KVPY Sample Paper KVPY Stream-SX Model Paper-24

The number of all positive integral roots of the equation, \[px=p+15x;\,\]\[p\in N\]is-

A) 3

B) 4

C) 5

D) none of these

Correct Answer: B

Solution :

\[x=\frac{p}{p-15}\]
\[x=1+\frac{15}{p-15}\]	?? (1)
Roots are positive \[\Rightarrow \frac{15}{p-15}>0\]i.e. \[p\in (-\,\infty ,0)\cup (15,\infty )\] But \[p\in N\]
\[\therefore p\in (15,\infty )\]
From (1), for x to be integer \[\frac{15}{p-15}\]Should be integer, it is possible, When \[0<p-15\le 15\] or \[0<p\le 30\] But \[p>15\]
\[\therefore \,\,\,p\in \,\,(15,\,\,\,30]\] So, \[p=16,\,\,18,\,\,20,\,\,30\]
\[\therefore \] No. of positive integral roots are 4.