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Question #40658

Difficulty: 8/10 | 🟢 Free

The Question

Let $C_r$ be the coefficient of $x^r$ in the expansion of $(1+x)^4(1+x)^6$ . Let $M$ be the maximum value among $C_r$ for $0 \le r \le 10$ . If $a = C_4$ and $b = C_6$ , consider the set $S$ of all positive integers $x \le 260$ that satisfy the inequality: $\frac{x^3 + a^3 + b^3 - 3xab}{x - M} \ge 0$ Find the sum of all elements in $S$ .

Options

A.
2262
B.
2052
C.
2304
D.
2514

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

Expansion of Products of Binomials|Greatest Binomial Coefficient|Arithmetic Mean Inequality|Sign Flipping Rule

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of binomial expansion properties and algebraic inequalities (AM-GM).

The primary trap is the 'Zero Cancellation Rule' combined with inequality denominators.

Students often use the AM-GM inequality ( $x^3+a^3+b^3 \ge 3xab$ ) to conclude the numerator is always non-negative, and then strictly solve for the denominator $x - M > 0$ .

By doing so, they completely miss the isolated discrete solution $x = 210$ where the numerator becomes exactly zero, satisfying the $\ge 0$ condition even when the denominator is negative.

This forces students to carefully reconcile continuous analytic bounds with discrete combinatorial values.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Expansion of Products of Binomials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free