Question

The equation $2{\tan }^{2}x-5\sec x=1$ holds true for exactly eleven distinct values of $x\in [0,\frac{n\pi }{2}],n\in N$. The greatest value of $n$ is

• A. $21$
• B. $22$
• C. $24$
• D. $23$

Answer 1

The correct option is D $23$

$2{\sec }^{2}x-2-5\sec x=1$
$\Rightarrow 2{\sec }^{2}x-5\sec x-3=0$
$\Rightarrow \sec x=3$
$[\because \sec x\ge 1\text{and}\sec x\le -1]$
$\Rightarrow \cos x=\frac{1}{3}$
It has two solutions in $[0,2\pi ]$
So, Eleven solutions in $[0,\frac{21\pi }{2}]$

Now, we have to find the greatest interval for which the equation has $11$ solutions
Eleven solutions in $[0,\frac{23\pi }{2}]$

Hence the maximum value of $n=23$