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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NTA Abhyas 2020 · Mathematics · Trigonometric Ratios & Identities

The difference between the greatest and the least possible value of...

The difference between the greatest and the least possible value of the expression 3−cosx+sin⁡2x3 - cos x + \sin^{2} x is&

Options

  1. A134\dfrac{13}{4}
  2. B174\dfrac{17}{4}
  3. C94\dfrac{9}{4}
  4. D14\dfrac{1}{4}

Correct answer

C. 94\dfrac{9}{4}

Step-by-step solution

3−cosx+1−cos⁡2x3 - cos x + 1 - \cos^{2} x
=−(cos⁡2x+cosx)+4= - \left(\cos^{2} x + cos x\right) + 4
=−(cos⁡2x+cosx+14)+4+14= - \left(\cos^{2} x + cos x + \dfrac{1}{4}\right) + 4 + \dfrac{1}{4}
=−(cosx+12)2+174= - {\left(cos x + \dfrac{1}{2}\right)}^{2} + \dfrac{17}{4}
=174−(cosx+12)2= \dfrac{17}{4} - {\left(cos x + \dfrac{1}{2}\right)}^{2}
Maximum value \left(atcos x = - \dfrac{1}{2}\right) = \dfrac{17}{4}<br/>Minimumvalue(atcosx=1)=174−94=2<br />Minimum value \left(atcos x = 1\right) = \dfrac{17}{4} - \dfrac{9}{4} = 2
Difference =174−2=94= \dfrac{17}{4} - 2 = \dfrac{9}{4}
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