Question

Consider, A, B, C be three sets of functions
$A=\left\{[x], \sqrt{1-x^2}, \frac{1}{x-1}, \frac{2}{1-\{x\}}\right\},$
$B=\left\{\tan x, \sin x+\cos x, \frac{3}{2+\cos x}\right\}, $
$C=\left\{x^{\frac{1}{3}}, \tan ^{-1} x, \operatorname{sgn}(1+[x]+[-x])\right\}$
A normal dice is thrown once, if it turns up a composite number, a function is selected from the set A, if it shows up a prime number, a function is selected from the set B, else a function is selected from set C. If the selected function is found to be a derivable in its domain and the probability it was selected from set $A$, is $\frac{p}{q}$, where $p, q \in N$, then find the least value of $(q-7 p)$.
[Note: $[ k ],\{ k \}$ and sgn $( k )$ denote greatest integer, fractional part and signum function of $k$ respectively.]

Answer 1

$ E _1$ :Composite number turns up.
$E _2:$ Prime number turns up.
$E _3$ : Neither prime nor composite number turns up.
A: Selected a derivable function

$\text { Required probability }=\frac{\frac{2}{6} \times \frac{1}{4}}{\frac{2}{6} \times \frac{1}{4}+\frac{3}{6} \times 1+\frac{1}{6} \times \frac{1}{3}}=\frac{3}{23} $
$\therefore q-7 p=23-21=2 $