The equation `tan^4x-2sec^2x+a=0` will have at least one solution if `1

Options

A. `|a|le4`
B. `|a| le 2`
C. ` |a| le sqrt(3)`
D. None of the above

Answer

C

Explanation

Given equation is
`tan^(4)x - 2 sec^(2)x + a^(2) = 0`
`rArr ( tan ^(2)x)^(2)- 2 [1+ tan^(2)x] + a^(2) = 0 `
`rArr (tan ^ (2)x)^(2) - 2 tan ^(2) x + a^(2) - 2 = 0`
This is the quadratic equation in tan x
So , roots of this will be real when discriminant `ge 0 " i.e., " b^(2) - 4c ge 0 `
`rArr (-2)^(2)-4(1) (a^(2)-2) ge 0 rArr 4- 4a^(2) + 8 ge0`
`rArr 12- 4a^(2) ge 0 rArr 4(3-a^(2)) ge 0`
`rArr 3- a^(2) ge 0 " "( :' 4 != 0)`
`rArr a^(2) le 3`
`rArr a le pm sqrt(3) rArr |a| le sqrt(3) `