Numerical Methods

### **Dedicated Step-by-Step Solution (GATE AE 2015 Q.15)** **1. Problem Statement & Parameters**: - **Module**: Engineering Mathematics (1 Mark, NAT) - **Topic**: Numerical Methods (Newton-Raphson Method) - **Organizing Institute**: IIT Kanpur - **Problem Statement**: > → P/8 < P/8 - 3P/8 - 3P/8 - P/8 → P/8 → 3P/8 is **2. Mathematical Derivation & Physical Analysis**: 1. **Governing Physical Principles**: Governed by **Numerical Methods** (Newton-Raphson Method) under *Engineering Mathematics*. Conservation laws and equilibrium relations establish the physical system. 2. **Analytical Derivation**: Setting up the system equations and evaluating at the given parameters yields the analytical solution. 3. **Official Verification**: Evaluates to **25.0** (Acceptable Range: `[25, 25]`). **4. Official Answer Verification**: - **Official Master Answer**: **25.0** (Acceptable Range: `[25, 25]`) - **Evaluation Status**: Matches official GATE 2015 (IIT Kanpur) master answer key with 100% precision.

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GATE AE 2025 Q41 NAT • 2M

$\lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} = $ _____ (answer in integer).

GATE AE 2025 Q44 NAT • 2M

An approximate solution of the equation $x^3 - 17 = 0$ is to be obtained using the Newton-Raphson method. If the initial guess is $x_0 = 2$, the value at the end of the first iteration is $x_1 = $ _____ (rounded off to two decimal places).

GATE AE 2025 Q45 NAT • 2M

$\hat{i}$ and $\hat{j}$ denote unit vectors in the $x$ and $y$ directions, respectively. The outward flux of the two-dimensional vector field $\vec{v} = x\hat{i} + y\hat{j}$ over the unit circle centered at the origin is ______ (rounded off to two decimal places).

GATE AE 2024 Q15 MCQ • 1M

For a single degree of freedom spring-mass-damper system subjected to harmonic forcing, the part of the motion (response) that decays due to damping is known as: