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JEE Main 202429 JanuaryEvening Shift
Exam
JEE Main
Year
2024
Date
29 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Trigonometric Equations

JEE Main 2024 · 29 January · Evening Shift · Mathematics · Trigonometric Equations

The sum of the solutions x ∈ R of the equation 3 2x + ^ 3 2x ^ 6 x...

The sum of the solutions x∈Rx \in R of the equation 3cos⁡2x+cos⁡32xcos⁡6x−sin⁡6x=x3−x2+6\dfrac{3\cos 2x + \cos^{3}2x}{\cos^{6}x - \sin^{6}x} = {x}^{3} - {x}^{2} + 6 is

Options

  1. A00
  2. B11
  3. C−1- 1
  4. D33

Correct answer

C. −1- 1

Step-by-step solution

Given: 3cos⁡2x+cos⁡32xcos⁡6x−sin⁡6x=x3−x2+6\dfrac{3\cos 2x + \cos^{3}2x}{\cos^{6}x - \sin^{6}x} = {x}^{3} - {x}^{2} + 6

⇒cos⁡2x(3+cos⁡22x)(cos⁡2x)3−(sin⁡2x)3=x3−x2+6⇒ \dfrac{\cos 2x\left(3 + \cos^{2}2x\right)}{{\left(\cos^{2}x\right)}^{3} - {\left(\sin^{2}x\right)}^{3}} = {x}^{3} - {x}^{2} + 6

⇒cos⁡2x(3+cos⁡22x)(cos⁡2x−sin⁡2x)(cos⁡4x+sin⁡4x+sin⁡2xcos⁡2x)=x3−x2+6⇒ \dfrac{\cos 2x\left(3 + \cos^{2}2x\right)}{\left(\cos^{2}x - \sin^{2}x\right)\left(\cos^{4}x + \sin^{4}x + \sin^{2}x\cos^{2}x\right)} = {x}^{3} - {x}^{2} + 6

⇒(3+cos⁡22x)(sin⁡2x+cos⁡2x)2−2sin⁡2xcos⁡2x+sin⁡2xcos⁡2x=x3−x2+6⇒ \dfrac{\left(3 + \cos^{2}2x\right)}{{\left(\sin^{2}x + \cos^{2}x\right)}^{2} - 2\sin^{2}x\cos^{2}x + \sin^{2}x\cos^{2}x} = {x}^{3} - {x}^{2} + 6

⇒(3+cos⁡22x)1−sin⁡2xcos⁡2x=x3−x2+6⇒ \dfrac{\left(3 + \cos^{2}2x\right)}{1 - \sin^{2}x\cos^{2}x} = {x}^{3} - {x}^{2} + 6

⇒(3+cos⁡22x)1−(2sin⁡xcos⁡x)24=x3−x2+6⇒ \dfrac{\left(3 + \cos^{2}2x\right)}{1 - \dfrac{{\left(2\sin x \cos x\right)}^{2}}{4}} = {x}^{3} - {x}^{2} + 6

⇒4(3+cos⁡22x4−sin⁡22x)=x3−x2+6⇒ 4\left(\dfrac{3 + \cos^{2}2x}{4 - \sin^{2}2x}\right) = {x}^{3} - {x}^{2} + 6

⇒4(3+1−sin⁡22x4−sin⁡22x)=x3−x2+6⇒ 4\left(\dfrac{3 + 1 - \sin^{2}2x}{4 - \sin^{2}2x}\right) = {x}^{3} - {x}^{2} + 6

⇒4=x3−x2+6⇒ 4 = {x}^{3} - {x}^{2} + 6

⇒x3−x2+2=0⇒ {x}^{3} - {x}^{2} + 2 = 0

⇒(x+1)(x2−2x+2)=0⇒ \left(x + 1\right)\left({x}^{2} - 2x + 2\right) = 0

⇒(x+1)((x−1)2+1)=0⇒ \left(x + 1\right)\left({\left(x - 1\right)}^{2} + 1\right) = 0

So, the sum of real roots is −1- 1.

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