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Question #60047

Difficulty: 5/10 | 🟢 Free

The Question

Let $A$ be the set of all real roots of the equation: $(x-2) \sum_{k=1}^{5} k (x-1)^{k-1} = 5(x-1)^5 - (x-1)^4 - 2(x-1)^3 + 5(x-1)^2 - 12(x-1) + 5$ Let $B = \{x \in \mathbb{R} : \sin(\pi x) = 0 \text{ and } 0 \le x \le 5\}$ . Let $D$ be the dataset consisting of the values of the function $g(x) = x^2 - x$ evaluated at each element of the set $A \cup (A \cap B)$ . What is the coefficient of range of the dataset $D$ ?

Options

A.
$\frac{5}{7}$
B.
$\frac{1}{3}$
C.
1
D.
$\frac{4}{7}$

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

Range and Coefficient of Range|Arithmetic-Geometric Progression (AGP)|The Zero-Factor Cancellation Trap

Why This Question Is Tricky

Examiner Levers

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Common Mistakes

This question is engineered to trigger the 'Zero-Factor Cancellation Trap' within the context of an Arithmetic-Geometric Progression (AGP).

To evaluate the AGP sum, students frequently divide both sides of the equation by the common ratio factor $(x-2)$ to apply the standard sum formula.

However, doing so implicitly assumes $x \neq 2$ , causing them to inadvertently reject $x=2$ (or $y=1$ ) as a root when solving the resulting polynomial.

If $x=2$ is missed, the student calculates the dataset as $D = \{6, 12\}$ , leading directly to the trap option of 1/3.

Additionally, the question incorporates a set theory distractor (the absorption law) to test conceptual focus under extraneous cognitive load.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Range and Coefficient of Range
Difficulty	5
Cognitive Load	6
Target Solve Time	150 sec
Access	Free