Class JEE Mathematics Sets, Relations, and Functions Q #1018
COMPETENCY BASED
APPLY
4 Marks 2025 JEE Main 2024 (Online) 6th April Morning Shift MCQ SINGLE
Let the relations $R_1$ and $R_2$ on the set $X = \{1, 2, 3, ..., 20\}$ be given by $R_1 = \{(x, y) : 2x - 3y = 2\}$ and $R_2 = \{(x, y) : -5x + 4y = 0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M + N$ equals
(A) 16
(B) 12
(C) 8
(D) 10
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Correct Answer: D

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Step-by-Step Solution

Step 1: Analyze R1

R1 = {(x, y) : 2x - 3y = 2}

We need to find pairs (x, y) that satisfy this equation, where x and y are in the set X = {1, 2, 3, ..., 20}.

If x = 1, 2(1) - 3y = 2 => -3y = 0 => y = 0, which is not in X.

If x = 2, 2(2) - 3y = 2 => 4 - 3y = 2 => 3y = 2 => y = 2/3, which is not in X.

If x = 4, 2(4) - 3y = 2 => 8 - 3y = 2 => 3y = 6 => y = 2. So, (4, 2) is in R1.

If x = 7, 2(7) - 3y = 2 => 14 - 3y = 2 => 3y = 12 => y = 4. So, (7, 4) is in R1.

If x = 10, 2(10) - 3y = 2 => 20 - 3y = 2 => 3y = 18 => y = 6. So, (10, 6) is in R1.

If x = 13, 2(13) - 3y = 2 => 26 - 3y = 2 => 3y = 24 => y = 8. So, (13, 8) is in R1.

If x = 16, 2(16) - 3y = 2 => 32 - 3y = 2 => 3y = 30 => y = 10. So, (16, 10) is in R1.

If x = 19, 2(19) - 3y = 2 => 38 - 3y = 2 => 3y = 36 => y = 12. So, (19, 12) is in R1.

R1 = {(4, 2), (7, 4), (10, 6), (13, 8), (16, 10), (19, 12)}

To make R1 symmetric, we need to add (2, 4), (4, 7), (6, 10), (8, 13), (10, 16), (12, 19).

M = 6

Step 2: Analyze R2

R2 = {(x, y) : -5x + 4y = 0}

We need to find pairs (x, y) that satisfy this equation, where x and y are in the set X = {1, 2, 3, ..., 20}.

4y = 5x => y = (5/4)x

If x = 4, y = (5/4)(4) = 5. So, (4, 5) is in R2.

If x = 8, y = (5/4)(8) = 10. So, (8, 10) is in R2.

If x = 12, y = (5/4)(12) = 15. So, (12, 15) is in R2.

If x = 16, y = (5/4)(16) = 20. So, (16, 20) is in R2.

R2 = {(4, 5), (8, 10), (12, 15), (16, 20)}

To make R2 symmetric, we need to add (5, 4), (10, 8), (15, 12), (20, 16).

N = 4

Step 3: Calculate M + N

M + N = 6 + 4 = 10

Correct Answer: 10

AI Suggestion: Option D

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply the concepts of relations and symmetry to determine the number of elements needed to make the given relations symmetric.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the concept of symmetric relations and how to manipulate relations to achieve symmetry. It requires understanding the underlying principles rather than just recalling facts.
Syllabus Audit: In the context of JEE, this is classified as COMPETENCY. It requires the application of the definition of symmetric relations to a specific problem, going beyond simple recall of definitions.

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