GATE 2025 CH – Question 46

Engineering Mathematics · Numerical Methods: Algebraic equations, integration, ODEs and finite-difference methods · 2 marks · Numerical answer

The Newton-Raphson method is used to find the root of $f(x) \equiv x^2 - x - 1 = 0$. Starting with an initial guess $x^{(0)} = 1$, the second iterate $x^{(2)}$ is ____ (rounded off to 2 decimal places).

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Correct answer: 1.66 to 1.68

Explanation

Here $f'(x) = 2x - 1$. First step: $x^{(1)} = 1 - \frac{1 - 1 - 1}{2 - 1} = 2$. Second step: $x^{(2)} = 2 - \frac{4 - 2 - 1}{4 - 1} = 2 - \frac{1}{3} = 1.67$.