Question

If the equation ${\cot }^{4}x-2{\text{cosec}}^{2}x+a^2=0$ has at least one real solution in $x$, then the number of possible integral values of $a$ is

• A. $4$
• B. $3$
• C. $2$
• D. $0$

Answer 1

The correct option is B $3$

${\cot }^{4}x-2{\text{cosec}}^{2}x+a^2=0$
$\Rightarrow {\cot }^{4}x-2(1+{\cot }^{2}x)+a^2=0$
$\Rightarrow {\cot }^{4}x-2{\cot }^{2}x+a^2-2=0$
$\Rightarrow ({\cot }^{2}x-1{)}^2=3-a^2$

For real solution,
$3-a^2\ge 0$
$\Rightarrow a^2-3\le 0$
$\Rightarrow a\in [-\sqrt{3},\sqrt{3}]$
Integral values are $-1,0,1$