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Question

If P=4sinx+cos2x, then which of the following is/are true ?

A
Maximum value of P is 5
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B
Minimum value of P is 4
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C
Maximum value of P occurs when sinx=0
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D
Minimum value of P occurs when sinx=1
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Solution

The correct options are
B Minimum value of P is 4
D Minimum value of P occurs when sinx=1
P=4sinx+cos2xP=4sinx+(1sin2x)
Assuming sinx=tt[1,1]
P=4t+1t2f(t)=(t2)2+5, t[1,1]
x coordinate of vertex is 2 and 2[1,1]
f(1)=4
f(1)=4
The mimimum value of P is 4 at sinx=1
The maximum value of P is 4 at sinx=1

Alternate Solution:
P=4sinx+cos2xP=4sinx+(1sin2x)P=(sin2x4sinx+4)+5
P=5(sinx2)2

1sinx13sinx211(sinx2)299(sinx2)2145(sinx2)244P4

P=45(sinx2)2=4sinx2=±1sinx=1 (sinx3)

P=45(sinx2)2=4sinx2=±3sinx=1 (sinx5)

The mimimum value of P is 4 when sinx=1.

The maximum value of P is 4 when sinx=1.

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