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
In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to ___________.
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
NUMERICAL
Let R1 and R2 be relations on the set {1, 2, ......., 50} such that R1 = {(p, pn) : p is a prime and n $\ge$ 0 is an integer} and R2 = {(p, pn) : p is a prime and n = 0 or 1}. Then, the number of elements in R1 $-$ R2 is _______________.
MCQ_SINGLE
Let $R_1$ and $R_2$ be two relation defined as follows: $R_1 = {(a, b) \in R^2 : a^2 + b^2 \in Q}$ and $R_2 = {(a, b) \in R^2 : a^2 + b^2 \notin Q}$, where $Q$ is the set of all rational numbers. Then :
MCQ_SINGLE
A survey shows that $73$% of the persons working in an office like coffee, whereas $65$% like tea. If $x$ denotes the percentage of them, who like both coffee and tea, then $x$ cannot be :
View All Questions