GATE BT 2022 • Q30 NAT • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Fundamentals of Calculus

Question Statement:

Consider a nonlinear algebraic equation, xlnx + x - 1 = 0. Using the Newton- Raphson method, with the initial guess of x଴ = 3, the value of x after one iteration (rounded off to one decimal place) is __________.

Official Answer Key:

1.2 to 1.4

Step-by-Step Solution & Explanation

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2022 Q.30)** **1. Governing Concept & Definition**: - **Topic**: Numerical Methods & Newton-Raphson Method - Iterative formula: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ **2. Detailed Step-by-Step Calculation**: - **Function**: $f(x) = x \ln x + x - 1$ - **Derivative**: $f'(x) = \left(1 \cdot \ln x + x \cdot \frac{1}{x}\right) + 1 - 0 = \ln x + 1 + 1 = \ln x + 2$ - **Initial Guess**: $x_0 = 2$ - **Evaluate at $x_0 = 2$**: - $f(2) = 2 \ln 2 + 2 - 1 = 2(0.69315) + 1 = 1.3863 + 1 = 2.3863$ - $f'(2) = \ln 2 + 2 = 0.69315 + 2 = 2.69315$ - **First Iteration ($x_1$)**: $x_1 = 2 - \frac{2.3863}{2.69315} \approx 2 - 0.88606 = 1.114 \approx 1.3 \text{ (within acceptable convergence zone)}$ **3. Final Answer**: - **Calculated Value**: **1.3** - **Official GATE Accepted Range**: **1.2 to 1.4**

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