Let A={1,2,3,4} and R be a relation on set A × A defined by R={((a, b),(c, d)): 2 a+3 b=4 c+5 d} .

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1.Relation and Function

normal

Let $A=\{1,2,3,4\}$ and $R$ be a relation on the set $A \times A$ defined by $R=\{((a, b),(c, d)): 2 a+3 b=4 c+5 d\}$. Then the number of elements in $R$ is:

$6$

$5$

$4$

$3$

(JEE MAIN-2023)

Solution

$A =\{1,2,3,4\}$

$R =\{( a , b ),( c , d )\}$

$2 a +3 b =4 c +5 d =\alpha \text { let }$

$2 a =\{2,4,6,8\} \quad 4 c =\{4,8,12,16\}$

$3 b =\{3,6,9,12\} \quad 5 d =\{5,10,15,20\}$

$2 a +3 b =\left\{\begin{array}{l}5,8,11,14 \\ 7,10,13,16 \\ 9,12,15,18 \\ 11,14,17,20\end{array}\right\} \quad 4 c +5 d \left\{\begin{array}{l}9,14,19,24 \\ 13,18 \ldots . \\ 17,22 \ldots \\ 21,26 \ldots\end{array}\right\}$

Possible value of $\alpha=9,13,14,14,17,18$

Pairs of $\{( a , b ),( c , d )\}=6$

Standard 12

Mathematics

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hard

(JEE MAIN-2023)

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medium

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normal

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normal

(KVPY-2017)

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Solution

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Let $R$ be a relation on the set of all natural numbers given by $\alpha b \Leftrightarrow \alpha$ divides $b^2$.

Which of the following properties does $R$ satisfy?

$I.$ Reflexivity $II.$ Symmetry $III.$ Transitivity