Before You Begin

Question #40654

Difficulty: 8/10 | 🟢 Free

The Question

Consider the function $f(x) = \left(1 + \frac{x}{2}\right)^{-3}$ and its quadratic Maclaurin polynomial $P(x) = \sum_{k=0}^{2} \binom{-3}{k} \left(\frac{x}{2}\right)^k$ . Let $S_1$ be the set of all integer solutions to the inequality $\frac{P(x) - 1 + \frac{3}{2}x}{x^2 - 2x - 15} \le 0$ Let $D$ be the interval of convergence of the infinite binomial series expansion of $f(x)$ . If $S_2 = S_1 \cap D$ , find the number of elements in the set $S_1 \setminus S_2$ .

Options

A.
2
B.
4
C.
5
D.
7

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

Properties of Binomial Coefficients|Approximation via Differentials|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question creates a 'Lethal Collision' between the Wavy Curve method and the domain of validity for binomial approximations.

Students routinely reduce the approximation to a polynomial and solve the algebraic inequality, finding $S_1$ .

The primary trap lies in the interval of convergence $D$ .

Many students either ignore $D$ entirely (yielding 7 elements) or incorrectly assume the boundary points are included in the domain of convergence (i.e., $D = [-2, 2]$ ).

If they make the latter mistake, they will incorrectly place $-2$ and $2$ into $S_2$ , leading to the trap answer of 2.

Furthermore, the wavy curve features a repeated root at $x=0$ , testing if students correctly maintain the sign across the even power.

Quick Facts About This Question

Question Information

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