Question

If the equation in $x$ given by $2\left({\frac{1}{\cos ^{-1}x}}\right)^{2 \pi}-a+\frac{1}{2} \left(2{\frac{1}{\cos ^{-1} x}}\right)^\pi$ $- a ^2=0$ has only one real solution then exhaustive set of values of ' $a$ ' is

• A. $\left(- 3 ,1\right)$
• B. $\left[- \infty , - 3\right]\cup\left[1 , \infty\right]$
• C. $\left(\right.-\infty ,-3\left.\right)\cup\left(\right.1,\infty\left.\right)$
• D. $\left[- 3 , \infty\right]$

Answer 1

Answer: (B)

Let $\frac{\pi }{2^{cos^{- 1} x}}=t\Rightarrow t\geq 2$
equation becomes $t^{2}-\left(a + \frac{1}{2}\right)t-a^{2}=0$
has one roots $2$ or greater than $2$ and other root less than $2,f\left(2\right)\leq 0$
$\Rightarrow 4-\left(a + \frac{1}{2}\right)2-a^{2}\leq 0$
$a^{2}+2a-3\geq 0$
$\left(a + 3\right)\left(a - 1\right)\geq 0$
$a\leq -3$ or $a\geq 1$