The number of distinct solutions of the equation 5 4 cos 2 2 x + cos 4 x + sin 4 x + cos 6 x + sin 6 x = 2 in the interval [ 0 , 2] is [2015]

Question

The number of distinct solutions of the equation $\frac{5}{4} \cos^{2} 2 x + \cos^{4} x + \sin^{4} x + \cos^{6} x + \sin^{6} x = 2$ in the interval $[0, 2 π]$ is [2015]

Smart Achievers · Accepted Answer

(8)

$\frac{5}{4} \cos^{2} 2 x + \cos^{4} x + \sin^{4} x + \cos^{6} x + \sin^{6} x = 2$

$\Rightarrow \frac{5}{4} \cos^{2} 2 x + 1 - \frac{1}{2} \sin^{2} 2 x + 1 - \frac{3}{4} \sin^{2} 2 x = 2$

$\Rightarrow \frac{5}{4} (\cos^{2} 2 x - \sin^{2} 2 x) = 0 \Rightarrow \cos 4 x = 0$

$\Rightarrow 4 x = (2 n + 1) \frac{π}{2} or x = (2 n + 1) \frac{π}{8}$

$For x \in [0, 2 π], n can take values 0 to 7$

$Hence, there are 8 solutions.$

Solution

Q.
The number of distinct solutions of the equation $\frac{5}{4} \cos^{2} 2 x + \cos^{4} x + \sin^{4} x + \cos^{6} x + \sin^{6} x = 2$ in the interval $[0, 2 π]$ is [2015]

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Solution

Q. The number of distinct solutions of the equation 54cos22x+cos4x+sin4x+cos6x+sin6x=2 in the interval [0,2π] is [2015]

Ans. (8) 54cos22x+cos4x+sin4x+cos6x+sin6x=2 ⇒54cos22x+1-12sin22x+1-34sin22x=2 ⇒54(cos22x-sin22x)=0⇒cos4x=0 ⇒4x=(2n+1)π2 or x=(2n+1)π8 For x∈[0,2π], n can take values 0 to 7 Hence, there are 8 solutions.

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Q.
The number of distinct solutions of the equation $\frac{5}{4} \cos^{2} 2 x + \cos^{4} x + \sin^{4} x + \cos^{6} x + \sin^{6} x = 2$ in the interval $[0, 2 π]$ is [2015]