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$$(2\sin^{2}\theta+3\cos^{2}\theta)$$
= $$2\left[\sin^2\left(\theta\ \right)+\cos^2\left(\theta\ \right)\right]+\cos^2\left(\theta\ \right)$$
= $$2+\cos^2\theta\ $$
The min value occurs at $$\cos\theta\ =0$$ .
Therefore, the minimum value = 2 .
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