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GATE 2024 CE (CE1) – Question 11

Engineering Mathematics · Numerical Methods: Error analysis, algebraic equations, interpolation, differentiation, integration, ODEs · 1 mark · Multiple choice

The smallest positive root of the equation $x^5 - 5x^4 - 10x^3 + 50x^2 + 9x - 45 = 0$ lies in the range

  1. $0 < x \le 2$
  2. $2 < x \le 4$
  3. $6 \le x \le 8$
  4. $10 \le x \le 100$

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Correct answer: (A) $0 < x \le 2$

Explanation

Group the terms: $x^4(x - 5) - 10x^2(x - 5) + 9(x - 5) = (x - 5)(x^4 - 10x^2 + 9) = (x - 5)(x^2 - 1)(x^2 - 9)$. The roots are $\pm 1$, $\pm 3$ and 5. The smallest positive root is 1, which lies in $0 < x \le 2$.