Question

If $2{\tan }^{2}x-5\sec x=1$ for exactly seven distinct values of $xϵ[0,\frac{n\pi }{2}],nϵN$, then find the greatest value of $n$.

Answer 1

Given,

$2{\tan }^{2}x-5\sec x=1$

$2({\sec }^{2}x-1)-5\sec x=1$

$2{\sec }^{2}x-2-5\sec x=1$

$2{\sec }^{2}x-5\sec x-3=0$

$2\sec x(\sec x-3)+1(\sec x-3)=0$

$(2\sec x+1)(\sec x-3)=0$

$\sec x=-\frac{1}{2},\sec x=3$

$=6\pi +\frac{\pi }{2}=\frac{13\pi }{2}$

$\frac{n\pi }{2}=\frac{13\pi }{2}$

$\therefore n=13$