MathsCBSE Class 10 · Basic · Sample paper, session 2024-25 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic sample paper 2024-25: question 38

From the official CBSE sample question paper for 2024-25, 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-basedSurface Areas and Volumes

Rinku was very happy to receive a fancy jumbo pencil from his best friend Rohan on his birthday. Pencil is a basic writing tool, when sharpened its shape is a combination of cylinder & cone as given in the picture.
Cylindrical pencil with conical head is a common shape worldwide since ages. Commonly pencils are made up of wood & plastic but we should promote pencils made up of eco-friendly material (many options available in the market these days) to save environment.

The dimensions of Rinku's pencil are given as follows: Length of cylindrical portion is 21cm. Diameter of the base is 1 cm and height of the conical portion is 1.2 cm.

Based on the above information, answer the following questions:

(i)
Find the slant height of the sharpened part.
1 mark
(ii)
Find curved surface area of sharpened part (in terms of $π$ ).
1 mark
(iii)
(A) Find the total surface area of the pencil (in terms of $π$ ).
2 marks
OR
(iii)
(B) The pencil's total height decreases by 8.2 cm after sharpening it many times, what is the volume of the cylindrical part of the shortened pencil (in terms of $π$ )?
2 marks

Figure in the paper: A photograph of a jumbo pencil with a cylindrical body and a sharpened conical tip at one end. Dimensions given in text: cylindrical portion length 21 cm, base diameter 1 cm, conical portion height 1.2 cm. See it in the official paper.

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

The radius is half the diameter. Height, radius and slant height of the cone form a right triangle.

Hint 2

The curved surface of a cone is πrl and the curved surface of a cylinder is 2πrH.

Hint 3

The surface of the pencil has three parts: the cylinder's side, the cone's side and one flat circular end. For (B), only the cylindrical length changes.

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.

Given: radius $r = \frac{1}{2} = 0.5$ cm, cone height $h = 1.2$ cm, cylinder length $H = 21$ cm.
(i) Slant height: $l^{2} = h^{2} + r^{2} = (1.2)^{2} + (0.5)^{2} = 1.44 + 0.25 = 1.69$ .
½ mark
(i) $l = 1.69 = 1.3$ cm.
½ mark
(ii) Curved surface area of the cone: $π r l = π \times 0.5 \times 1.3$ .
½ mark
(ii) This gives $0.65 π$ cm².
½ mark
(iii)(a) TSA of pencil = CSA of cylinder + CSA of cone + area of the flat circular base $= 2 π r H + π r l + π r^{2}$ .
½ mark
(iii)(a) Substitute: $2 π (0.5) (21) + 0.65 π + π (0.5)^{2} = 21 π + 0.65 π + 0.25 π$ .
1 mark
(iii)(a) TSA $= 21.9 π$ cm².
½ mark
(iii)(b) The conical tip stays the same after sharpening, so the cylindrical part shortens by 8.2 cm. New length $= 21 - 8.2 = 12.8$ cm.
½ mark
(iii)(b) Volume of the cylinder: $π r^{2} H^{'} = π \times 0.5 \times 0.5 \times 12.8$ .
1 mark
(iii)(b) This gives $3.2 π$ cm³.
½ mark

Answer: (i) 1.3 cm; (ii) 0.65π cm²; (iii)(A) 21.9π cm²; (iii)(B) 3.2π cm³

Where the marks are

Uses $l^{2} = h^{2} + r^{2}$ with the correct values (½ mark)
Gets $l = 1.3$ cm (½ mark)
Uses $π r l$ for the cone (½ mark)
Gets $0.65 π$ cm² (½ mark)
(iii)(A): writes TSA as cylinder CSA + cone CSA + base area (½ mark)
(iii)(A): substitutes correctly, with cylinder CSA $= 21 π$ (1 mark)
(iii)(A): gets $21.9 π$ cm² (½ mark)
(iii)(B): finds the new cylinder length 12.8 cm (½ mark)
(iii)(B): substitutes into $π r^{2} H$ (1 mark)
(iii)(B): gets $3.2 π$ cm³ (½ mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)
Topic: Surface Area of a Combination of Solids

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

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CBSE Class 10 Mathematics Basic sample paper 2024-25: question 38

Hints

Step-by-step solution

Revise the chapter

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