If the equation cot 4 x-2 operatornamecosec2 x+a2=0 has atleast one solution then, sum of all possible integral values of ' a ' is equal to

Question

If the equation cot 4 x-2 operatornamecosec2 x+a2=0 has atleast one solution then, sum of all possible integral values of ' a ' is equal to (A) 4 (B) 3 (C) 2 (D) 0

Tardigrade Admin · Accepted Answer

cot 4 x-2(1+ cot 2 x)+a2=0 ⇒ cot 4 x-2 cot 2 x+a2-2=0 ⇒ ( cot 2 x-1)2=3-a2 to have atleast one solution 3-a2 ≥ 0 ⇒ a2-3 ≤ 0 a ∈[-√3, √3] integral values -1,0,1 ∴ text sum =0

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

4

3

2

0

$cot^{4} x - 2 (1 + cot^{2} x) + a^{2} = 0$
$\Rightarrow cot^{4} x - 2 cot^{2} x + a^{2} - 2 = 0$
$\Rightarrow (cot^{2} x - 1)^{2} = 3 - a^{2}$
to have atleast one solution
$3 - a^{2} \geq 0$
$\Rightarrow a^{2} - 3 \leq 0$
$a \in [- 3, 3]$
integral values $- 1, 0, 1$
$∴ sum = 0$

Q. If the equation cot4x−2cosec2x+a2=0 has atleast one solution then, sum of all possible integral values of ' a ' is equal to

4

3

2

0

cot4x−2(1+cot2x)+a2=0 ⇒cot4x−2cot2x+a2−2=0 ⇒(cot2x−1)2=3−a2 to have atleast one solution 3−a2≥0 ⇒a2−3≤0 a∈[−3​,3​] integral values −1,0,1 ∴ sum =0

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

$cot^{4} x - 2 (1 + cot^{2} x) + a^{2} = 0$
$\Rightarrow cot^{4} x - 2 cot^{2} x + a^{2} - 2 = 0$
$\Rightarrow (cot^{2} x - 1)^{2} = 3 - a^{2}$
to have atleast one solution
$3 - a^{2} \geq 0$
$\Rightarrow a^{2} - 3 \leq 0$
$a \in [- 3, 3]$
integral values $- 1, 0, 1$
$∴ sum = 0$