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Question #40325

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{Px + Q}{2x^2 - 5x + 2}$ be expressed as a series expansion $f(x) = \sum_{n=0}^{\infty} a_n x^n + \sum_{n=1}^{\infty} b_n x^{-n}$ valid in the region $\frac{1}{2} < |x| < 2$ . If $a_2 = -1$ and $b_2 = 1$ , find the ordered pair $(P, Q)$ .

Options

A.
$(20, -16)$
B.
$(-12, 0)$
C.
$(18, -12)$
D.
$(12, -16)$

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

Ordered Pair Equality|Partial Fractions Decomposition|Binomial Theorem for Negative/Fractional Indices

Why This Question Is Tricky

Common Mistakes

This question tests the 'Lethal Collision' between partial fraction decomposition and the convergence criteria of the Negative Index Binomial Theorem.

A common trap for students is to mechanically expand $(1-x)^{-1}$ without verifying the domain of convergence.

Here, the specified annulus $\frac{1}{2} < |x| < 2$ forces the term $\frac{A}{2x-1}$ to be expanded in negative powers of $x$ (converging for $|x| > \frac{1}{2}$ ) and $\frac{B}{x-2}$ in positive powers of $x$ (converging for $|x| < 2$ ).

Failing to extract the correct factors (e.g., factoring out $-1$ instead of $2x$ ) leads to incorrect series representations, swapping the $a_n$ and $b_n$ coefficients, or missing crucial negative signs, directly penalizing a lack of mathematical rigor regarding convergence intervals.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Ordered Pair Equality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free