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Question #40645

Difficulty: 8/10 | 🟢 Free

The Question

Let $a_n(x)$ be the coefficient of $t^n$ in the expansion of $\left(1 + t + \frac{x}{|x|} t^2\right)^n$ for $x \neq 0$ . Let $D$ be the set of all $x \in \mathbb{R} \setminus \{0\}$ for which the series $\sum_{n=1}^{\infty} a_n(x) x^n$ converges absolutely. If $D = (-a, 0) \cup (0, b)$ , find the value of $\frac{1}{a^2} + \frac{1}{b^2}$ .

Options

A.
10
B.
12
C.
14
D.
16

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

Multinomial Theorem|Modulus Sign Dependency|Real-Valued Function Domain

Why This Question Is Tricky

Common Mistakes

This question tests the convergence of a multinomial series where the base depends on the sign of the variable.

The trap lies in assuming the radius of convergence is symmetric or determined purely by real roots.

For $x < 0$ , the alternating signs in the multinomial expansion cause massive cancellation, leading to a complex singularity that dictates the radius of convergence.

Students failing to account for the modulus split or the complex roots of the discriminant will incorrectly evaluate the domain.

Quick Facts About This Question

Question Information

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