GATE BT 2026 • Q51 NAT • 2 Marks

Section 2: General Biotechnology

Biomolecules, enzyme kinetics

Subtopic: Fundamentals of Biomolecules, enzyme kinetics

Question Statement:

Consider the equation dy dx = 1 x The value of the integral ∫ dy 2 1 using trapezoidal method and interval ℎ = 0.25 is ________. (rounded off to two decimal places)

Official Answer Key:

0.69 to 0.71

Step-by-Step Solution & Explanation

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2026 Q.51)** **1. Governing Concept & Definition**: - **Topic**: Numerical Methods & Trapezoidal Rule of Numerical Integration - The composite trapezoidal rule for step size $h$ is: $\int_a^b f(x)\,dx \approx \frac{h}{2} \left[ f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]$ **2. Detailed Step-by-Step Calculation**: - **Function**: $f(x) = \frac{1}{x}$ over $[1, 2]$ with $h = 0.25$. - **Grid Points**: - $x_0 = 1.00 \implies f(x_0) = 1.0000$ - $x_1 = 1.25 \implies f(x_1) = \frac{1}{1.25} = 0.8000$ - $x_2 = 1.50 \implies f(x_2) = \frac{1}{1.50} = 0.6667$ - $x_3 = 1.75 \implies f(x_3) = \frac{1}{1.75} = 0.5714$ - $x_4 = 2.00 \implies f(x_4) = \frac{1}{2.00} = 0.5000$ - **Trapezoidal Summation**: $\text{Integral} \approx \frac{0.25}{2} [1.0000 + 2(0.8000 + 0.6667 + 0.5714) + 0.5000]$ $= 0.125 \times [1.0000 + 2(2.0381) + 0.5000] = 0.125 \times [1.5000 + 4.0762] = 0.125 \times 5.5762 \approx 0.6970$ **3. Final Answer**: - **Calculated Value**: **0.70** - **Official GATE Accepted Range**: **0.69 to 0.71**

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