MathsCBSE Class 12 · Sample paper, session 2023-24 · 1 mark · Assertion-reason

CBSE Class 12 Mathematics sample paper 2023-24: question 19

From the official CBSE sample question paper for 2023-24, 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-basedApplication of Derivatives

Let $f (x)$ be a polynomial function of degree 6 such that $\frac{d}{d x} (f (x)) = (x - 1)^{3} (x - 3)^{2}$ , then

ASSERTION (A): $f (x)$ has a minimum at $x = 1$ .

REASON (R): When $\frac{d}{d x} (f (x)) < 0, \forall x \in (a - h, a)$ and $\frac{d}{d x} (f (x)) > 0, \forall x \in (a, a + h)$ ; where 'h' is an infinitesimally small positive quantity, then $f (x)$ has a minimum at $x = a$ , provided $f (x)$ is continuous at $x = a$ .

(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 2023-24.

Hints

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

Hint 1

Check the sign of $f^{'} (x)$ just left and right of $x = 1$ .

Hint 2

Which test does R describe?

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.

Near $x = 1$ the factor $(x - 3)^{2}$ is positive, so the sign of $f^{'} (x)$ is the sign of $(x - 1)^{3}$ . Thus $f^{'} < 0$ just left of 1 and $f^{'} > 0$ just right of 1.
By the first derivative test, which is exactly R, $f$ has a minimum at $x = 1$ . Both statements are true and R explains A: option (A).
1 mark

Answer: Option (A)

Where the marks are

Shows the sign change of $f^{'}$ from $-$ to $+$ at $x = 1$ using R (1 mark)

Revise the chapter

Chapter: Application of Derivatives (notes, formulas, where marks are lost and every past question)
Topic: Maxima and Minima

Practise this chapterBoard-style questions from Application of Derivatives, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2023-24: every question, by section.

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CBSE Class 12 Mathematics sample paper 2023-24: question 19

Hints

Step-by-step solution

Revise the chapter

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