Class CBSE Class 12 Mathematics Continuity and Differentiability Q #795
KNOWLEDGE BASED
UNDERSTAND
1 Marks 2023 MCQ SINGLE
6. The function $f(x)=|x|$ is
(A) continuous and differentiable everywhere.
(B) continuous and differentiable nowhere.
(C) continuous everywhere, but differentiable everywhere except at $x=0$.
(D) continuous everywhere, but differentiable nowhere.
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Step-by-Step Solution

  1. Continuity: The function $f(x) = |x|$ is continuous everywhere. This can be visualized by its graph, which has no breaks.
  2. Differentiability: The function $f(x) = |x|$ is not differentiable at $x=0$ because the left-hand derivative and the right-hand derivative at $x=0$ are not equal. For $x < 0$, $f(x) = -x$, so $f'(x) = -1$. For $x > 0$, $f(x) = x$, so $f'(x) = 1$. At $x=0$, the derivative is undefined.
  3. Conclusion: Therefore, $f(x) = |x|$ is continuous everywhere, but differentiable everywhere except at $x=0$.

Correct Answer: continuous everywhere, but differentiable everywhere except at $x=0$<\/strong>

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because it requires students to recall the properties of the modulus function, specifically its continuity and differentiability.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the concept of continuity and differentiability of a modulus function. It requires the student to know the general properties of such functions rather than specific facts or procedures.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly assesses the understanding of continuity and differentiability, which are core concepts covered in the textbook.

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