Question

Consider, A, B, C be three sets of functions
$A=\left\{[x],\sqrt{1-x^2},\frac{1}{x-1},\frac{2}{1-\{x\}}\right\},$
$B=\left\{\tan x,\sin x+\cos x,\frac{3}{2+\cos x}\right\},$
$C=\left\{x^{\frac{1}{3}},{\tan }^{-1}x,\mathrm{sgn}(1+[x]+[-x])\right\}$
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 $(q-7p)$.
[Note: $[k],\{k\}$ and sgn $(k)$ denote greatest integer, fractional part and signum function of $k$ respectively.]

Answer 1

Answer: 2

$E_1$ :Composite number turns up.
$E_2:$ Prime number turns up.
$E_3$ : Neither prime nor composite number turns up.
A: Selected a derivable function

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