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Prove that 7 - 3√(5) is… (Class 10 Maths Basic 2024, Q28)

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MathsCBSE Class 10 · Basic · Board exam 2024, set 430/2/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2024: question 28

Set in the CBSE Class 10 board exam of 2024, paper 430/2/1, section C, question 28, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28 · 3 marks · Short answerReal Numbers

Prove that is an irrational number, given that is an irrational number.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/2/1.

Hints

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

Hint 1

Use proof by contradiction: start by supposing the number is rational.

Hint 2

Rearrange the equation so that is alone on one side.

Hint 3

What kind of number is the other side? Does this agree with the given fact?

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.

  1. Assume, to the contrary, that is rational. Then for some rational number .

    1 mark

  2. Then , so .

    1 mark

  3. Since is rational, is rational, so would be rational. This contradicts the given fact that is irrational. So our assumption is wrong, and is irrational.

    1 mark

Answer: is irrational (proved by contradiction)

Where the marks are

  • Assumes is rational (1 mark)
  • Expresses (1 mark)
  • States the contradiction and concludes the number is irrational (1 mark)

Revise the chapter

Practise this chapterBoard-style questions from Real Numbers, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 430/2/1 (Basic): 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.