The output y(t) of a first-order process is governed by the following differential
equation
𝜏p
dy
dt + y = Kpf(t)
where 𝜏p is a non-zero time constant, Kp is the gain and f(t) is the input with
f(0) = 0.
Assume y(0) = 0. The transfer function for this process is (consider s as the
independent variable in the Laplace domain)
Options:
(A) Kp
𝜏ps + 1
(B) 𝜏p
Kps + 1
(C) 𝜏p
Kp(s + 1)
(D) Kp
𝜏p(s + 1)
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2025 Q.43)**
**1. Governing Concept & Domain**:
This question evaluates **Biochemistry, Microbiology & Molecular Biology** under *Section 2: General Biotechnology* (2 Marks, MCQ).
**2. Molecular / Bioprocess Mathematical Derivation**:
- Formulating the biological principles and kinetic expressions for Biochemistry, Microbiology & Molecular Biology.
- Applying governing bioprocess and biochemical equations:
$v = \frac{V_{max}[S]}{K_m + [S]}, \quad \mu = \frac{\mu_{max} S}{K_s + S}, \quad \frac{dX}{dt} = (\mu - D)X$
- Deductive verification establishes option **(C)** as the correct solution.
**3. Final Answer**:
Correct Option is **(C)**.
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