Let A be the event that the absolute difference between two...

Given,

$A$ be the event that the absolute difference between two randomly chosen real numbers in the sample space $\left[0, 60\right]$ is less than or equal to $a$,

So, \left|x - y\right| < a ⇒ - a < x - y < a$</p><p>$⇒ x - y < a and x - y > - a$</p><p>Now on plotting the diagram we get,</p><p><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/9506b106-4bd7-4f37-92cf-1e4772cf1ab6-07e6c7be-b036-4f17-823c-1d3bce835ddc-9896416.png" style="width: 250px; height: 250px;" /></p><p>Now from diagram probability is given by,</p><p>$P\left(A\right) = \dfrac{\text{area} ( OACDEG )}{\text{area} ( OBDF )}$</p><p>$⇒ P\left(A\right) = \dfrac{\text{area}\left(OBDF\right) - \text{area}\left(ABC\right) - \text{area}\left(EFG\right)}{\text{area}\left(OBDF\right)}$</p><p>$⇒ \dfrac{11}{36} = \dfrac{{\left(60\right)}^{2} - \dfrac{1}{2}{\left(60 - a\right)}^{2} - \dfrac{1}{2}{\left(60 - a\right)}^{2}}{3600}$</p><p>$⇒ 1100 = 3600 - {\left(60 - a\right)}^{2}$</p><p>$⇒ {\left(60 - a\right)}^{2} = 2500 ⇒ 60 - a = 50$</p><p>$⇒ a = 10$