Q31(a). Prove that the lengths of tangents drawn from an external point to a circle are equal.

In right triangles OQP and ORP: OQ = OR (radii), OP is common and ∠OQP = ∠ORP = 90°, so △OQP ≅ △ORP by RHS; hence PQ = PR by CPCT, i.e. the tangents drawn from an external point to a circle are equal in length.

Q32(b). From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.

Height of the tower =7(√3+1) m ≈ 19.12 m.

CBSE Class 10 Maths (Basic) 2023 — Paper 430/4/1

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

1 of its 44 questions has been set in other papers too (each one says where); the other 43 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

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Questions 21–25 · 2 marks each

Q21 2 marks

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Q22(a) 2 marks

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Q22(b) 2 marks

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Q23(a) 2 marks

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Q23(b) 2 marks

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Questions 26–31 · 3 marks each

Q26 3 marks

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Q27 3 marks

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Q28(a) 3 marks

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Q29 3 marks

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Q31(a) 3 marks

Also in 2026 · 30/2/1 (Q29(a)), 2026 · 430/1/1 (Q30), 2024 · 430(B)/S (Q32), 2020 · 430(B) (Q37(a)), 2018 · 30/1/1 (Q18), 2018 · 30/2/1 (Q15), 2018 · 30/3/1 (Q16), 2017 · 30/1/1 (Q22), 2017 · 30(B) (Q24), 2017 · 530/1 (Q24), 2016 · 30/1/1 (Q22), 2016 · 30/2/1 (Q22), 2016 · 30(B) (Q24), 2016 · 530/1 (Q24), 2015 · 30(B) (Q24)

Prove that the lengths of tangents drawn from an external point to a circle are equal.

Show answer

In right triangles $O QP$ and $O R P$ : $O Q = O R$ (radii), $O P$ is common and $∠ O QP = ∠ O R P = 9 0^{\circ}$ , so $△ O QP ≅ △ O R P$ by RHS; hence $P Q = P R$ by CPCT, i.e. the tangents drawn from an external point to a circle are equal in length.

Step-by-step solution →

OR

Q31(b) 3 marks

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Questions 32–35 · 5 marks each

Q32(a) 5 marks

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Q32(b) 5 marks

From the top of a $7$ m high building, the angle of elevation of the top of a cable tower is $6 0^{\circ}$ and the angle of depression of its foot is $4 5^{\circ}$ . Determine the height of the tower.

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Height of the tower $= 7 (3 + 1)$ m $\approx 19.12$ m.

Step-by-step solution →
Q33(a) 5 marks

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Q33(b) 5 marks

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Q34 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 →