Chapter 9: Some Applications of Trigonometry

VVI questions — the ones the board asked most (12)

1. Q1Long answer5 marksAsked: 2026

   The angle of elevation of the top of a building from the foot of a tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 60 m high then find the height of the building.

   

   Medha Ma’am

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   Model answer · heights and distances, two right triangles

   1. 1.Let the height of the tower AB be 60 m and the height of the building CD be h m, and let the distance between the foot of the tower and the foot of the building be x m. The angle of elevation of the top of the building from the foot of the tower is 30°, and the angle of elevation of the top of the tower from the foot of the building is 60°. Using trigonometric ratios in the right-angled triangles formed, we get two equations connecting the height of the tower, the height of the building and the distance x between them.1 mark
   2. 2.In the right triangle formed with the tower AB, using tan 60° = AB/x, we have tan 60° = 60/x. Since tan 60° = √3, this gives √3 = 60/x, so x = 60/√3 = 60√3/3 = 20√3 m. Thus the base x, which is the distance between the foot of the tower and the foot of the building, is 20√3 metres.2 marks
   3. 3.In the right triangle formed with the building CD, using tan 30° = h/x, we have tan 30° = h/x, where h is the height of the building and x is the distance between the foot of the tower and the foot of the building found above. Since tan 30° = 1/√3, this gives 1/√3 = h/x.1 mark
   4. 4.Given: x = 20√3 m Formula: 1/√3 = h/x Substitution: 1/√3 = h/(20√3) Calculation: h = 20√3 × 1/√3 = 20 Therefore, the height of the building is 20 m.1 mark

   Diagram: Draw the tower AB (60 m) vertical at one end and the building CD (h m) vertical at the other end, joined by a horizontal base line of length x m; mark the angle 60° at the foot of the building (point D) looking up to the top of the tower A, and mark the angle 30° at the foot of the tower (point B) looking up to the top of the building C.

   • Write both equations tan 60° = 60/x and tan 30° = h/x clearly, using the actual tower height 60 m given in this question.
   • Show the working x = 60/√3 = 20√3 m before using it in the second equation.
   • End with the final substitution h = 20√3 × 1/√3 = 20 and state the answer as 20 m with the unit.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

2. Q2Long answer5 marksAsked: 2026

   From a point P on the ground the angle of elevation of the top of a 10 m tall building is 30°. A flag is hoisted at the top of the building and angle of elevation of the top of the flagstaff from P is 45°. Find the length of the flagstaff and the distance of the building from the point P. ( where sqrt(3)=1.732 )

   

   Medha Ma’am

   AskMedha’s AI teacher

   Hear it from Medha Ma’am · 160 s

   Model answer · heights and distances, flagstaff

   1. 1.Let AB denote the height of the building, BD the flagstaff and P the given point on the ground. Here AB = 10 m is the height of the building and AP metres is the distance of the building from the point P. Since there are two right triangles PAB and PAD, and we know the height of the building AB, we first consider the right triangle PAB, in which the angle of elevation of the top of the building from P is 30°. In this right-angled triangle, we have tan 30° = AB/AP.1.5 marks
   2. 2.Given: tan 30° = AB/AP, AB = 10 m Formula: tan 30° = 1/√3 Substitution: 1/√3 = 10/AP Calculation: AP = 10√3 Answer: AP = 10√3 m = 17.32 m, so the distance of the building from P is 17.32 m.1 mark
   3. 3.Now let DB = h metres be the length of the flagstaff, so that AD = (10 + h) metres, since AB = 10 m is the height of the building. We next consider the larger right triangle PAD, in which the angle of elevation of the top of the flagstaff from P is 45°. In this right-angled triangle, we have tan 45° = AD/AP.1 mark
   4. 4.Given: tan 45° = AD/AP, AD = (10 + h) m, AP = 10√3 m Formula: tan 45° = 1 Substitution: 1 = (10 + h)/10√3 Equation: 10 + h = 10√31 mark
   5. 5.Given: 10 + h = 10√3, √3 = 1.732 Calculation: h = 10√3 − 10 = 10(√3 − 1) = 10 × 0.732 Answer: h = 7.32 m So the length of the flagstaff is 7.32 m.0.5 marks

   Diagram: Draw point P on the ground, with vertical line AB representing the building (A at the bottom on the ground level, B at the top of the building) and BD as the flagstaff above B, with D at its top; mark P on the ground joined to A, B and D; label AB = 10 m, BD = h, angle APB = 30°, angle APD = 45°, and AP = 10√3 m.

   • Write 'tan 30°' and 'tan 45°' explicitly as the trigonometric ratios used in each right-angled triangle.
   • State the final distance as 17.32 m and the flagstaff height as 7.32 m with units.
   • Show the substitution step 1 = (10 + h)/10√3 before solving for h, since this equation itself is marked.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

3. Q3Long answer5 marksAsked: 2022

   A ladder rests on a vertical wall such that upper part of ladder just touches the top of the wall. The ladder makes an angle of 60 degree with the ground. The lower part of the ladder is at a distance 1.5 m from the wall. Find the length of the ladder and height of the wall.

   

   Medha Ma’am

   AskMedha’s AI teacher

   Hear it from Medha Ma’am · 140 s

   Model answer · heights and distances, ladder problem

   1. 1.Let AB be the ladder resting against the vertical wall BC, with the foot of the ladder at point A on the ground. The ladder makes an angle of 60° with the ground, so ∠BAC = 60°. The distance of the foot of the ladder from the wall is AC = 1.5 m, which is the base of the right triangle ABC, right-angled at C. The length of the ladder AB is the hypotenuse. Since the base and the hypotenuse are involved, we use the trigonometric ratio cos 60° = base/hypotenuse, that is, cos 60° = AC/AB.1.5 marks
   2. 2.Substituting the known values in cos 60° = AC/AB, we get cos 60° = 1.5/AB. Since cos 60° = 1/2, we have 1/2 = 1.5/AB, which gives AB = 1.5 × 2 = 3. Hence, the length of the ladder AB is 3 meters.1 mark
   3. 3.The height of the wall BC is the side opposite to the angle of 60°, that is, the perpendicular of the right triangle ABC, while AC = 1.5 m is the base. Since the perpendicular and the base are involved, we use the trigonometric ratio tan 60° = perpendicular/base, that is, tan 60° = BC/AC.1.5 marks
   4. 4.Substituting the known values in tan 60° = BC/AC, we get tan 60° = BC/1.5. Since tan 60° = √3, we have √3 = BC/1.5, which gives BC = 1.5√3. Hence, the height of the wall BC is 1.5√3 meters, which is approximately 2.598 meters.1 mark

   Diagram: Draw a right triangle ABC with the vertical wall BC, the ground AC, and the ladder AB as the hypotenuse; label A at the foot of the ladder, B at the top of the wall, C at the base of the wall, mark ∠BAC = 60°, AC = 1.5 m, AB = ladder length, and BC = height of wall, with a right angle shown at C.

   • Write cos 60° = AC/AB while finding the ladder length, and tan 60° = BC/AC while finding the wall height, labelling AB, BC, AC clearly.
   • Show the substitution step for both cos 60° and tan 60° before writing the final numeric answer.
   • State the final answers with units: ladder length as 3 meters and wall height as 1.5√3 meters (approximately 2.598 meters).

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

4. Q4Long answer5 marksAsked: 2022

   A vertical flagstaff and a 50 m high tower are situated on the same horizontal base. From the top of the tower, angles of depression of upper and lower ends of flagstaff are 30 degree and 45 degree respectively. Find the height of the flagstaff.

   How the model answer begins

   1. The height of the flagstaff is found as follows. Let AB be the tower of height 50 m standing on the horizontal ground at B, with A as the top of the tower. Let CD be the vertical flagstaff standing on the same…

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5. Q5Long answer5 marksAsked: 2022

   Two ships are approaching a lighthouse from opposite directions. The angles of depression of the two ships from the top of lighthouse are 60° and 45°. If the distance between two ships is 100 m, find the height of the lighthouse.

   How the model answer begins

   1. Let AB be the lighthouse of height h metres, with A at the top. Let C and D be the positions of the two ships on opposite sides of the lighthouse, with the angles of depression of the two ships from the top of the…

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6. Q6Long answer5 marksAsked: 2023

   Two poles of equal heights are standing opposite to each other on either sides of the road which is 80 m wide. From a point P between them on the road the angle of elevation of the top of one pole is 60° and the angle of depression of the point P from the top of another pole is 30°. Find the height of each pole and distances of the point P from the poles.

   How the model answer begins

   1. Let AB and CD be the two poles of equal height standing on either sides of the road. Let the road AC be 80 metres wide and let the height of each pole be h metres. Let P be a point between the two poles on the road.

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7. Q7Long answer5 marksAsked: 2023

   Two poles of equal heights are standing opposite to each other on either sides of the road which is 80 m wide. From a point P between them on the road the angle of elevation of the top of one pole is 60° and the angle of depression of the point P from the top of another pole is 30°. Find the height of each pole and distances of the point P from the poles.

   How the model answer begins

   1. Let the two poles AB and CD be of equal height h metres, standing on opposite sides of a road of width 80 m. Let P be a point on the road between the two poles such that the angle of elevation of the top of one pole…

   The full answer from Medha Ma’am →
8. Q8Long answer5 marksAsked: 2021

   The angle of elevation of the top of a tower from a point O on the ground is 30 degree and the angle of elevation of the top of the flagstaff on the top of the tower is 60 degree. If the length of the flagstaff is 5 m, then tell the height of the tower.

   How the model answer begins

   1. Let AB be the tower with height h metres standing on the ground, and let BC be the flagstaff fixed on top of the tower with BC = 5 m. Let O be a point on the ground such that OA = x metres is the distance from O to the…

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9. Q9Long answer5 marksAsked: 2021

   As observed from the top of a light house 95 m high above the sea level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships.

   How the model answer begins

   1. AB = 95 m is the height of the lighthouse, and C and D are the positions of the two ships on the sea, with angles of depression 45° and 30° respectively, so that right triangle ABC is formed with ship C and right…

   The full answer from Medha Ma’am →
10. Q10Short answer2 marksAsked: 2025

    A ladder 7 m long makes an angle of 30° with the wall. Find the height of the point on the wall where the ladder touches the wall.

    How the model answer begins

    1. The ladder (7 m) is the hypotenuse and makes 30° with the wall, so the height h on the wall is the side next to that angle: cos 30° = h/7.

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11. Q11Short answer2 marksAsked: 2026

    If the length of shadow of a pole is equal to sqrt(3) times of its height then find the angle of elevation of the sun.

    How the model answer begins

    1. Let the height of the pole be h and the length of its shadow be h√3.

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12. Q12Short answer2 marksAsked: 2022

    From a point on the ground, which is at a distance of 30 m from the foot of a tower, the angle of elevation of the top of the tower is 30°. Find the height of the tower.

    How the model answer begins

    1. Let AB be the tower and P be the point on the ground 30 m away from the foot of the tower, so AP = 30 m. The angle of elevation of the top of the tower from P is 30°. In the right triangle PAB, right-angled at A, we…

    The full answer from Medha Ma’am →

Objective questions with the board’s key (10)

1. 1. Bihar Board 2026

   If the length of the shadow of a tree is equal to its height then angle of elevation of the sun is

   • (A) 30°
   • (B) 45° ✓
   • (C) 60°
   • (D) 75°

   Answer: (B)

2. 2. Bihar Board 2026

   A 15 m long ladder is resting at the top of a vertical wall forming an angle 60° with the wall ; then the height of the wall is

   • (A) 15/2 m ✓
   • (B) 15 m
   • (C) 30 m
   • (D) 15/4 m

   Answer: (A)

3. 3. Bihar Board 2026

   The angle of elevation of the top of a tower at a distance of 25 m from its foot is 60°, then the height of the tower is

   • (A) 5 sqrt(3) m
   • (B) 15 sqrt(3) m
   • (C) 25 sqrt(3) m ✓
   • (D) 25 m

   Answer: (C)

4. 4. Bihar Board 2026

   A kite is at a height of 20 m from the earth and its string makes an angle 45° with the earth, then the length of the string is

   • (A) 20 sqrt(2) m ✓
   • (B) 10 sqrt(2) m
   • (C) 5 sqrt(2) m
   • (D) 10 m

   Answer: (A)

5. 5. Bihar Board 2025

   The length of the shadow of a pillar is √3 times the height of the pillar. The angle of elevation of the sun is

   • (A) 30° ✓
   • (B) 45°
   • (C) 60°
   • (D) 90°

   Answer: (A)

6. 6. Bihar Board 2023

   If the angle of elevation of the sun is 60°, then the height of the vertical tower that casts 30 m long shadow is

   • (A) 30/sqrt(3) m
   • (B) 15*sqrt(3) m
   • (C) 30*sqrt(3) m ✓
   • (D) 15 m

   Answer: (C)

7. 7. Bihar Board 2022

   If the length of the shadow of a pole is 2sqrt(3) m and angle of elevation of sun is 60°, then the height of the pole will be

   • (A) 4sqrt(3) m
   • (B) 6 m
   • (C) 12 m
   • (D) 3 m ✓

   Answer: (D)

8. 8. Bihar Board 2022

   The ratio of the length of a rod to its shadow is 1:sqrt(3), so the angle of elevation of the sun is

   • (A) 30° ✓
   • (B) 45°
   • (C) 60°
   • (D) 90°

   Answer: (A)

9. 9. Bihar Board 2022

   The length of a vertical wall and length of its shadow are in the ratio 1:sqrt(3). The angle of elevation of sun is

   • (A) 30° ✓
   • (B) 45°
   • (C) 60°
   • (D) 90°

   Answer: (A)

10. 10. Bihar Board 2021

    If the height of a vertical pillar is sqrt(3) times the length of its shadow on the ground, then the angle of elevation of the sun will be

    • (A) 30°
    • (B) 45°
    • (C) 60° ✓
    • (D) 75°

    Answer: (C)

Questions parents and students ask

How many marks does "Some Applications of Trigonometry" carry in Bihar Board Class 10 Maths?

In the Bihar Board papers of 2021–2026, this chapter carried about 3 marks and about 2 questions a year; by marks it ranks 14 in Maths.

Which are the most important (VVI) questions of "Some Applications of Trigonometry"?

The 12 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 12 written questions of this chapter, with answers, are in the AskMedha app.

Do the answers follow the board’s marking scheme?

BSEB does not publish a marking scheme. Each answer is written point by point on a structure derived from past papers and checked by a second, independent model; it is never called the board’s official scheme.