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JEE Advanced 2006
Exam
JEE Advanced
Year
2006
Subject
Mathematics
Chapter
Trigonometric Equations

JEE Advanced 2006 · Mathematics · Trigonometric Equations

The set of values of θ satisfying the inequation 2 ^2 θ-5 θ+2>0 ,...

The set of values of θ\theta satisfying the inequation 2sin⁡2θ−5sin⁡θ+2>02 \sin ^2 \theta-5 \sin \theta+2>0, where 0<θ<2π0 < \theta < 2 \pi, isThe set of values of θ satisfying the inequation 2 ^2 θ-5 θ+2>0 , where 0 < θ < 2 π , isThe set of values of θ satisfying the inequation 2 ^2 θ-5 θ+2>0 , where 0 < θ < 2 π , is

Options

  1. A(0,π6)∪(5π6,2π)\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right)
  2. B[0,π6]∪[5π6,2π]\left[0, \frac{\pi}{6}\right] \cup\left[\frac{5 \pi}{6}, 2 \pi\right]
  3. C[0,π3]∪[2π3,2π]\left[0, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, 2 \pi\right]
  4. DNone of these

Correct answer

A. (0,π6)∪(5π6,2π)\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right)

Step-by-step solution

 As, 2sin⁡2θ−5sin⁡θ+2>0⇒(2sin⁡θ−1)(sin⁡θ−2)>0[ where, (sin⁡θ−2)<0 for all θ∈R]∴(2sin⁡θ−1)<0\begin{aligned} \text{ As, }2 \sin ^2 \theta-5 \sin \theta+2>0 \\ & \Rightarrow(2 \sin \theta-1)(\sin \theta-2)>0 \\ & \quad[\text{ where, }(\sin \theta-2) < 0 \text{ for all } \theta \in \mathbb{R}] \\ & \therefore \quad(2 \sin \theta-1) < 0 \end{aligned}The set of values of θ satisfying the inequation 2 ^2 θ-5 θ+2>0 , where 0 < θ < 2 π , is\\⇒sin⁡θ<12, shown as ∴θ∈(0,π6)∪(5π6,2π)\begin{array}{ll} \Rightarrow& \sin \theta< \frac{1}{2}, \text{ shown as }\\ \therefore & \theta \in\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6},2 \pi\right) \end{array}$$
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