The teacher hasn't uploaded a solution for this question yet.
AI Tutor Explanation
Powered by Gemini
Step-by-Step Solution
Recall the properties of definite integrals: Specifically, the property related to even and odd functions integrated over symmetric intervals.
Consider the case when f(x) is an odd function: An odd function satisfies the condition \(f(-x) = -f(x)\). The integral of an odd function over a symmetric interval \([-a, a]\) is zero. That is, \(\int_{-a}^{a}f(x)dx = 0\) if \(f(-x) = -f(x)\).
Consider the case when f(x) is an even function: An even function satisfies the condition \(f(-x) = f(x)\). The integral of an even function over a symmetric interval \([-a, a]\) is twice the integral from 0 to a. That is, \(\int_{-a}^{a}f(x)dx = 2\int_{0}^{a}f(x)dx\) if \(f(-x) = f(x)\). This is not equal to zero unless the integral from 0 to a is zero.
Analyze the other options: Options (C) and (D) involve the property \(f(a-x)\), which relates to symmetry about the line \(x = a/2\), and doesn't directly imply that the integral from -a to a is zero.
Conclusion: The correct condition for \(\int_{-a}^{a}f(x)dx = 0\) is when \(f(-x) = -f(x)\), which means f(x) is an odd function.
Correct Answer: f(-x)=-f(x)
AI Suggestion: Option B
AI generated content. Review strictly for academic accuracy.
Pedagogical Audit
Bloom's Analysis:
This is an UNDERSTAND question because it requires students to recall and explain the properties of definite integrals, specifically how symmetry affects the integral's value.
Knowledge Dimension:CONCEPTUAL
Justification:The question tests the understanding of concepts related to definite integrals and properties of even and odd functions. It requires the student to relate the symmetry of a function to the value of its definite integral over a symmetric interval.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly assesses the knowledge of properties of definite integrals as covered in the textbook.