Which of the following statements is true for the function f(x) = cases x² + 3, & x ≠ 0 \\ 1, & x = 0 cases ? (A) f(x) is continuous and differentiable x R (B) f(x) is continuous x R (C) f(x) is continuous and differentiable x R - \0\ (D) f(x) is discontinuous at infinitely many points

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 1 mark · MCQ

CBSE Class 12 Mathematics 2024: question 3

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section A, question 3, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q3 · 1 mark · MCQContinuity and Differentiability

Which of the following statements is true for the function $f (x) = {x^{2} + 3, 1, x \neq = 0 x = 0$ ?

(A)f(x) is continuous and differentiable $\forall x \in R$
(B)f(x) is continuous $\forall x \in R$
(C)f(x) is continuous and differentiable $\forall x \in R - {0}$
(D)f(x) is discontinuous at infinitely many points

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/1/1.

Hints

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

Hint 1

Compare $lim_{x \to 0} f (x)$ with $f (0)$ .

Hint 2

Away from 0, the function is a polynomial.

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.

$lim_{x \to 0} f (x) = 0 + 3 = 3$ , but $f (0) = 1$ , so $f$ is discontinuous at $x = 0$ and therefore not differentiable there.
½ mark
For $x \neq = 0$ , $f (x) = x^{2} + 3$ is a polynomial, so it is continuous and differentiable. Hence (C) is true.
½ mark

Answer: (C)

Where the marks are

Shows the discontinuity at 0 and continuity elsewhere, giving (C) (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 paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 3

Hints

Step-by-step solution

Revise the chapter

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