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Question

# If A=4sinΘ+cos2Θ,then which of the following is true?

A
maximum value of A is 5.
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B
minimum value of A is 4
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C
maximum value of A occurs when sinΘ=12
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D
minimum value of A occurs when sinΘ=1
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Solution

## The correct option is B minimum value of A is −4Consider f(x)=4sinx+cos2x f(x)=4sinx+1−sin2(x) Therefore f(x)=−sin2(x)+4sin(x)+1 Now sin(x)ϵ[−1,1] Thus the function must attain a minimum value at sin(x)=−1. That is , at x=−π2,3π2 Hence f(−π2)=f(3π2)=−1+4(−1)+1 =−4 Hence minimum value of the function is −4. Similarly, it must attain a maximum value at x=π2. Therefore f(π2)=−1+4+1=4 Thus the range of f(x) is [−4,4].

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