MathsCBSE Class 12 · Board exam 2023, set 65/1/1 · 4 marks · Case-based

CBSE Class 12 Mathematics 2023: question 38

Set in the CBSE Class 12 board exam of 2023, paper 65/1/1, section E, question 38, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q38 · 4 marks · Case-basedCompetency-basedContinuity and Differentiability

The equation of the path traced by a roller-coaster is given by the polynomial $f (x) = a (x + 9) (x + 1) (x - 3)$ . If the roller-coaster crosses y-axis at a point $(0, - 1)$ , answer the following :

(i)
Find the value of 'a'.
2 marks
(ii)
Find $f^{''} (x)$ at $x = 1$ .
2 marks

Figure in the paper: Picture of a roller-coaster track drawn over coordinate axes, with a large arch-shaped hill on the left of the y-axis and smaller undulations to the right; y-axis marked with dots. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/1/1.

Hints

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

Hint 1

Crossing the y-axis at $(0, - 1)$ means $f (0) = - 1$ .

Hint 2

Expand the product of the three factors into a cubic before differentiating.

Hint 3

Differentiate twice and then put $x = 1$ .

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.

(i) The curve passes through $(0, - 1)$ , so $f (0) = - 1$ , i.e. $a (0 + 9) (0 + 1) (0 - 3) = - 1$ .
1 mark
(i) $- 27 a = - 1$ , so $a = \frac{1}{27}$ .
1 mark
(ii) $f (x) = \frac{1}{27} (x + 9) (x + 1) (x - 3) = \frac{1}{27} (x^{3} + 7 x^{2} - 21 x - 27)$ .
½ mark
(ii) $f^{'} (x) = \frac{1}{27} (3 x^{2} + 14 x - 21)$ .
½ mark
(ii) $f^{''} (x) = \frac{6 x + 14}{27}$ .
½ mark
(ii) $f^{''} (1) = \frac{6 + 14}{27} = \frac{20}{27}$ .
½ mark

Answer: (i) $a = \frac{1}{27}$ ; (ii) $f^{''} (1) = \frac{20}{27}$

Where the marks are

(i) Substitutes $(0, - 1)$ into f(x) (1 mark)
(i) $a = 1/27$ (1 mark)
(ii) Expands f(x) to $\frac{1}{27} (x^{3} + 7 x^{2} - 21 x - 27)$ (½ mark)
(ii) Correct $f^{'} (x)$ (½ mark)
(ii) Correct $f^{''} (x) = (6 x + 14) /27$ (½ mark)
(ii) $f^{''} (1) = 20/27$ (½ mark)

Revise the chapter

Chapter: Continuity and Differentiability (notes, formulas, where marks are lost and every past question)
Topic: Parametric and Second Order Derivatives

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 paper2023 board paper, set 65/1/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2023: question 38

Hints

Step-by-step solution

Revise the chapter

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