MathsCBSE Class 10 · Standard · Board exam 2023, set 30/6/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2023: question 33

Set in the CBSE Class 10 board exam of 2023, paper 30/6/1, section D, question 33, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33 · 5 marks · Long answerSurface Areas and Volumes

A solid is in the shape of a right-circular cone surmounted on a hemisphere, the radius of each of them being 7 cm and the height of the cone is equal to its diameter. Find the volume of the solid.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/6/1.

Hints

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

Hint 1

The height of the cone equals its diameter, which is 2 × 7.

Hint 2

Add $\frac{2}{3} π r^{3}$ and $\frac{1}{3} π r^{2} h$ , taking $\frac{1}{3} π r^{2}$ common.

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.

Radius of cone = radius of hemisphere $r = 7$ cm. Height of cone = diameter $= 14$ cm.
1 mark
Volume of solid = volume of hemisphere + volume of cone $= \frac{2}{3} π r^{3} + \frac{1}{3} π r^{2} h = \frac{2}{3} π (7)^{3} + \frac{1}{3} π (7)^{2} (14)$ .
3 marks
$= \frac{1}{3} \times \frac{22}{7} \times 7 \times 7 (14 + 14) = \frac{154}{3} \times 28 = \frac{4312}{3} \approx 1437.33$ cm³.
1 mark

Answer: Volume = $\frac{4312}{3}$ cm³ ≈ 1437.33 cm³

Where the marks are

Takes r = 7 cm and cone height 14 cm (1 mark)
Correct volume of hemisphere and of cone substituted (1½ + 1½ marks)
Final volume $\frac{4312}{3}$ cm³ ≈ 1437.33 cm³ (1 mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)
Topic: Volume 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 paper2023 board paper, set 30/6/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 33

Hints

Step-by-step solution

Revise the chapter

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