GATE BT 2025 • Q33 NAT • 1 Mark

Section 2: General Biotechnology

Biomolecules, enzyme kinetics

Subtopic: Fundamentals of Biomolecules, enzyme kinetics

Question Statement:

Consider a nonlinear algebraic equation, ex - 2 = 0. Using the Newton-Raphson method, with the initial guess of x0 = 1, the approximated value of the root of the equation after one iteration is________. (Round off to two decimal places)

Official Answer Key:

0.72 to 0.75

Step-by-Step Solution & Explanation

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2025 Q.33)** **1. Governing Concept & Definition**: - **Topic**: Numerical Methods & Newton-Raphson Method - For a non-linear algebraic equation $f(x) = 0$, the Newton-Raphson iterative scheme is: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ **2. Detailed Step-by-Step Calculation**: - **Given Equation**: $f(x) = e^x - 2 = 0$ $f'(x) = e^x$ - **Initial Guess**: $x_0 = 1$ - **Evaluate at $x_0 = 1$**: $f(1) = e^1 - 2 = e - 2 \approx 2.71828 - 2 = 0.71828$ $f'(1) = e^1 \approx 2.71828$ - **First Iteration ($x_1$)**: $x_1 = 1 - \frac{e - 2}{e} = 1 - \left(1 - \frac{2}{e}\right) = \frac{2}{e}$ $x_1 = \frac{2}{2.71828} \approx 0.735759 \approx 0.74$ **3. Final Answer**: - **Calculated Value**: **0.74** - **Official GATE Accepted Range**: **0.72 to 0.75**

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