Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Trigonometry

JEE Advanced · Mathematics · Trigonometry

If the largest possible value of x such that 0 < x < π satisfying...

If the largest possible value of xx such that 0<x<π0 < x < \pi satisfying the equation 2sin⁡2x+1cos⁡x+sin⁡x=2\frac{2 \sin 2 x+1}{\cos x+\sin x}=\sqrt{2} is aπb\frac{a \pi}{b}. Find (a+b)(a+b)

Correct answer

37

Step-by-step solution

2((cos⁡x+sin⁡x)2−1)=2(cos⁡x+sin⁡x)C+S=t2t2−1=2t⇒2t2−2t−1=0t=2±2+84=2±10412(cos⁡x+sin⁡x)=2(1−5)4sin⁡(x+π4)=−(5−14)x+π4=−π10 or 11π10x=11π10−π4=17π20≡aπb⇒a+b=37\begin{aligned} 2\left((\cos x+\sin x)^2-1\right) & =\sqrt{2}(\cos x+\sin x) \\ C+S & =t \\ 2 t^2-1 & =\sqrt{2} t \Rightarrow 2 t^2-\sqrt{2} t-1=0 \\ t & =\frac{\sqrt{2} \pm \sqrt{2+8}}{4}=\frac{\sqrt{2} \pm \sqrt{10}}{4} \\ \frac{1}{\sqrt{2}}(\cos x+\sin x) & =\frac{\sqrt{2}(1-\sqrt{5})}{4} \\ \sin \left(x+\frac{\pi}{4}\right) & =-\left(\frac{\sqrt{5}-1}{4}\right) \\ x+\frac{\pi}{4} & =\frac{-\pi}{10} \text { or } \frac{11 \pi}{10} \\ x & =\frac{11 \pi}{10}-\frac{\pi}{4}=\frac{17 \pi}{20} \equiv \frac{a \pi}{b} \Rightarrow a+b=37\end{aligned}
Practice Trigonometry on Quantrex Academy →

More from Trigonometry

The number of real root(s) of the equation x^2 x=1 lie(s) between 0 and 2 π is/are Most Important Selected Qs for JEE AdvancedIf |x^2-(π)/(4)+ x-2|=|x^2-2|+| x|+(π)/(4) then Most Important Selected Qs for JEE AdvancedIf 3 x+ (2 x-(7 π)/(6))=-2 , then x is equal to ( where m ∈ I) Most Important Selected Qs for JEE AdvancedWhat is the minimum value of ( ^4 α)/( ^2 β)+( ^4 β)/( ^2 α) for α, B ≠ (k π)/(2), k ∈ Z ? Most Important Selected Qs for JEE AdvancedIf x+ y ≥ α x x ∈ R then y+ α is equal to Most Important Selected Qs for JEE AdvancedIf A, B are the roots of the quadratic equation a b x^2-c^2 x+a b=0 where a, b, c are… Most Important Selected Qs for JEE AdvancedIf θ and - θ are the roots of the equation a x^2-b x-c=0 , where a, b, c are the sides of… Most Important Selected Qs for JEE AdvancedIf 2 ^2((π)/(2) ^2 x)=1- (π 2 x), x ≠(2 n+1) (π)/(2), n ∈ I , then Most Important Selected Qs for JEE AdvancedConsider the equation 1+ 3 2 x ^4 x+4 ^2 x + ^4 x+4 ^2 x = 2 ^ -1 ( 2 x) x ∈(-π, π) Most Important Selected Qs for JEE AdvancedSet of values of x of the equation 2 x= ( ^3 x- ^3 x) is - Most Important Selected Qs for JEE AdvancedIf 4+ 8- 32+ 768 =a 2 \ left((11 π)/(b)), where a and b are natural numbers then (b)/(a)… Most Important Selected Qs for JEE AdvancedIf a x+ x+1=cosec x then the possible values of x can be Most Important Selected Qs for JEE AdvancedFor the equation 2( x- 2 x)- 2 x(1+2 x)+2 x=0, x ∈[0,2 π]- Most Important Selected Qs for JEE AdvancedWhich of the following set of values of x satisfies the equation 2^ (2 ^2 x-3 x+1) +2^… Most Important Selected Qs for JEE AdvancedIf α satisfies the equation 2 2 ^3 x-54 2 ^3 x=19 , then possible value of ( .2 ^2 α+ 2… Most Important Selected Qs for JEE AdvancedIf f(x)=(3)/(1+ ^2 x)+(9)/(1+ ^2 x) , then : Most Important Selected Qs for JEE Advanced Full Trigonometry list Trigonometry JEE Maths PYQsJEE Mathematics PYQs All Most Important Selected Qs for JEE Advanced PYQs