Question

If the equation ${\cot }^{4}x-2{cosec}^{2}x+a^2=0$ has at least one solution , then the sum of all possible integral values of a is equal to

• A. $4$
• B. $3$
• C. $2$
• D. $0$

Answer 1

The correct option is D $0$

${\cot }^{4}x-2{cosec}^{2}x+a^2=0$
${\cot }^{4}x-2{\cot }^{2}x+a^2-2=0$
Put ${\cot }^{2}x=y$
$\Rightarrow y^2-2y+a^2-2=0$
For at least one solution ,
$D\ge 0$
$4-4(a^2-2)\ge 0$
$\Rightarrow a^2-2\le 1$
$\Rightarrow a^2-3\le 0$
$\Rightarrow a\in [-\sqrt{3},\sqrt{3}]$
Sum of all possible integral values $=-1+0+1=0$