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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Trigonometric Equations

NTA Abhyas 2020 · Mathematics · Trigonometric Equations

The number of solutions of the equation (3 + cos x) ^ 2 = 4 - 2 ^ 8...

The number of solutions of the equation (3+ cos x){ \left(3 +\text{ cos } x\right)}^{2} = 4 - 2\sin^{8} xin \left[0, 9\pi\right)is equal to

Options

  1. A44
  2. B55
  3. C66
  4. D77

Correct answer

A. 44

Step-by-step solution

−1≤cosx≤1⇒2≤3+cosx≤4- 1 \le cos x \le 1 ⇒2 \le 3 + cos x \le 4
⇒4≤(3+cosx)2≤16⇒ 4 \le {\left(3 + cos x\right)}^{2} \le 16
Also, −1≤sinx≤1- 1 \le sin x \le 1
⇒0≤2sin⁡8x≤2⇒ 0 \le 2\sin^{8} x \le 2
⇒2≤4−2sin⁡8x≤4⇒ 2 \le 4 - 2\sin^{8} x \le 4 so, equality is possible when LHS=RHS=4LHS = RHS = 4
⇒cosx=−1⇒ cos x = - 1 and sinx=0sin x = 0
⇒x=(2n+1)π⇒ x = ( 2n + 1 ) \pi
⇒x=π,3π,5π,7π⇒ x = \pi, 3\pi, 5\pi, 7\pi
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