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Question

cos2θ+2cosθ is always


A

greater than -32

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B

less than or equal to 32

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C

greater than or equal to -32 and less than or equal to 3

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D

None of these

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Solution

The correct option is C

greater than or equal to -32 and less than or equal to 3


Explanation for the correct option:

Step 1. Let f(θ)=cos2θ+2cosθ

=2cos2θ-1+2cosθ ; cos2θ=2cos2θ-1

Step 2. Differentiate f(θ) with respect to θ for minimum value:

f'θ=-4cosθsinθ-2sinθ

=0

cosθ=-12

minimum of f(θ)=2×-122-1+2×-12

=2×14-1-1

=-32

f(θ)-32

Step 3. when θ=0°, the given expression will be maximum.

That is, the maximum value will be 3.

Therefore, cos2θ+2cosθ is greater than or equal to -32 and less than or equal to 3.

Hence, Option ‘C’ is Correct.


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