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If a = 2 Sin 2 X − Cos 2 X , Then a Lies in the Interval

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Question

If \[A = 2 \sin^2 x - \cos 2x\] , then A lies in the interval

Options

  • \[\left[ - 1, 3 \right]\]

  • \[\left[ 1, 2 \right]\] 

  • \[\left[ - 2, 4 \right]\]

  •  none of these 

MCQ
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Solution

\[\left[ - 1, 3 \right]\] 

\[A = 2 \sin^2 x - \cos2x\]

\[ = 2 \sin^2 x - \left( 1 - 2 \sin^2 x \right)\]

\[ = 4 \sin^2 x - 1\]

\[ \because 0 \leq \sin^2 x \leq 1\]

\[ \Rightarrow 4 \times 0 \leq 4 \times \sin^2 x \leq 4 \times 1\]

\[ \Rightarrow 0 \leq 4 \sin^2 x \leq 4\]

\[ \Rightarrow 0 - 1 \leq 4 \sin^2 x - 1 \leq 4 - 1\]

\[ \Rightarrow - 1 \leq 4 \sin^2 x - 1 \leq 3\]

\[ \Rightarrow - 1 \leq A \leq 3\]

\[ \Rightarrow A \in \left[ - 1, 3 \right]\]

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Values of Trigonometric Functions at Multiples and Submultiples of an Angle
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Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle - Exercise 9.5 [Page 44]

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R.D. Sharma Mathematics [English] Class 11
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.5 | Q 17 | Page 44

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