GATE BT 2016 • Q35 NAT • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Fundamentals of Calculus

Question Statement:

The equilibrium potential of a biological membrane for Na + is 55 mV at 37 ⁰C. Concentration of Na+ inside the cell is 20 mM. Assuming the membrane is permeable to Na+ only, the Na+ concentration outside the membrane will be ______________ mM. (Faraday constant: 23062 cal.V-1.mol-1, Gas constant: 1.98 cal.mol-1.K-1)

Official Answer Key:

1.65 - 1.75

Step-by-Step Solution & Explanation

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2016 Q.35)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Definite Integrals via Partial Fractions - Evaluating: $I = \int_0^{0.9} \frac{dx}{(1 - x)(2 - x)}$ **2. Detailed Step-by-Step Calculation**: - **Partial Fraction Decomposition**: $\frac{1}{(1 - x)(2 - x)} = \frac{1}{1 - x} - \frac{1}{2 - x}$ - **Integrate**: $\int \left( \frac{1}{1 - x} - \frac{1}{2 - x} \right) dx = -\ln(1 - x) + \ln(2 - x) = \ln\left(\frac{2 - x}{1 - x}\right)$ - **Evaluate at Limits ($0$ to $0.9$)**: - At $x = 0.9$: $\ln\left(\frac{2 - 0.9}{1 - 0.9}\right) = \ln\left(\frac{1.1}{0.1}\right) = \ln(11)$ - At $x = 0$: $\ln\left(\frac{2 - 0}{1 - 0}\right) = \ln(2)$ - **Definite Integral Value**: $I = \ln(11) - \ln(2) = \ln\left(\frac{11}{2}\right) = \ln(5.5) \approx 1.7047 \approx 1.70$ **3. Final Answer**: - **Calculated Value**: **1.70** - **Official GATE Accepted Range**: **1.65 to 1.75**

Practice This in Real CBT Exam Interface

Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.

Launch CBT Mock Test