Class JEE Mathematics Sets, Relations, and Functions Q #1041
KNOWLEDGE BASED
APPLY
4 Marks 2022 JEE Main 2022 (Online) 28th July Morning Shift MCQ SINGLE
For $\alpha \in N$, consider a relation $R$ on $N$ given by $R = \{(x, y) : 3x + \alpha y$ is a multiple of $7\}$. The relation $R$ is an equivalence relation if and only if :
(A) $\alpha = 14$
(B) $\alpha$ is a multiple of $4$
(C) $4$ is the remainder when $\alpha$ is divided by $10$
(D) $4$ is the remainder when $\alpha$ is divided by $7$
Prev Next
Correct Answer: D
Explanation
For $R$ to be an equivalence relation, it needs to be reflexive, symmetric and transitive.

Reflexive: $(a, a) \in R$ for all $a \in N$.
So, $3a + \alpha a = 7k$, where $k$ is an integer.
This implies $(3 + \alpha)a = 7k$.
Thus, $3 + \alpha$ must be a multiple of $7$, so $3 + \alpha = 7k_1$.
Therefore, $\alpha = 7k_1 - 3 = 7k_1 + 4$ for some integer $k_1$.
This indicates that when $\alpha$ is divided by $7$, the remainder is $4$.

Symmetric: If $(a, b) \in R$, then $(b, a) \in R$.
Given $3a + (7k - 3)b = 7m$, then $3(a - b) + 7kb = 7m$.
If $3(b - a) + 7ka = 7m$, then $aRb \implies bRa$.
Thus, $R$ will be symmetric for $\alpha = 7k_1 - 3$.

Transitive: If $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$.
So, $3a + (7k - 3)b = 7k_1$ and $3b + (7k_2 - 3)c = 7k_3$.
Adding $3a + 7kb + (7k_2 - 3)c = 7(k_1 + k_3)$.
Then $3a + (7k_2 - 3)c = 7m$.
Thus, $(a, c) \in R$.
Therefore, $R$ is transitive.

Since $\alpha = 7k - 3 = 7k + 4$, the remainder when $\alpha$ is divided by $7$ is $4$.

More from this Chapter

NUMERICAL
The number of elements in the set $\{ n \in Z:|{n^2} - 10n + 19| < 6\} $ is _________.
MCQ_SINGLE
Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $P(m,n)$ from the point $Q(-2,-3)$ is :
MCQ_SINGLE
Let $R$ be the set of real numbers. Statement I: $A = \{(x, y) \in R \times R: y - x \text{ is an integer }\}$ is an equivalence relation on $R$. Statement II: $B = \{(x,y) \in R \times R: x = \alpha y \text{ for some rational number } \alpha\}$ is an equivalence relation on $R$.
NUMERICAL
Let $S=\left\{p_1, p_2 \ldots, p_{10}\right\}$ be the set of first ten prime numbers. Let $A=S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x, y), x \in S$, $y \in A$, such that $x$ divides $y$, is ________ .
NUMERICAL
Let $\mathrm{A}=\{-4,-3,-2,0,1,3,4\}$ and $\mathrm{R}=\left\{(a, b) \in \mathrm{A} \times \mathrm{A}: b=|a|\right.$ or $\left.b^{2}=a+1\right\}$ be a relation on $\mathrm{A}$. Then the minimum number of elements, that must be added to the relation $\mathrm{R}$ so that it becomes reflexive and symmetric, is __________
View All Questions