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

NTA Abhyas 2020 · Mathematics · Trigonometric Equations

The equation tan^ 4 x - 2sec^ 2 x + a^ 2 = 0 will have at least one...

The equation tan4x−2sec2x+a2=0tan^{4}x - 2sec^{2}x + a^{2} = 0 will have at least one solution, if

Options

  1. A∣a∣≤4\left|a\right| \le 4
  2. B∣a∣≤2\left|a\right| \le 2
  3. C∣a∣≤3\left|a\right| \le \sqrt{3}
  4. D∣a∣>2\left|a\right| > 2

Correct answer

C. ∣a∣≤3\left|a\right| \le \sqrt{3}

Step-by-step solution

tan4x−2sec2x+a2=0tan^{4}x - 2sec^{2}x + a^{2} = 0
tan4x−2(1+tan2x)+a2=0tan^{4}x - 2\left(1 + tan^{2}x\right) + a^{2} = 0
tan4x−2tan2x+1=3−a2tan^{4}x - 2tan^{2}x + 1 = 3 - a^{2}
(tan2x−1)2≥0⇒3−a2≥0{\left(tan^{2}x - 1\right)}^{2} \ge 0 ⇒ 3 - a^{2} \ge 0
⇒a2−3≤0⇒∣a∣≤3⇒ a^{2} - 3 \le 0 ⇒ \left|a\right| \le \sqrt{3}
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