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Question #40606

Difficulty: 8/10 | 🟢 Free

The Question

Let $V$ be the maximum absolute value among all terms in the binomial expansion of $(5 + 2x)^{100}$ for a given integer $x$ . Let $S$ be the set of all integers $x$ such that $E_3(V) = 4$ , where $E_p(N)$ denotes the exponent of the highest power of prime $p$ dividing $N$ . Find the value of $\sum_{x \in S} x^2$ .

Options

A.
298
B.
149
C.
426
D.
0

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

Exponent of Prime in n!|Numerically Greatest Term|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of the Numerically Greatest Term (NGT) concept and prime divisibility (Legendre's/Kummer's Theorem).

The primary trap lies in the modulus function inherent in the NGT formula: $r = \lfloor \frac{(n+1)|bx|}{|a|+|bx|} \rfloor$ .

Students often solve for $r$ and find valid positive integers $x = 7, 10$ , but forget that the modulus allows negative solutions $x = -7, -10$ , leading them to incorrectly sum only the positive squares (149 instead of 298).

Additionally, the problem forces students to recognize that $E_3(V) = E_3(\binom{100}{r}) + r E_3(|x|)$ , meaning any multiple of 3 for $x$ will blow up the exponent, elegantly restricting the search space.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Exponent of Prime in n!
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free