The number of roots of ex + 0.5x2 - 2 = 0 in the range [-5, 5] is A) 0 B) 1 C) 2 D) 3

Official answer: C) 2. Explanation: Let f(x) = ex + 0.5x2 - 2. 1. First and Second Derivatives: f'(x) = ex + x f''(x) = ex + 1 Since ex > 0 for all real x, f''(x) = ex + 1 > 0 for all x R. Because f(x) is strictly convex everywhere, it can have at most 2 real roots. 2. Evaluating f(x) at specific points: - f(-5) = e-5 + 0.5(-5)2 - 2 0.0067 + 12.5 - 2 = 10.5067 > 0 - f(0) = e0 + 0 - 2 = 1 - 2 = -1 0 3. Applying Intermediate Value Theorem: - Since f(-5) > 0 and f(0) 0, there exists at least one root in (0, 2) [-5, 5]. Since f(x) can have at most 2 real roots, there are exactly 2 roots in the interval [-5, 5]. Hence, option (C) is correct.

General Aptitude · Calculus

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GATE CSE 2017 Set 2 Q64 · MCQ · 2 marks

The number of roots of $e^x + 0.5x^2 - 2 = 0$ in the range $[-5, 5]$ is

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In General Aptitude, Calculus questions test the fundamental qualitative properties of functions, such as convexity, derivative tests, and root counting via the Intermediate Value Theorem. Rather than requiring complex symbolic integration or tedious differentiation, items focus on finding the exact number of real roots within a specified interval by analyzing sign changes and curvature.

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

General Aptitude · Calculus

Official IIT answer key · IIT Roorkee · Audited Aug 2026

Take topic test

GATE CSE 2017 Set 2 Q64 · MCQ · 2 marks

The number of roots of $e^x + 0.5x^2 - 2 = 0$ in the range $[-5, 5]$ is

Select an option to check

Key concept

Calculus

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Calculus

Full Calculus guide →

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

The number of roots of $e^x + 0.5x^2 - 2 = 0$ in the range $[-5, 5]$ is

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The number of roots of $e^x + 0.5x^2 - 2 = 0$ in the range $[-5, 5]$ is

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The number of roots of ex+0.5x2−2=0e^x + 0.5x^2 - 2 = 0ex+0.5x2−2=0 in the range [−5,5][-5, 5][−5,5] is

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The number of roots of ex+0.5x2−2=0e^x + 0.5x^2 - 2 = 0ex+0.5x2−2=0 in the range [−5,5][-5, 5][−5,5] is

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Check you can still solve Calculus under a clock

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The number of roots of $e^x + 0.5x^2 - 2 = 0$ in the range $[-5, 5]$ is

The number of roots of $e^x + 0.5x^2 - 2 = 0$ in the range $[-5, 5]$ is