MathsCBSE Class 10 · Basic · Board exam 2024, set 430/5/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2024: question 38

Set in the CBSE Class 10 board exam of 2024, paper 430/5/1, 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

Two poles of different heights stand on level ground and at a distance of 40 m. Both poles are supported by wires attached from the top of each pole to the bottom of the other. A coupling is placed at point C, where the two wires cross (as shown in the figure).

Based on the above information, answer the following questions :

(i)
Find the height of pole AB.
1 mark
(ii)
Find the height of pole PQ.
1 mark
(iii)
(a) If the angle of elevation of the top of the pole PQ from the top of the pole AB is $3 0^{\circ}$ , find the distance BQ.
2 marks
OR
(iii)
(b) If the coupling is at a height of 20 m from the ground, how far down the wire from the smaller pole AB is the coupling ?
2 marks

Figure in the paper: Two poles AB and PQ stand on level ground, with A and P at the base, 40 m apart (marked on the ground line). AB is the shorter pole (B at top) and PQ is the taller pole (Q at top). A wire runs from A to Q and another from P to B; the two wires cross at point C. At A, the angle between ground AP and wire AQ is $6 0^{\circ}$ . At P, the angle between ground PA and wire PB is $3 0^{\circ}$ . Right angles are at A and P. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/5/1.

Hints

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

Hint 1

Each pole, the ground and a wire form a right triangle, and the 40 m base is known.

Hint 2

Use tan of the given angle at the far end to get each pole's height: opposite side over the 40 m base.

Hint 3

For (iii), draw a horizontal line from B to PQ, or a perpendicular from C to the ground, to make a new right triangle with a known side.

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 $△ B A P$ (right angle at A): $tan 3 0^{\circ} = \frac{A B}{A P}$
½ mark
(i) $\frac{1}{3} = \frac{A B}{40} \Rightarrow A B = \frac{40}{3} = \frac{40 3}{3}$ m
½ mark
(ii) In right $△ QP A$ (right angle at P): $tan 6 0^{\circ} = \frac{P Q}{A P}$
½ mark
(ii) $3 = \frac{P Q}{40} \Rightarrow P Q = 403$ m
½ mark
(iii)(a) Figure: draw $B M ⊥ P Q$ , so ABMP is a rectangle and $B M = A P = 40$ m. Then $∠ QB M = 3 0^{\circ}$ in right $△ B M Q$ .
½ mark
(iii)(a) $cos 3 0^{\circ} = \frac{B M}{B Q}$
½ mark
(iii)(a) $\frac{3}{2} = \frac{40}{B Q} \Rightarrow B Q = \frac{80}{3} = \frac{80 3}{3}$ m
1 mark
(iii)(b) Figure: draw $C D ⊥ A P$ , so $C D = 20$ m. In right $△ A D C$ , $∠ C A D = 6 0^{\circ}$ .
½ mark
(iii)(b) $sin 6 0^{\circ} = \frac{C D}{A C}$
½ mark
(iii)(b) $\frac{3}{2} = \frac{20}{A C} \Rightarrow A C = \frac{40}{3} = \frac{40 3}{3}$ m $(\approx 23.09$ m $)$ . So the coupling is $\frac{40 3}{3}$ m along the wire from the foot A of pole AB.
1 mark

Answer: (i) AB = 40√3/3 m; (ii) PQ = 40√3 m; (iii)(a) BQ = 80√3/3 m; (iii)(b) AC = 40√3/3 m

Where the marks are

(i) Uses $tan 3 0^{\circ} = A B / A P$ (½ mark)
(i) Gets $A B = 403 /3$ m (½ mark)
(ii) Uses $tan 6 0^{\circ} = P Q / A P$ (½ mark)
(ii) Gets $P Q = 403$ m (½ mark)
(iii)(a) Figure with $B M ⊥ P Q$ , $B M = 40$ m (½ mark)
(iii)(a) Uses $cos 3 0^{\circ} = B M / B Q$ (½ mark)
(iii)(a) Gets $B Q = 803 /3$ m (1 mark)
(iii)(b) Figure with $C D = 20$ m (½ mark)
(iii)(b) Uses $sin 6 0^{\circ} = C D / A C$ (½ mark)
(iii)(b) Gets $A C = 403 /3$ m (1 mark)

Revise the chapter

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

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 paper2024 board paper, set 430/5/1 (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 2024: question 38

Hints

Step-by-step solution

Revise the chapter

More questions from Some Applications of Trigonometry