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Question #40601

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{10}{x^2+1}$ and $K = \{ [f(x)] : x \in [1, 2] \}$ , where $[ \cdot ]$ denotes the greatest integer function. For each $k \in K$ , a set $S_k$ consisting of exactly $k$ distinct elements is given. Let $W_k$ be the number of ordered pairs $(A, B)$ of subsets of $S_k$ such that $A \cap B \neq \emptyset$ . Find the value of $\sum_{k \in K} W_k$ .

Options

A.
1000
B.
1360
C.
993
D.
360

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

Restricted Selection (At least one common)|Exhaustive Events Concept|Greatest Integer Function (GIF)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to combine calculus-based range finding with combinatorial complementary counting.

The primary trap lies in the Greatest Integer Function (GIF) acting on a continuous function.

Students often incorrectly evaluate the boundary values of $[f(x)]$ , either missing $k=5$ by assuming open intervals or failing to recognize that $f(x)$ achieves the exact value of 5 at $x=1$ .

Furthermore, the combinatorial trap requires realizing that calculating 'at least one common element' directly is computationally heavy, whereas the complementary approach (Total pairs $4^k$ minus Disjoint pairs $3^k$ ) yields a clean geometric progression.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Restricted Selection (At least one common)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free