GATE AG 2026 • Q50 NAT • 2 Marks

Section 1: Engineering Mathematics

Numerical Methods

Subtopic: Solutions of linear and non-linear algebraic equations

Question Statement:

While solving an equation e⁻ˣ − x = 0 for x, using the Newton-Raphson method with initial guess x₀ = 2, the solution after the first iteration (x₁) is ________ (Rounded off to two decimal places)

Official Answer Key:

0.33 to 0.38

Step-by-Step Solution & Explanation

f(x) = e⁻ˣ − x ⇒ f′(x) = −e⁻ˣ − 1 Newton-Raphson formula: x₁ = x₀ − f(x₀)/f′(x₀) At x₀ = 2: f(2) = e⁻² − 2 = 0.13534 − 2 = −1.86466 f′(2) = −e⁻² − 1 = −0.13534 − 1 = −1.13534 x₁ = 2 − (−1.86466)/(−1.13534) = 2 − 1.64239 = 0.36 SECTION 2: FARM MACHINERY

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