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Question #40312

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, 2, 3, \dots, 20\}$ . Consider the vectors $\vec{a} = 2\hat{i} - 6\hat{j} + 3\hat{k}$ and $\vec{b}_n = n\hat{i} + n^2\hat{j} + n^3\hat{k}$ for $n \in S$ . Two sets $A$ and $B$ are defined as follows: $A = \{ n \in S \mid (\vec{a} \times \vec{b}_n) \cdot \hat{k} \le 80 \}$ $B = \{ n \in S \mid (\vec{b}_n \times \hat{a}) \cdot \hat{k} \ge -20 \}$ where $\hat{a}$ is the unit vector in the direction of $\vec{a}$ . If $A \Delta B$ denotes the symmetric difference of sets $A$ and $B$ , find the sum of all elements in $A \Delta B$ .

Options

A.
12
B.
13
C.
15
D.
195

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

Difference vs. Symmetric Difference|Vector Product Anti-Commutativity|Unit Vector Normalization

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of vector product properties within set-builder notation.

The primary trap lies in the definition of Set B, which requires both the anti-commutative property of the cross product ( $\vec{b}_n \times \vec{a} = -\vec{a} \times \vec{b}_n$ ) and the normalization of vector $\vec{a}$ into a unit vector $\hat{a}$ .

Students frequently ignore the negative sign (assuming commutativity) or forget to divide by the magnitude $|\vec{a}| = 7$ .

Failing either condition drastically alters the elements of Set B, leading to an incorrect symmetric difference.

The symmetric difference operation further penalizes those who mistakenly compute the intersection or union.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Difference vs. Symmetric Difference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free