0 and r=201.

The critical trap is that students will accept these values and answer 2.

However, the original function's exponents contain square roots in the denominators, strictly requiring 0 < r < 201.

Thus, the algebraic solutions are extraneous, and the correct answer is 0.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Selection vs Arrangement
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free