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Question #40907

Difficulty: 8/10 | 🟢 Free

The Question

Let $k$ be the sum of the infinite series $1 + 4\left(\frac{x-2}{2}\right) + 7\left(\frac{x-2}{2}\right)^2 + 10\left(\frac{x-2}{2}\right)^3 + \dots \infty$ . A point $P$ divides the line segment joining $A(2, 3)$ and $B(6, 3)$ internally in the ratio $k:1$ . If the $x$ -coordinate of $P$ is at most $4$ , then the complete set of values of $x$ is:

Options

A.
$(1, 2]$
B.
$(1, 2] \cup [10, \infty)$
C.
$(0, 2]$
D.
$(-\infty, 2] \cup [10, \infty)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Ratio of Division (k:1)|Arithmetic-Geometric Progression (AGP)|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of coordinate geometry constraints with the convergence criteria of an infinite Arithmetic-Geometric Progression (AGP).

The primary trap lies in blindly applying the AGP sum formula and solving the resulting rational inequality via the Wavy Curve method, which yields extraneous solutions (like $x \ge 10$ ).

Students must recognize that the sum formula is only valid when the common ratio strictly satisfies $|r| < 1$ .

Additionally, the problem subtly filters out negative ratios by specifying 'internal division' ( $k > 0$ ), requiring a multi-layered intersection of intervals to reach the correct domain.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Ratio of Division (k:1)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free