Biomolecules, enzyme kinetics

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2023 Q.60)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Functional Equations - Given the functional relation: $f(x) \cdot f(x + 1) = 3$ $f(x + 1) = \frac{3}{f(x)}$ **2. Detailed Step-by-Step Calculation**: - **Evaluate consecutive integer terms starting from $x = 2$**: - $f(2) = 5$ - $f(3) = \frac{3}{f(2)} = \frac{3}{5}$ - $f(4) = \frac{3}{f(3)} = \frac{3}{3/5} = 5$ - $f(5) = \frac{3}{f(4)} = \frac{3}{5}$ - **Observation of Periodicity**: The function oscillates with period 2 for integer values: - For all even integers $n$: $f(n) = 5$ - For all odd integers $n$: $f(n) = \frac{3}{5} = 0.6$ - Since $10$ is an even integer: $f(10) = 5$ **3. Final Answer**: - **Calculated Value**: **5** - **Official GATE Accepted Range**: **5 to 5**