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QJEEM24Mab69c: JEE Main 2024 Mathematics - Class 11 JEEM24Mab69c Solution: Let . Let be a relation on defined by if and only if . Let be…

Let A={1,2,3,4,5}\mathrm{A}=\{1,2,3,4,5\} . Let R\mathrm{R} be a relation on A\mathrm{A} defined by xRyx \mathrm{R} y if and only if 4x≤5y4 x \leq 5 \mathrm{y} . Let m\mathrm{m} be the number of elements in R\mathrm{R} and n\mathrm{n} be the minimum number of elements from A×A\mathrm{A} \times \mathrm{A} that are required to be added to R to make it a symmetric relation. Then m + n is equal to :

Multiple Choice Options

  • A 23
  • B 26
  • C 25
  • D 24

Step-by-Step Verified Solution

amp;A={1,2,3,4,5}amp;xRy⇔4x≤5yamp;4x≤5y⇒xy≤54⇒xy≤1.25\begin{aligned}& A=\{1,2,3,4,5\} \\& x R y \Leftrightarrow 4 x \leq 5 y \\& 4 x \leq 5 y \quad \Rightarrow \quad \frac{x}{y} \leq \frac{5}{4} \quad \Rightarrow \frac{x}{y} \leq 1.25\end{aligned}

amp;R={(1,2),(1,3),(1,4),(1,5),(1,1),(2,2),(2,3),(2,4),amp;(2,5),(3,3),(3,4),(3,5),(4,4),(4,5),(5,4),(5,5)}amp;∴n(R)=m=16\begin{aligned}& R=\{(1,2),(1,3),(1,4),(1,5),(1,1),(2,2),(2,3),(2,4), \\& (2,5),(3,3),(3,4),(3,5),(4,4),(4,5),(5,4),(5,5)\} \\& \therefore \quad n(R)=m=16\end{aligned}

Elements to be added to RR to make it symmetric

(1,2)∈R⇒(2,1)(1,2) \in R \quad \Rightarrow \quad(2,1) should be added

Similarly, (3,1),(4,1),(5,1),(3,2),(4,2),(5,2),(4,3),(5,3)(3,1),(4,1),(5,1),(3,2),(4,2),(5,2),(4,3), (5,3)

∴9\therefore 9 elements should be added

∴amp;n=9∴amp;m+n=25\begin{array}{ll}\therefore & n=9 \\\therefore & m+n=25\end{array}

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