- Amylase: This is a polysaccharide found in plants, with \( \alpha \)-1,4 and \( \alpha \)-1,6 linkages between glucose units, so the correct linkage type is \( \alpha \)-C1-C4 plant (IV).
- Cellulose: Cellulose is a polysaccharide made of \( \beta \)-1,4 linkages, commonly found in plants, so the correct linkage is \( \beta \)-C1-C4 plant (I).
- Glycogen: Glycogen is a polysaccharide found in animals, consisting of \( \alpha \)-1,4 and \( \alpha \)-1,6 linkages between glucose units, so the correct linkage is \( \alpha \)-C1-C4 \( \alpha \)-C1-C6 plant (III).
- Amylopectin: Similar to glycogen, it is found in plants and is made of \( \alpha \)-1,4 and \( \alpha \)-1,6 linkages, so the correct linkage is \( \alpha \)-C1-C4 plant (II). Thus, the correct answer is (2).
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: