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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 kcos x - 3sin x = k + 1 is solvable only if

The equation kcosx−3sinx=k+1kcos x - 3sin x = k + 1 is solvable only if

Options

  1. Ak∈(−∞,4)k \in \left(- \infty,4\right)
  2. Bk∈(−∞,4]k \in \left(- \infty,4\right]
  3. Ck∈(4,∞)k \in \left(4, \infty\right)
  4. Dk∈[4,∞)k \in \left[4, \infty\right)

Correct answer

B. k∈(−∞,4]k \in \left(- \infty,4\right]

Step-by-step solution

kcos x - 3sin x = k + 1<br/><br/>Forasolution∣k+1∣≤k2+(−3)2<br /><br />For a solution \left|k + 1\right|\le \sqrt{k^{2} + {\left(- 3\right)}^{2}}<br/><br/><br /><br />⇒ \left|k + 1\right|\le \sqrt{k^{2} + 9}<br/><br/>Onsquaringbothsides,weget,<br/><br/><br /><br />On squaring both sides, we get,<br /><br />k^{2} + 1 + 2k \le k^{2} + 9<br/><br/><br /><br />⇒ 2k \le 8 ⇒ k \le 4<br/><br/><br /><br />⇒ k \in \left(- \infty,4\right]
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