Subtopic: Fundamentals of Downstream processing (chromatography, filtration)
Question Statement:
Let m and n be fixed real numbers. If the function y(t) = C1et + C2e-t is a
solution of
d2y
dt2 + m dy
dt + ny = 0
for any constants C1 and C2, then m + n is equal to
Options:
(A) -2
(B) -1
(C) 0
(D) 1
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2025 Q.45)**
**1. Governing Concept**:
- **Topic**: Downstream Processing & Centrifugal Sedimentation (Stokes' Law)
- The terminal settling velocity ($v_g$) of a spherical particle of diameter $d_p$ and density $\rho_p$ in a fluid of density $\rho_f$ and viscosity $\mu$ under gravity is given by **Stokes' Law**:
$v_g = \frac{d_p^2 (\rho_p - \rho_f) g}{18 \mu}$
- In a centrifuge with centrifugal acceleration $\omega^2 r = G g$:
$v_c = \frac{d_p^2 (\rho_p - \rho_f) \omega^2 r}{18 \mu}$
- Sedimentation velocity is directly proportional to the square of particle diameter ($d_p^2$) and the density difference $(\rho_p - \rho_f)$, and inversely proportional to fluid viscosity ($\mu$).
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Settling velocity increases with the square of diameter ($d_p^2$), not linearly.
- **Option (B) is CORRECT**: Sedimentation velocity is inversely proportional to the liquid viscosity ($\mu$).
- **Option (C) is INCORRECT**: Velocity is proportional to the density difference $(\rho_p - \rho_f)$, not the sum.
- **Option (D) is INCORRECT**: Settling velocity increases with centrifugal field ($\omega^2 r$), not decreases.
**3. Final Answer**:
- **Correct Choice**: **B**
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.