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CASE STUDY 3 A boy… (Class 10 Maths Basic SQP 2023-24, Q38)

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MathsCBSE Class 10 · Basic · Sample paper, session 2023-24 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic sample paper 2023-24: question 38

From the official CBSE sample question paper for 2023-24, 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-basedSome Applications of Trigonometry

CASE STUDY 3

A boy is standing on the top of light house. He observed that boat P and boat Q are approaching the light house from opposite directions. He finds that angle of depression of boat P is and angle of depression of boat Q is . He also knows that height of the light house is 100 m.

Based on the above information, answer the following questions.

  1. (i)

    What is the measure of ?

    1 mark
  2. (ii)

    If , then is also . Why?

    1 mark
  3. (iii)

    Find length of PD.

    2 marks

  4. OR

    (iii)

    Find length of DQ.

    2 marks

Figure in the paper: Left: a lighthouse on a rock with boats P (left) and Q (right) on water; a dashed horizontal line from the top of the lighthouse and the angle of depression marked. Right: schematic with a horizontal line XAY through point A at the top of the lighthouse; angle XAP = 45° and angle YAQ = 30° marked as angles of depression. A vertical line from A to D on the base PQ is marked 100 m. P and Q are on the base line, with D between them. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

The horizontal line XY at the top of the lighthouse is parallel to the ground PQ.

Hint 2

An angle of depression equals the angle of elevation at the boat, because they are alternate angles.

Hint 3

In the right triangles with AD = 100 m, use tan = opposite/adjacent with 45° or 30°.

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. (i) and AP is a transversal, so [alternate interior angles].

    ½ mark

  2. Hence .

    ½ mark

  3. (ii) and AQ is a transversal, so [alternate interior angles are equal].

    1 mark

  4. (iii)(a) In right ():

    ½ mark

  5. ½ mark

  6. m, so boat P is 100 m from the lighthouse.

    1 mark

  7. (iii)(b) In right ():

    ½ mark

  8. ½ mark

  9. m ( m).

    1 mark

Answer: (i) 45°; (ii) they are alternate interior angles, since XY ∥ PQ; (iii)(a) PD = 100 m; (iii)(b) DQ = 100√3 m ≈ 173.2 m

Where the marks are

  • States ∠APD = ∠XAP as alternate angles (½ mark)
  • Gets ∠APD = 45° (½ mark)
  • Gives the reason: alternate interior angles, since XY ∥ PQ (1 mark)
  • Writes tan 45° = 100/PD (½ mark)
  • Simplifies using tan 45° = 1 (½ mark)
  • Gets PD = 100 m (1 mark)
  • OR: writes tan 30° = 100/DQ (½ mark), uses 1/√3 = 100/DQ (½ mark), gets DQ = 100√3 m (1 mark)

Revise the chapter

Practise this chapterBoard-style questions from Some Applications of Trigonometry, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2023-24 (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.