The number of distinct real roots of |(cosec x,sec x,sec x)(sec x cosec x sec x)(sec x, sec x, cosec x)| = 0 lies in the interval -π/4 ≤​ x ≤ π/4 is

Question

Mark the correct alternative in the following :

The number of distinct real roots of \(\begin{vmatrix} cosec\,x &sec\,x & sec\,x \\[0.3em] sec\,x & cosec\,x & sec\,x \\[0.3em] sec\,x &sec\,x & cosec\,x \end{vmatrix}\) = 0 lies in the interval \(-\frac{\pi}{4}≤x≤\frac{\pi}{4}\) is

A. 1 

B. 2 

C. 3 

D. 0

Answer 1

Correct answer : (A)

∆ = \(\begin{vmatrix} cosec\,x &sec\,x & sec\,x \\[0.3em] sec\,x & cosec\,x & sec\,x \\[0.3em] sec\,x &sec\,x & cosec\,x \end{vmatrix}\)

C₁ → C₁ + C₂ + C₃

∆ = \(\begin{vmatrix} cosec\,x+2\,sec\,x &sec\,x & sec\,x \\[0.3em] 2\,sec\,x+cosec\,x & cosec\,x & sec\,x \\[0.3em] 2\,sec\,x +cosec\,x&sec\,x & cosec\,x \end{vmatrix}\)

∆ = \(\begin{vmatrix} 1 &sec\,x & sec\,x \\[0.3em] 1 & cosec\,x & sec\,x \\[0.3em] 1&sec\,x & cosec\,x \end{vmatrix}\)

∆ = (cosec x+ 2sec x )[(cosec x - sec x)²] 

Case1 : 

(cosecc x + 2sec x ) = 0

tan x = \(-\frac{1}{2}\)

(1^st real root)

Case : 2 

(cosec x - sec x)² =0 

tan x = 1 

(2^nd real root)