If sum of the roots of the equation 2 (x+(π)/(4))( ^2 x-(1+ 3 ) x+2+ 3 )= 9+18 x · x in [0,2 π] is (p)/(q) π where p, q ∈ N , then find the least value of |p-q| .

Correct answer: 25 aligned & (25)( x+ x)( ^2 x-(1+ 3 ) x+3+ 3 )=3| x+ x| & Case-1: x+ x ≥ 0 & ( x+ x)( ^2 x-(1+ 3 ) x+ 3 )=0 & x=-1,1, 3 & x=(3 π)/(4), (π)/(4), (π)/(3), (7 π)/(4) & Case-2: x+ x < 0 & ( x+ x) ( ^2 x-(1+ 3 ) x+6+ 3 ) _ no solution =0 & aligned aligned & x=-1 x=(3 π)/(4), (7 π)/(4) & Required sum =(37 π)/(12) aligned

JEE Advanced

Exam: JEE Advanced
Subject: Mathematics
Chapter: Trigonometry

JEE Advanced · Mathematics · Trigonometry

If sum of the roots of the equation 2 (x+(π)/(4))( ^2 x-(1+ 3 )...

If sum of the roots of the equation

\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)\left(\sec ^2 x-(1+\sqrt{3}) \tan x+2+\sqrt{3}\right)=\sqrt{9+18 \sin x \cdot \cos x} \text { in }[0,2 \pi] \text { is } \frac{p}{q} \pi

where

p, q \in N

, then find the least value of

|p-q|

Correct answer

Step-by-step solution

\begin{aligned} & (25)(\sin x+\cos x)\left(\tan ^2 x-(1+\sqrt{3}) \tan x+3+\sqrt{3}\right)=3|\sin x+\cos x| \\ & \text { Case-1: } \sin x+\cos x \geq 0 \\ & (\sin x+\cos x)\left(\tan ^2 x-(1+\sqrt{3}) \tan x+\sqrt{3}\right)=0 \\ & \tan x=-1,1, \sqrt{3} \\ & \Rightarrow \quad x=\frac{3 \pi}{4}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{7 \pi}{4} \\ & \text { Case-2: } \sin x+\cos x < 0 \\ & (\sin x+\cos x) \underbrace{\left(\tan ^2 x-(1+\sqrt{3}) \tan x+6+\sqrt{3}\right)}_{\text {no solution }}=0 \\ & \end{aligned}

\begin{aligned} & \tan x=-1 \Rightarrow x=\frac{3 \pi}{4}, \frac{7 \pi}{4} \\ & \therefore \text { Required sum }=\frac{37 \pi}{12}\end{aligned}

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