Q27. Prove that √5 is an irrational number.

Assuming √5 = a/b with a, b coprime and b ≠ 0 leads to 5 dividing both a and b, contradicting that they are coprime. Hence the assumption is wrong and √5 is an irrational number.

Q28. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

∠APB+∠AOB=180°, so the angle between the tangents is supplementary to the angle the chord of contact subtends at the centre. Hence proved.

CBSE Class 10 Maths (Standard) 2023 — Paper 30/4/1

80 marks · 3 hours · 38 questions (44 with the OR choices)

3 of its 44 questions have been set in other papers too (each one says where); the other 41 are only in this paper.

Take this paper as a timed mock

Free with an account: the board's time limit, marking at the end, and the answer to every question.

Questions 1–20 · 1 mark each

Q1 1 mark

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Q2 1 mark

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Q3 1 mark

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Q4 1 mark

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Q5 1 mark

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Q6 1 mark

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Q7 1 mark

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Q8 1 mark

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Q9 1 mark

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Q10 1 mark

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Q11 1 mark

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Q12 1 mark

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Q13 1 mark

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Q14 1 mark

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Q15 1 mark

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Q16 1 mark

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Q17 1 mark

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Q18 1 mark

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Q19 1 mark

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Q20 1 mark

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Questions 21–25 · 2 marks each

Q21 2 marks

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Q22 2 marks

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Q23(a) 2 marks

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Q23(b) 2 marks

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Q24 2 marks

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Q25(a) 2 marks

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Q25(b) 2 marks

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Questions 26–31 · 3 marks each

Q26(a) 3 marks

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OR

Q26(b) 3 marks

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Q27 3 marks

Also in 2026 · 30/2/1 (Q26), 2026 · 30/3/1 (Q28), 2026 · 430/3/1 (Q26), 2026 · 430/4/1 (Q29), 2025 · 30/6/1 (Q26(a)), 2025 · 30(B) (Q26), 2025 · 430/2/1 (Q26(a)), 2024 · 30/2/1 (Q27), 2024 · 30(B)/S (Q29), 2024 · 430/S/1 (Q26(a)), 30/1/1 (Q27), 30/C/B (Q26), 430/C/1 (Q29(b)), 2020 · 30/2/1 (Q35), 2020 · 30/4/1 (Q27), 2019 · 30/5/1 (Q13)

Prove that $5$ is an irrational number.

Show answer

Assuming $5 = \frac{a}{b}$ with $a, b$ coprime and $b \neq = 0$ leads to $5$ dividing both $a$ and $b$ , contradicting that they are coprime. Hence the assumption is wrong and $5$ is an irrational number.

Step-by-step solution →
Q28 3 marks

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Show answer

$∠ A P B + ∠ A O B = 18 0^{\circ}$ , so the angle between the tangents is supplementary to the angle the chord of contact subtends at the centre. Hence proved.

Step-by-step solution →
Q29(a) 3 marks

Also in 2018 · 30/1/1 (Q27), 2018 · 30/2/1 (Q30), 2018 · 30/3/1 (Q25)

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OR

Q29(b) 3 marks

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Q30 3 marks

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Q31 3 marks

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Questions 32–35 · 5 marks each

Q32(a) 5 marks

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Q32(b) 5 marks

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Q33(a) 5 marks

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Q33(b) 5 marks

Also in 2024 · 430/2/1 (Q34(b)), 430/C/1 (Q34)

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Q34 5 marks

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Q35 5 marks

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Questions 36–38 · 4 marks each

Q36 4 marks

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Q37 4 marks

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Q38 4 marks

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Other 2023 papers

All CBSE Class 10 Maths papers →