The number of real solutions of the equation | vedclass

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(B) Given the equation $2 \sin 3x + \sin 7x - 3 = 0$.This can be rewritten as $2 \sin 3x + \sin 7x = 3$.Since the maximum value of $\sin 	heta$ is $1

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$2$

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(B) Given the equation $2 \sin 3x + \sin 7x - 3 = 0$.This can be rewritten as $2 \sin 3x + \sin 7x = 3$.Since the maximum value of $\sin 	heta$ is $1$,the sum $2 \sin 3x + \sin 7x$ can be $3$ only if $\sin 3x = 1$ and $\sin 7x = 1$ simultaneously.For $\sin 3x = 1$,$3x = 2n\pi + \frac{\pi}{2} \Rightarrow x = \frac{2n\pi}{3} + \frac{\pi}{6} = \frac{(4n+1)\pi}{6}$ for $n \in \mathbb{Z}$.For $\sin 7x = 1$,$7x = 2m\pi + \frac{\pi}{2} \Rightarrow x = \frac{2m\pi}{7} + \frac{\pi}{14} = \frac{(4m+1)\pi}{14}$ for $m \in \mathbb{Z}$.Equating the two expressions for $x$: $\frac{4n+1}{6} = \frac{4m+1}{14}$ $\Rightarrow 14(4n+1) = 6(4m+1)$ $\Rightarrow 56n + 14 = 24m + 6$ $\Rightarrow 56n + 8 = 24m$ $\Rightarrow 7n + 1 = 3m$.Testing values for $n$ in the interval $[-2\pi, 2\pi]$ (i.e.,$x \in [-6.28, 6.28]$):If $n = 0, x = \frac{\pi}{6} \approx 0.52$ (Valid,$3m = 1$ not integer).If $n = 1, x = \frac{5\pi}{6} \approx 2.61$ (Valid,$3m = 8$ not integer).If $n = 2, x = \frac{9\pi}{6} = 1.5\pi$ (Valid,$3m = 15 \Rightarrow m = 5$).If $n = -2, x = -1.5\pi$ (Valid,$3m = -13$ not integer).Checking common solutions: $x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{9\pi}{6}, \dots$ and $x = \frac{\pi}{14}, \frac{5\pi}{14}, \dots$. The common solutions in $[-2\pi, 2\pi]$ are $x = \frac{3\pi}{2}$ and $x = -\frac{3\pi}{2}$.

The number of real solutions of the equation $2 \sin 3x + \sin 7x - 3 = 0$ which lie in the interval $[-2\pi, 2\pi]$ is

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