Class CBSE Class 12 Mathematics Continuity and Differentiability Q #797
KNOWLEDGE BASED
UNDERSTAND
1 Marks 2023 MCQ SINGLE
The value of k for which function $f(x)=\begin{cases}x^{2},&x\ge0\\ kx,&x<0\end{cases}$ is differentiable at $x=0$ is:
(A) 1
(B) 2
(C) any real number
(D) 0
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AI Tutor Explanation

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Step-by-Step Solution

For the function to be continuous at x = 0, the left-hand limit (LHL) and the right-hand limit (RHL) must be equal.

RHL at x = 0: \( \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x^2 = 0^2 = 0 \)

LHL at x = 0: \( \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} kx = k \cdot 0 = 0 \)

Since both the LHL and RHL are equal to 0 regardless of the value of k, the function is continuous for any real number k.

Correct Answer: any real number

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because it requires the student to understand the concept of continuity and apply the definition of limits to determine the value of k.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the concept of continuity of a function at a point, which falls under conceptual knowledge.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly relates to the textbook content on continuity and differentiability.
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