MathsCBSE Class 10 · Basic · Board exam 2025, set 430/3/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2025: question 37

Set in the CBSE Class 10 board exam of 2025, paper 430/3/1, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedSome Applications of Trigonometry

Case Study 2

Two motorboats A and B are waiting at the opposite banks of a river in order to reach the opposite side. From a point P on the bridge, 20 m above the river, the angles of depression of the boats are $3 0^{\circ}$ and $4 5^{\circ}$ respectively, as shown in the figure given below. Both the boats leave at the same time at the speed of 10 m/s and 5 m/s, respectively.

Based on the above information, answer the following questions:

(i)
Find the distance travelled by boat A to reach point D in the river, vertically below the point P. (Use $3 = 1.73$ )
1 mark
(ii)
What is the width of the river?
1 mark
(iii)
(a) Which boat will reach point D first, and how much earlier, than the other boat?
2 marks
OR
(iii)
(b) What is the distance between the two boats after 3 seconds?
2 marks

Figure in the paper: Side view of a river with banks. P is a point on the bridge 20 m above the river. D is directly below P on the river surface, with a dashed vertical line PD of length 20 m. Boat A is at point A on the left, and boat B is at point B on the right, both on the water line AB with D between them. The angle of depression from P to A is 30° and from P to B is 45°. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/3/1.

Hints

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

Hint 1

The angle of depression from P equals the angle of elevation of P from the boat, because they are alternate angles.

Hint 2

Use $tan θ = \frac{P D}{horizontal distance}$ in triangles PDA and PDB. The width of the river is AD + DB.

Hint 3

Time = distance ÷ speed. After 3 s, subtract both boats' distances from the river's width, since they move towards each other.

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.

(i) PD = 20 m and PD ⟂ AB. The angle of depression of A is 30°, so $∠ P A D = 30°$ [alternate angles].
In right $△ P D A$ : $tan 30° = \frac{P D}{A D} = \frac{20}{A D}$
½ mark
(i) $\frac{1}{3} = \frac{20}{A D} \Rightarrow A D = 203 = 20 \times 1.73 = 34.6$ m.
Boat A travels 34.6 m to reach D.
½ mark
(ii) The angle of depression of B is 45°, so $∠ P B D = 45°$ .
In right $△ P D B$ : $tan 45° = \frac{20}{D B} \Rightarrow 1 = \frac{20}{D B} \Rightarrow D B = 20$ m.
½ mark
(ii) Width of the river $= A D + D B = 34.6 + 20 = 54.6$ m.
½ mark
(iii)(a) Time taken by boat A to reach D $= \frac{distance}{speed} = \frac{34.6}{10} = 3.46$ s.
1 mark
(iii)(a) Time taken by boat B to reach D $= \frac{20}{5} = 4$ s.
½ mark
(iii)(a) Boat A reaches D first, earlier by $4 - 3.46 = 0.54$ s.
½ mark
(iii)(b) Distance covered by boat A in 3 s $= 10 \times 3 = 30$ m. This is less than 34.6 m, so A has not yet reached D.
½ mark
(iii)(b) Distance covered by boat B in 3 s $= 5 \times 3 = 15$ m. This is less than 20 m, so B has not yet reached D.
½ mark
(iii)(b) The boats move towards each other, so the distance between them $= 54.6 - (30 + 15) = 9.6$ m.
1 mark

Answer: (i) 34.6 m; (ii) 54.6 m; (iii)(a) Boat A, 0.54 s earlier; (iii)(b) 9.6 m

Where the marks are

(i) Writes $tan 30° = 20/ A D$ (½ mark)
(i) $A D = 203 = 34.6$ m (½ mark)
(ii) $tan 45° = 20/ D B$ , giving $D B = 20$ m (½ mark)
(ii) Width of river $= 54.6$ m (½ mark)
(iii)(a) Time for boat A $= 3.46$ s (1 mark)
(iii)(a) Time for boat B $= 4$ s (½ mark)
(iii)(a) Boat A is earlier by 0.54 s (½ mark)
(iii)(b) A covers 30 m and B covers 15 m in 3 s (½ + ½ marks)
(iii)(b) Distance between the boats $= 54.6 - 45 = 9.6$ m (1 mark)

Revise the chapter

Chapter: Some Applications of Trigonometry (notes, formulas, where marks are lost and every past question)
Topic: Angle of Depression

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 paper2025 board paper, set 430/3/1 (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.

CBSE Class 10 Mathematics Basic 2025: question 37

Hints

Step-by-step solution

Revise the chapter

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