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Question

# If A=2sin θ+cos2θ, then which of the following is/are true?

A

Maximum value of A=5.
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B

Minimum value of A=2.
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C

Maximum value of A occurs when sin θ=1.
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D

Minimum value of A occurs when sin θ=1.
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Solution

## The correct option is D Minimum value of A occurs when sin θ=−1.For solving these types of questions, we first try to simplify the expression. Here, A can be simplified as: A=2sin θ+cos2θ⇒A=2sin θ+1−sin2θ⇒A=−(sin2θ−2sin θ−1]⇒A=−(sin2θ−2sin θ−1+1−1]⇒A=−(sin2θ−2sin θ+1]+2⇒A=2−(1−sin θ )2Now the maximum value of A occurs when (1−sin θ)2 is minimum. And the minimum value of (1−sin θ)2 occurs when sin θ=1, then the maximum value of A is 2−(1−1)2=2. Also, A will be minimum when (1−sin θ)2 is maximum. And (1−sin θ)2 is maximum at sin θ=−1 Thus, the minimum value of A=2−(1−(−1))2=2−22=−2 Hence, Options b. c. and d. are correct.

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