Question

# Let $$\alpha$$ be the root of the equation $$25\cos^{2}\theta + 5\cos \theta - 12 = C$$, where $$\dfrac {\pi}{2} < \alpha < \pi$$.What is $$\sin 2\alpha$$ equal to?

A
2425
B
2425
C
512
D
2125

Solution

## The correct option is D $$\dfrac{-24}{25}$$Consider that $$\cos\theta=x$$$$\Rightarrow 25x^2+5x-12=0$$$$\Rightarrow 25x^2+20x-15x-12=0$$$$\Rightarrow 5x(5x+4)-3(5x+4)=0$$$$\Rightarrow (5x-3)(5x+4)=0$$$$\Rightarrow x=-\dfrac{4}{5}$$ or $$\dfrac{3}{5}$$Now, as $$\alpha\in\left(\dfrac{\pi}{2},\pi\right)$$, we have $$\cos\alpha=-\dfrac{4}{5}$$$$\Rightarrow \sin2\alpha=2\sin\alpha\cos\alpha=2\times-\dfrac{4}{5}\times\dfrac{3}{5}=-\dfrac{24}{25}$$This is the required solution.Mathematics

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