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. 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 h→0^-(f(c+h)-f(c))/(h)≠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.

MathsCBSE Class 12 · Sample paper, session 2024-25 · 1 mark · Assertion-reason

CBSE Class 12 Mathematics sample paper 2024-25: question 19

From the official CBSE sample question paper for 2024-25, section A, question 19, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q19 · 1 mark · Assertion-reasonCompetency-basedContinuity and Differentiability

Assertion (A): Consider the function defined as $f (x) = ∣ x ∣ + ∣ x - 1∣$ , $x \in R$ . Then $f (x)$ is not differentiable at $x = 0$ and $x = 1$ .

Reason (R): Suppose $f$ be defined and continuous on $(a, b)$ and $c \in (a, b)$ , then $f (x)$ is not differentiable at $x = c$ if $lim_{h \to 0^{-}} \frac{f ( c + h ) - f ( c )}{h} \neq = lim_{h \to 0^{+}} \frac{f ( c + h ) - f ( c )}{h}$ .

(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.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

Open one hint at a time and try again before the next.

Hint 1

Write $f$ piecewise on $x < 0$ , $0 \leq x < 1$ , $x \geq 1$ .

Hint 2

Compare slopes on either side of 0 and 1.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

At $x = 0$ : LHD $= - 2$ , RHD $= 0$ ; at $x = 1$ : LHD $= 0$ , RHD $= 2$ . So $f$ is not differentiable at 0 and 1; (A) true.
(R) is the definition-based criterion (unequal one-sided derivatives) used to prove (A). Option (A).
1 mark

Answer: Option (A)

Where the marks are

Shows unequal LHD and RHD at 0 and 1 and links to (R) (1 mark)

Revise the chapter

Chapter: Continuity and Differentiability (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Continuity and Differentiability, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25: every question, by section.

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CBSE Class 12 Mathematics sample paper 2024-25: question 19

Hints

Step-by-step solution

Revise the chapter

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