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Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10 PDF Download

Time: 1 hour

M.M. 30

Attempt all questions.

  • Question numbers 1 to 5 carry 1 mark each.
  • Question numbers 6 to 8 carry 2 marks each.
  • Question numbers 9 to 11 carry 3 marks each.
  • Question number 12 & 13 carry 5 marks each.

Q1: _______________________ is drawn from the eye of an observer to the targeted object. (1 Mark)

(a) A parallel line
(b) Line of sight
(c) Elevation line
(d) Depression line

Ans: (b)

The line of sight is a line that is drawn from the eye of an observer to the targeted object viewed by the observer and the line of sight is required to form the angle of elevation.

Q2: A slope is built against a wall which makes an angle 30° with the ground and the height of the wall is 2 meters. Find the length of the slope in meters. (1 Mark)

a) 2
b) 4
c) 1
d) 3

Ans: (b)

AC is the length of the slope in ∆ABC.Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10

Q3: The shadow of a tower is equal to its height at 10-45 a.m. The sun’s altitude is (1 Mark)

(a) 30°
(b) 45°
(c) 60°
(d) 90°

Ans: (b)

Let the height of tower BC = x m and sun’s altitude = θ

Then Length of its shadow, AB = x mUnit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10In rt. ∆ABC, tan θ = BC/AB = x/x = 1
⇒ tan θ = tan 450
∴ θ = 45° 

Q4: In given figure, the value of CE is (1 Mark)Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10

(a) 12 cm
(b) 6 cm
(c) 9 cm
(d) 6√3 cm 

Ans: (a)

In rt. ∆EBC, cos 60° = BC/CE

⇒ 
1/2 = 6/CE

⇒ 
CE = 12cm

Q5: When the length of shadow of a vertical pole is equal to √3 times of its height, the angle of elevation of the Sun’s altitude is __________________________. (1 Mark)

Ans: 30°

Let the height of the vertical pole, BC = h m

∴ Shadow AB = √3h m and the angle of elevation angle BAC = θ Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10

In rt ∆ABC, tan θ = BC/AB = h/√3h = 1/√3 = tan 30°

∴ θ = 30°

Hence the Sun’s altitude is 30°.

Q6: The ratio of the height of a tower and length of its shadow on the ground is √3 : 1 what is the angle of elevation of the sun? (2 Marks)

Ans: Let TW be tower and WS be its shadow.

Let ∠TSW = θ (angle of elevation)Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10

tanθ = √3/1

θ=60∘

Therefore, the angle of elevation of the sun is 60º.

Q7: A ladder, leaning against a wall, makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder. (2 Marks)

Ans: Let AC be the ladder and the foot C is 2.5 m away from the wall AB.
Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10
cos 60° = BC/AC = 2.5/h

1/2 = 2.5/h

h = 5m

Q8: An observer 1.5 m tall is 28.5 m away from a tower of height 30 m. Find the angle of elevation of the top of tower from his eye. (2 Marks)

Ans: In right ∆ABC,Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10

tan θ = 1 = tan 45°

θ = 45°

Q9: A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/h. (3 Marks)

Ans: Let C & D be two positions of the boat,& AB be the cliff & let speed of boat be xm/ min.
Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10Let BC = y
∴ CD = 2x (∵ Distance = speed × time) Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10

Q10: The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of elevation of the top of the tower from the foot of the hill is 30°. If height of the tower is 50 m, find the height of the hill. (3 Marks)

Ans: Let HL be Hill and TW be Tower angle of elevations ∠WLT = 30°

∠ LWH = 60°, let WL = xUnit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10In \Delta WLTΔWLT, use tan30º to find x 

 tan30º = 50/x

⇒ x = 50√3 ……. (i)

In rt. ∠d ∆WLH

⇒ h/x = tan 60°

⇒ h = 50√3 × √3 = 150 [Using (i)]

∴ Height of hill = 150 m

Q11: A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of hill as 30°. Find the distance of the hill from the ship and the height of the hill. (3 Marks)

Ans:
Let AB be height of ship’s deck and CD be hill with C as top and D as base. Here angle of elevation ∠CAE = 60°.
Draw AE ⊥ CD and angle of depression ∠EAD = 30°
Let AE = BD = x
CE = y and ED = 10
Also, ∠BDA = ∠EAD = 30° [alternate angles]
In ∆ABD Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10

Now, CD = x + y
⇒ CD = 30 + 10 = 40 m
∴ The height of the hill is 40m. 

Q12: At the foot of a mountain the elevation of its summit is 45°, after ascending 1000 m towards the mountain up a slope of 30° inclination, the elevation is found to be 60°. Find the height of the mountain. (5 Marks)

Ans: Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10Let F be the foot and T be the summit of the mountain TFH such that ∠TFH = 45°
∴ In right ∆TBF,
∠BTF = 90° – 45° = 45°
Let height of mountain be h i.e., TB = h
since ∠TFH = ∠BTF = 45° ⇒ BF = BT = h
∠DFE = 30° and FD = 1000 m, ∠TDP = 60°
Draw DE ⊥BF and DP ⊥ BT
In right ∆DEF,
cos 30° = FE/DF = √3/2 = FE/1000
⇒ FE = 500√3 m
Also sin 30° = DE/DF = 1/2 = DE/1000
⇒ DE = 500 m
∴ BP = DE ⇒ BF = 500 m
Now TP = TB – BP = h – 500
In right ∆TPD, Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10

Q13: The angles of elevation and depression of the top and the bottom of a tower from the top of a building, 60 m high, are 30° and 60° respectively. Find the difference between the heights of the building and the tower and the distance between them. (5 Marks)

Ans: In right ∆ABE, Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10

∴ Difference between heights of the building and tower = y = 20 m
Distance between tower and building = x
= 20√3 m

The document Unit Test (Solutions): Some Applications of Trigonometry | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Unit Test (Solutions): Some Applications of Trigonometry - Mathematics (Maths) Class 10

1. What are some real-life applications of trigonometry in hobbies?
Ans.Trigonometry is widely used in various hobbies such as photography, where it helps in calculating angles for shots, and in sports like archery or skiing, where angles and distances are crucial for performance. Additionally, in music, trigonometric functions can describe sound waves and help in tuning instruments.
2. How can trigonometry help in navigation for outdoor hobbies?
Ans.Trigonometry aids in navigation by allowing individuals to determine their position using angles and distances. For example, hikers can use triangulation with landmarks to find their way, while sailors use trigonometric principles to chart their courses based on the angles to celestial bodies.
3. What role does trigonometry play in designing models for hobbies like model building?
Ans.Trigonometry is essential in model building as it helps in calculating dimensions and angles to create accurate representations. Whether building scale models of buildings or vehicles, understanding the relationships between angles and lengths ensures that components fit together properly.
4. Can trigonometry be used in digital hobbies such as gaming or animation?
Ans.Yes, trigonometry is crucial in gaming and animation. It is used to calculate object movements, camera angles, and lighting effects, enabling realistic graphics and smooth gameplay. Techniques like rotation and scaling often rely on trigonometric functions.
5. How does trigonometry enhance the experience in outdoor activities like climbing or cycling?
Ans.Trigonometry enhances experiences in activities like climbing or cycling by allowing individuals to assess slopes and angles of terrain. By understanding these measurements, climbers can evaluate the difficulty of routes, while cyclists can optimize their performance on inclines by calculating the best path and energy expenditure.
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