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JEE Main 202131 AugustMorning Shift
Exam
JEE Main
Year
2021
Date
31 August
Shift
Morning Shift
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

JEE Main 2021 · 31 August · Morning Shift · Mathematics · Trigonometric Ratios & Identities

cosec 18 ° is a root of the equation:

cosec18°cosec 18 ° is a root of the equation:

Options

  1. Ax2−2x−4=0{x}^{2} - 2x - 4 = 0
  2. B4x2+2x−1=04{x}^{2} + 2x - 1 = 0
  3. Cx2+2x−4=0{x}^{2} + 2x - 4 = 0
  4. Dx2−2x+4=0{x}^{2} - 2x + 4 = 0

Correct answer

A. x2−2x−4=0{x}^{2} - 2x - 4 = 0

Step-by-step solution

We know that

sin⁡18°=5−14\sin 18 ° = \dfrac{\sqrt{5} - 1}{4}

cosec18°=1sin⁡18°=45−1cosec 18 ° = \dfrac{1}{\sin 18 °} = \dfrac{4}{\sqrt{5} - 1}

Rationalizing denominator

=45−1×5+15+1= \dfrac{4}{\sqrt{5} - 1} \times \dfrac{\sqrt{5} + 1}{\sqrt{5} + 1}

=4(5+1)4= \dfrac{4\left(\sqrt{5} + 1\right)}{4}

=5+1= \sqrt{5} + 1

Hence equation x=5+1x = \sqrt{5} + 1

⇒(x−1)2=5⇒ ( x - 1{)}^{2} = 5

⇒x2−2x+1=5⇒ {x}^{2} - 2x + 1 = 5
⇒x2−2x−4=0⇒ {x}^{2} - 2x - 4 = 0

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