Question 19 [Assertion Reasoning] - CBSE Class 12 Sample Paper for 2025 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

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Question 19 Statement A Assertion (A): Consider the function defined as 𝑓(𝑥)=|𝑥|+|𝑥−1|,𝑥∈𝑅. Then 𝑓(𝑥) is not differentiable at 𝒙=𝟎 and 𝒙=𝟏. Statement R Reason (R): Suppose 𝑓 be defined and continuous on (𝑎,𝑏) and 𝑐∈(𝑎,𝑏), then 𝑓(𝑥) is not differentiable at 𝑥=𝑐 if lim_(ℎ→0^− ) (𝑓(𝑐+ℎ)−𝑓(𝑐))/ℎ≠lim_(ℎ→0^+ ) (𝑓(𝑐 + ℎ)−𝑓(𝑐))/ℎ. (A) Both (A) and (R) are true and (R) is the correct explanation of (A). (B) Both (A) and ( 𝑅 ) are true but (𝑅) is not the correct explanation of (𝐴). (C) (A) is true but (R) is false. (D) (A) is false but (𝑅) is true.Checking Assertion Assertion (A): Consider the function defined as 𝑓(𝑥)=|𝑥|+|𝑥−1|,𝑥∈𝑅. Then 𝑓(𝑥) is not differentiable at 𝒙=𝟎 and 𝒙=𝟏. Given 𝑓(𝑥)= |𝑥|+ |𝑥−1|. Here, we have 2 critical points x = 0 and x − 1 = 0 i.e. x = 0, and x = 1 Let’s draw its graph From graph, we can see that At x = 0 and x = 1, It is an edge point And, the function is not differentiable at edge points. Since at x = 0, and x = 1 is not differentiable ∴ Assertion is true Checking Reason Reason (R): Suppose 𝑓 be defined and continuous on (𝑎,𝑏) and 𝑐∈(𝑎,𝑏), then 𝑓(𝑥) is not differentiable at 𝑥=𝑐 if lim_(ℎ→0^− ) (𝑓(𝑐+ℎ)−𝑓(𝑐))/ℎ≠lim_(ℎ→0^+ ) (𝑓(𝑐 + ℎ)−𝑓(𝑐))/ℎ.. Here, reasoning is describing the differentiability test lim_(ℎ→0^− ) (𝑓(𝑐+ℎ)−𝑓(𝑐))/ℎ is Left hand derivative at x = c lim_(ℎ→0^+ ) (𝑓(𝑐 + ℎ)−𝑓(𝑐))/ℎ is Right hand derivative at x = c And, since the function is not differentiable at x = c, LHD ≠ RHD This is true Hence, Reason is true Is Reason a Correct explanation for Assertion? Since Derivative test can be used to check differentiability at x = 0, and x = 1 Thus, we used the concept mentioned in Reasoning to check Assertion Therefore, Reasoning is a correct explanation for Assertion So, Assertion is true Reasoning is true And, Reasoning is a correct explanation for Assertion So, the correct answer is (a)