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JEE Mathematics 2024 Question with Solution

Let f(x)f(x) be defined as:

f(x)=x−2, for −2<x<0f(x) = x - 2, \text{ for } -2 < x < 0f(x)=1−2x, for 0≤x≤2f(x) = 1 - 2x, \text{ for } 0 \le x \le 2

Let h(x)=f(x)+f(x)h(x) = f(x) + f(x). Then the integral of h(x) dxh(x)\,dx is equal to:

  • A

    22

  • B

    44

  • C

    11

  • D

    66

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: The question defines a piecewise function f(x)f(x) and asks for an integral involving h(x)=f(x)+f(x)h(x) = f(x) + f(x).

Find: The required value of the integral.

The solution is unrelated to this question. It discusses the domain of a composite function f(g(x))f(g(x)) instead of the given piecewise function and integral. Therefore, no valid step-by-step working for this question could be extracted from the solution.

Since the solution content is mismatched, the answer is taken from the provided correct answer field. The correct option is A.

Common mistakes

  • Treating the solution as if it belongs to this question. That is wrong because the extracted solution discusses a different composite-function domain problem. Use only the given piecewise definition of f(x)f(x) for this question.

  • Misreading h(x)=f(x)+f(x)h(x) = f(x) + f(x) as a composition such as f(f(x))f(f(x)) or as two different functions. It means h(x)=2f(x)h(x) = 2f(x), so the integrand must be formed directly from the given piecewise function.

  • Ignoring the piecewise intervals while integrating. That is wrong because f(x)f(x) has different expressions on different parts of its domain. The integral, if definite over the full domain, must be split according to the interval boundaries.

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