given-y-sin-pi-3-2x-2cos-pi-12-x-1-where-x-0-2pi-has-maximum-and-minimum-value-is-p-and-q-find-p-2-q-2-A-18-B-16-C-63-4-D-16-

Question Number 81354 by jagoll last updated on 12/Feb/20

given y = \sin (\frac{π}{3} - 2 x) + 2 \cos (\frac{π}{12} + x) + 1

where x \in (0, 2 π) has maximum and

minimum value is p and q .

find p^{2} - q^{2} ?

(A) - 18 (B) - 16

(C) \frac{63}{4} (D) 16

Commented by john santu last updated on 12/Feb/20

y = \sin (- 2 x + \frac{π}{3}) + 2 \cos (x + \frac{π}{12}) + 1

\sin (- 2 x + \frac{π}{3}) = \cos (\frac{π}{2} - (- 2 x + \frac{π}{3}))

\cos (2 x + \frac{π}{6}) = \cos 2 (x + \frac{π}{12})

l e t x + \frac{π}{12} = θ

y = \cos 2 θ + 2 \cos θ + 1

y = 2 \cos^{2} θ + 2 \cos θ

f o r \cos θ = 1 \Rightarrow f_{1} = 4 (m a x)

f o r \cos θ = - 1 \Rightarrow f_{2} = 0

f o r \cos θ = - \frac{1}{2} \Rightarrow f_{3} = 2 \times \frac{1}{4} + (- 1) = - \frac{1}{2} (m i n)

p^{2} - q^{2} = (p + q) (p - q) = \frac{7}{2} \times \frac{9}{2} = \frac{63}{4}

Commented by jagoll last updated on 12/Feb/20

thank you

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