MathsCBSE Class 10 · Basic · Board exam 2026, set 430/3/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2026: question 37

Set in the CBSE Class 10 board exam of 2026, 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: Points A and B on the river's opposite banks on a horizontal line; D lies on this line vertically below P, with AD and DB along it. P is on a bridge 20 m above the river, with angle of depression 30° to A and 45° to B, marked at P. PD = 20 m is shown as a vertical dashed line. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 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 equals the angle of elevation at the boat (alternate angles).

Hint 2

Use tan in each right triangle with PD = 20 m as the perpendicular.

Hint 3

Time = distance ÷ speed. After 3 s, find how far each boat still is from D.

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) In right $△ P D A$ , $∠ P A D = 3 0^{\circ}$ (alternate to the angle of depression), so $tan 3 0^{\circ} = \frac{P D}{A D} \Rightarrow A D = 203 = 20 \times 1.73 = 34.6$ m.
1 mark
(ii) In right $△ P D B$ , $tan 4 5^{\circ} = \frac{P D}{D B} \Rightarrow D B = 20$ m. Width of the river $= A D + D B = 34.6 + 20 = 54.6$ m.
1 mark
(iii)(a) Time for boat A $= \frac{34.6}{10} = 3.46$ s; time for boat B $= \frac{20}{5} = 4$ s.
1 mark
(iii)(a) Boat A reaches D first, $4 - 3.46 = 0.54$ s earlier than boat B.
1 mark
(iii)(b) In 3 s, boat A covers $10 \times 3 = 30$ m, so it is $34.6 - 30 = 4.6$ m from D. Boat B covers $5 \times 3 = 15$ m, so it is $20 - 15 = 5$ m from D on the other side.
1 mark
(iii)(b) Distance between the boats $= 4.6 + 5 = 9.6$ 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

AD = 20√3 = 34.6 m using tan 30° (1 mark)
DB = 20 m and width = 54.6 m (1 mark)
(iii)(a) Times 3.46 s and 4 s (1 mark); boat A first by 0.54 s (1 mark)
(iii)(b) Remaining distances 4.6 m and 5 m (1 mark); separation 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 paper2026 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 2026: question 37

Hints

Step-by-step solution

Revise the chapter

More questions from Some Applications of Trigonometry