MathsCBSE Class 12 · Sample paper, session 2024-25 · 4 marks · Case-based

CBSE Class 12 Mathematics sample paper 2024-25: question 36

From the official CBSE sample question paper for 2024-25, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedApplication of Derivatives

Ramesh, the owner of a sweet selling shop, purchased some rectangular card board sheets of dimension $25 c m$ by $40 c m$ to make container packets without top. Let $x c m$ be the length of the side of the square to be cut out from each corner to give that sheet the shape of the container by folding up the flaps.
Based on the above information answer the following questions.

(i)
Express the volume (V) of each container as function of $x$ only.
1 mark
(ii)
Find $\frac{d V}{d x}$
1 mark
(iii)
(a) For what value of $x$ , the volume of each container is maximum?
2 marks
OR
(iii)
(b) Check whether V has a point of inflection at $x = \frac{65}{6}$ or not?
2 marks

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

After cutting squares of side $x$ from each corner, each side of the sheet shrinks by $2 x$ , and the height of the box is $x$ .

Hint 2

Expand $V$ before differentiating, then factorise $d V / d x$ to find the critical points. Check which values keep all dimensions positive.

Hint 3

A point of inflection is where $d^{2} V / d x^{2}$ changes sign. Check the sign of $4 (6 x - 65)$ on either side of $65/6$ .

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) After cutting squares of side $x$ and folding, the base is $(40 - 2 x)$ by $(25 - 2 x)$ and the height is $x$ . So $V = x (40 - 2 x) (25 - 2 x) = 4 x^{3} - 130 x^{2} + 1000 x$ cm³, where $0 < x < 12.5$ .
1 mark
(ii) Differentiate: $\frac{d V}{d x} = 12 x^{2} - 260 x + 1000 = 4 (3 x^{2} - 65 x + 250) = 4 (3 x - 50) (x - 5)$ .
1 mark
(iii)(a) For extreme values put $\frac{d V}{d x} = 0$ , i.e. $4 (3 x - 50) (x - 5) = 0$ .
½ mark
(iii)(a) This gives $x = 5$ or $x = \frac{50}{3}$ . Reject $x = \frac{50}{3}$ , since then $25 - 2 x < 0$ ; so $x = 5$ .
½ mark
(iii)(a) Second derivative: $\frac{d ^{2} V}{d x ^{2}} = 24 x - 260$ .
½ mark
(iii)(a) At $x = 5$ : $\frac{d ^{2} V}{d x ^{2}} = 120 - 260 = - 140 < 0$ . So $V$ is maximum when $x = 5$ cm.
½ mark
(iii)(b) We have $\frac{d V}{d x} = 4 (3 x^{2} - 65 x + 250)$ .
½ mark
(iii)(b) So $\frac{d ^{2} V}{d x ^{2}} = 4 (6 x - 65)$ , which is $0$ at $x = \frac{65}{6}$ . This point lies in $(0, 12.5)$ .
½ mark
(iii)(b) For $x$ slightly less than $\frac{65}{6}$ , $6 x - 65 < 0$ , so $\frac{d ^{2} V}{d x ^{2}} < 0$ (concave down). For $x$ slightly more than $\frac{65}{6}$ , $\frac{d ^{2} V}{d x ^{2}} > 0$ (concave up).
½ mark
(iii)(b) The concavity changes sign at this point. Hence $V$ has a point of inflection at $x = \frac{65}{6}$ .
½ mark

Answer: (i) $V = x (40 - 2 x) (25 - 2 x)$ cm³; (ii) $\frac{d V}{d x} = 4 (3 x - 50) (x - 5)$ ; (iii)(a) $x = 5$ cm; (iii)(b) Yes, $x = \frac{65}{6}$ is a point of inflection

Where the marks are

(i) Writes $V = x (40 - 2 x) (25 - 2 x)$ (1 mark)
(ii) Gets $d V / d x = 4 (3 x - 50) (x - 5)$ (1 mark)
(iii)(a) Sets $d V / d x = 0$ (1/2 mark)
(iii)(a) Finds the critical points $x = 5$ and $50/3$ , rejecting $50/3$ (1/2 mark)
(iii)(a) Finds $d^{2} V / d x^{2} = 24 x - 260$ (1/2 mark)
(iii)(a) Shows $d^{2} V / d x^{2} = - 140 < 0$ at $x = 5$ , so the volume is maximum (1/2 mark)
(iii)(b) Writes $d V / d x$ (1/2 mark)
(iii)(b) Shows $d^{2} V / d x^{2} = 4 (6 x - 65) = 0$ at $x = 65/6$ (1/2 mark)
(iii)(b) Shows the sign change of $d^{2} V / d x^{2}$ about $65/6$ (1/2 mark)
(iii)(b) Concludes it is a point of inflection (1/2 mark)

Revise the chapter

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

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 2024-25: 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 sample paper 2024-25: question 36

Hints

Step-by-step solution

Revise the chapter

More questions from Application of Derivatives