GATE AG 2022 • Q14 NAT • 1 Mark

Section 1: Engineering Mathematics

Numerical Methods

Subtopic: Solutions of linear and non-linear algebraic equations

Question Statement:

The root of the equation sin x − 4x + 1 = 0 after its first iteration, using Newton-Raphson method with an initial guess of x₀ = 0.2, is __________. [round off to three decimal places]

Official Answer Key:

0.250 to 0.350

Step-by-Step Solution & Explanation

• Newton-Raphson: x₁ = x₀ − f(x₀)/f'(x₀); f(x)=sin x−4x+1, f'(x)=cos x−4 • f(0.2) = sin(0.2)−0.8+1 = 0.398669; f'(0.2) = cos(0.2)−4 = −3.019933 • x₁ = 0.2 − (0.398669/−3.019933) = 0.332015 Computed value: x₁ = 0.332 (within official range 0.250–0.350)

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