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X - CBSE - MATHSVOLUME 2

Table of Contents

Volume 2 of 2 Universal Tutorials – X CBSE – Mathematics

Table of Contents CHAPTER 09: SOME APPLICATIONS OF TRIGONOMETRY ................................ 1

CHAPTER MAP: ........................................................................................................................................... 1 Angle of Elevation: (Angle of inclination) ........................................................................................... 1 Angle of Depression: (Angle of declination) ...................................................................................... 1

Solved Examples 9.1: .............................................................................................................. 2 Unsolved Exercise 9.1: ............................................................................................................ 4 Miscellaneous Exercise: .......................................................................................................... 7 Multiple Choice Questions: ...................................................................................................... 9 Previous Year Board Questions: ........................................................................................... 12 Answers to Unsolved Exercises: ........................................................................................... 13

CHAPTER 10: CIRCLES ......................................................................................... 14

CHAPTER MAP: ......................................................................................................................................... 14 Introduction: ..................................................................................................................................... 14 Tangent to a circle: .......................................................................................................................... 14

Unsolved Exercise 10.1: ........................................................................................................ 15 Number of Tangents from a point on a circle: ................................................................................. 15

Solved Examples 10.2: .......................................................................................................... 16 Unsolved Exercise 10.2: ........................................................................................................ 17 Miscellaneous Exercise: ........................................................................................................ 20 Multiple Choice Questions: .................................................................................................... 21 Previous Year Board Questions: ........................................................................................... 24 Answers to the Unsolved Exercise: ....................................................................................... 26

CHAPTER 11: CONSTRUCTIONS ......................................................................... 27

CHAPTER MAP: ......................................................................................................................................... 27 BASIC CONSTRUCTIONS (REVISION) ........................................................................................................... 27

Division of a Line Segment: ............................................................................................................. 28 Construction of similar triangles: ..................................................................................................... 29

Unsolved Exercise 11.1: ........................................................................................................ 30 Construction of Tangents to a Circle: .............................................................................................. 31

Unsolved Exercise 11.2: ........................................................................................................ 32 Miscellaneous Exercise: ........................................................................................................ 33 Previous Year Board Questions: ........................................................................................... 34

CHAPTER 12: AREAS RELATED TO CIRCLES ................................................... 35

CHAPTER MAP: ......................................................................................................................................... 35 Introduction: ..................................................................................................................................... 35

Perimeter and Area of a circle: ................................................................................................... 35 Table of Perimeter, Area of different figures: ............................................................................. 35

Universal Tutorials – X CBSE – Mathematics Volume 2 of 2

Solved Examples 12.1: .......................................................................................................... 37 Unsolved Exercise 12.1: ........................................................................................................ 37

Areas of Sector and Segment of a circle: ........................................................................................ 39 Solved Examples 12.2: .......................................................................................................... 40 Unsolved Exercise 12.2: ........................................................................................................ 41

Areas of combination of plane figures: ............................................................................................ 43 Solved Examples 12.3: .......................................................................................................... 43 Unsolved Exercise 12.3: ........................................................................................................ 44 Miscellaneous Exercise: ........................................................................................................ 46 Multiple Choice Questions: .................................................................................................... 48 Previous Year Board Questions: ........................................................................................... 49 Answers to Unsolved Exercises: ........................................................................................... 50

CHAPTER 13: SURFACE AREAS AND VOLUMES .............................................. 52

CHAPTER MAP: ......................................................................................................................................... 52 Introduction: ..................................................................................................................................... 52 Surface Area of Combination of Solids: .......................................................................................... 53


Volume of Combination of Solids: ................................................................................................... 56 Solved Examples 13.2: .......................................................................................................... 56 Unsolved Exercise 13.2: ........................................................................................................ 57

Conversion of Solid from one shape to another: ............................................................................. 58 Solved Examples 13.3: .......................................................................................................... 58 Unsolved Exercise 13.3: ........................................................................................................ 59

Frustum of a Cone: .......................................................................................................................... 61 Solved Examples 13.4: .......................................................................................................... 61 Unsolved Exercise 13.4: ........................................................................................................ 63 Miscellaneous Exercise: ........................................................................................................ 64 Multiple Choice Questions: .................................................................................................... 66 Previous Year Board Questions: ........................................................................................... 68 Answer to Unsolved Exercises: ............................................................................................. 69

CHAPTER 14: STATISTICS.................................................................................... 71

CHAPTER MAP: ......................................................................................................................................... 71 Introduction: ..................................................................................................................................... 71 Calculation of Central tendencies for grouped data ........................................................................ 71

14.1 Mean of grouped data: ....................................................................................................... 71 Solved Examples 14.1: .......................................................................................................... 72 Unsolved Exercise 14.1: ........................................................................................................ 74

To find mean by Assumed mean method: ................................................................................. 75 To determine mean by step deviation method: .......................................................................... 76

Solved Examples 14.2: .......................................................................................................... 76

Table of Contents

Volume 2 of 2 Universal Tutorials – X CBSE – Mathematics

Unsolved Exercise 14.2: ........................................................................................................ 79 14.2 Mode of grouped data: ............................................................................................................ 81


14.3 Median of Grouped Data: ........................................................................................................ 83 Solved Examples 14.4: .......................................................................................................... 83 Unsolved Exercise 14.4: ........................................................................................................ 86

Comparative Study: .................................................................................................................... 88 Graphical Representation of Cumulative frequency Distribution: ................................................... 88

Ogive of Less than type: ............................................................................................................ 88 Ogive of more than type: ............................................................................................................ 89

Solved Examples 14.5: .......................................................................................................... 90 Unsolved Exercise 14.5: ........................................................................................................ 91 Miscellaneous Exercise: ........................................................................................................ 92 Multiple Choice Questions: .................................................................................................... 93 Previouse Year Board Questions .......................................................................................... 94 Answer to the Unsolved Exercise: ......................................................................................... 96

CHAPTER 15: PROBABILITY ................................................................................ 98

CHAPTER MAP .......................................................................................................................................... 98 Introduction: ..................................................................................................................................... 98 Elementary Event: ........................................................................................................................... 98 Sample Space: ................................................................................................................................ 98 Equally likely outcomes (events): .................................................................................................... 99 Definition: Probability: ...................................................................................................................... 99 Complementary Event: .................................................................................................................... 99 Impossible Event: .......................................................................................................................... 100 Sure Event: .................................................................................................................................... 100

Solved Examples 15.1: ........................................................................................................ 100 Unsolved Exercise 15.1: ...................................................................................................... 101 Solved Examples 15.2: ........................................................................................................ 102 Unsolved Exercise 15.2: ...................................................................................................... 102 Solved Examples 15.3: ........................................................................................................ 103 Unsolved Exercise 15.3: ...................................................................................................... 104 Miscellaneous Exercise: ...................................................................................................... 105 Multiple Choice Questions: .................................................................................................. 107 Previous Year Board Questions: ......................................................................................... 109 Answers to Unsolved Exercise: ........................................................................................... 110

SAMPLE QUESTION PAPER [2017–18) .............................................................. 112

Chapter 09: Some Applications of Trigonometry 1

Volume 2 of 2 Universal Tutorials – X CBSE – Mathematics 1

Chapter 09: Some Applications of Trigonometry

Chapter Map:

Angle of Elevation: (Angle of inclination) Consider a student is looking at the top of a minar. The angle formed by the line of sight with the

horizontal is called the angle of elevation of top of the minar from the eyes of the student. Thus the line of sight is the line drawn from eyes of an observer to the point on the object viewed by

the observer. The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal

when the point being viewed is above the horizontal level that is the case when we raise our head to look at the object.

The angle of elevation in the given case is θ°.

Angle of Depression: (Angle of declination) A Student sitting on the balcony is looking down at a flower pot placed on a stair of the temple. In this case, the line of sight is below the horizontal level. The angle so formed by the line of sight

with the horizontal is called the angle of depression. Thus the angle of depression of a point on the object being viewed is the angle formed by the line of

sight with the horizontal when the point is below the horizontal level, i.e. the case when we lower our head to look at the point being viewed.

The angle of depression in the given case is θ°.

Horizontal level (θ)° Angle of elevation

Object

Reference Point on Ground (Eye of the observer)

Line of sight

Horizontal level

Reference Point on Ground (Eye of the observer)

Object

(θ°) Angle of depression

Line of sight

Elevation (Angle of Inclination)

Heights and Distances

Depression (Declination)

Problems on height and distance

2

2 Universal Tutorials – X CBSE – Mathematics Volume 2 of 2

SOLVED EXAMPLES 9.1:

1) A ladder leaning against a wall makes an angle of 60° with the ground. If the length of the ladder is 19m, find the distance of the foot of the ladder from the wall.

Sol: AB is the wall, AC is the ladder = 19 m ∠ACB = 60°

To find BC, cos 60° = ACBC =

21

19BC =

21 ⇒ 2BC = 19 ⇒ BC = 9.5 m

2) An electrician has to repair an electric fault on a pole of height 5 m. She needs to reach a point 1.3 m below the top of the pole to undertake the repair work. What should be the length of the ladder that she should use which, when inclined at an angle of 60° to the horizontal, would enable her to reach the required position? [ 3 = 1.73]

Sol: In Fig. the electrician is required to reach the point B on the pole AD. So, BD = AD – AB = (5 – 1.3) m = 3.7 m Here, BC represents the ladder. We need to find its length, i.e. the hypotenuse of the right ΔBDC. Now, can you think which trigonometric ration should we consider? It should be sin 60°.

So, BCBD = sin 60° or

BC7.3 =

23

∴ BC = 3

27.3 × = 4.28 m (approx.)

i.e. the length of the ladder should be 4.28 m.

Now, BDDC = cot 60° =

31 i.e. DC =

37.3 = 2.14 m (approx.)

∴ She should place the foot of the ladder at a distance of 2.14 m from the pole. 3) From a window (h m high above the ground) of a house in a street, the angles of elevation

and depression of the top and foot of another house on the opposite side of the street are θ and φ respectively. Show that the height of the opposite house is h [1 + tan θ. cot φ] metres

Sol: Let the window be at height h m from the ground ∠CAE = θ ∠EAD = φ

In ΔAEC, θ= tanAECE , CE = AE × tan θ

In rectangle ABDE, AB = h = DE BD = AE = x

In ΔAED, φ= cotEDAE , AE = ED. cot φ = h cot φ; x = h. cot φ

CE = AE tan θ = x tan θ Height of the opposite house, CD = CE + ED = h cot φ tanθ + h = h [1 + cot φ tan θ] 4) Two pillars of equal height stand on either side of a roadway, which is 100 m wide. At a

point in the roadway between the pillars the elevations of the tops of the pillars are 60° and 45°. Find the height of each pillar and the position of the point.

A

BC

60°

A

D C60°

B

θA

B

E

D

C

h hφ

φ



Sol: Let AB and CD be two pillars of the same height h and let E be the point of observation in the roadway. Then, ∠AEB = 60°, ∠CED = 45° and AC = 100 m.

Now, 3160cot =°=

ABAE

33

1 hAEh

AE=⇒=⇒ ––– (i)

and hCEh

CECDCE

=⇒=⇒=°= 1145cot ––– (ii)

But, AE + CE = AC = 100 m

∴ 3

h + h = 100 ⇔ ( 3 + 1)h = 100 3

⇒ h = ( )( )( ) )13(350

1313

133100

−=−

−×

+ ⇒ h = (150 – 50 3 ) = (150 – 50 × 1.732) = 63.4

∴ Height of each pillar = 63.4 m.

Also, AE = )13(503

)13(3503

−=−

=h = 50 × 0.732 = 36.6 m.

Thus, the observer is at a distance of 36.6 m from one pillar and 63.4 m from the other. 5) From the top of a building 96m high, the angle of depression of two vehicles on a road at

the same level and in the same line with the foot of the building and on the same side of it are x° and y°, where tan x° = 3/4 and tan y° = 1/3 Calculate the distance between the vehicles.

Sol: AB is the building C and D represent the position of vehicles. According to the question, AB = 96 cm ∠ACB = ∠EBC = x ∠ADB = ∠EBD = y

∴ tan x = ACAB ⇒

AC96 =

43 ⇒ AC = 128 m

tan y = ADAB ⇒

AD96 =

31 ⇒ AD = 288 m

∴ CD = AD – AC = 288 – 128 = 160 m 6) A man on the cliff observes a boat at an angle of depression of 30° which is approaching

the shore to the point immediately beneath the observer with a uniform speed. Six minutes later the angle of depression of the boat is found to be 60°. Find the time taken by the boat to reach the shore.

Sol: Let P be the point on the cliff. Q is the shore. A and B are the two positions of the boat. Now, ∠PAB = ∠XPA = 30° and ∠PBQ = ∠XPB = 60° We have, ∠APB = 60° – 30° = 30° [Ext. ∠PBA = ∠PAB + ∠APB

⇒ ∠APB = ∠PBA – ∠PAB] Now in ΔAPB, ∠APB = ∠PAB [Each = 30°] ⇒ PB = AB = x [Say]

Now in right ΔPBQ, PBBQ = cos 60°

D C A

BE

x

y

y

x

A

BC

D

E

45º 60º

A B Q

PX

60°

30°

30°

60°

30° x

x

4


⇒ BQ = PB cos 60° = x × 21 =

2x

The boat sailing at uniform speed takes 6 minutes to cover the distance AB i.e. x.

Since BQ = 21 x

∴ Time taken by the boat to cover the distance BQ is 2x

x6

× = 3 minutes

Hence the time taken by the boat to reach the shore = 6 + 3 = 9 minutes

UNSOLVED EXERCISE 9.1:

CW Exercise: 1) A ladder is placed against a wall such that it just reaches the top of the wall. The foot of the

ladder is 1.5 m away from the wall and the ladder is inclined at an angle of 60° with the ground. Find the height of the wall.

2) The length of a string between a kite and a point on the ground is 68m. If the string makes an angle θ with the level ground such that tan θ = (15/8), how high is the kite?

3) The angle of elevation of the sun at certain time was found to be 45°. What was the length of the shadow of a boy of height 1.75 m at that time?

4) A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree. [NCERT]

5) A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming that there is no slack in the string. [NCERT]

6) The horizontal distance between the two trees of different height is 60m. The angle of depression of the first tree when seen from the top of the other is 45°. If the height of the second tree is 80m, find the height of the first tree.

7) An electric pole is 10 m high. A steel wire tied to top of the pole is affixed at a point on the ground to keep the pole upright. If the wire makes an angle of 45° with the horizontal through the foot of the pole, find the length of the wire.

8) At a point on level ground, the angle of elevation of a vertical tower is found to be such that its tangent is 5/12. On walking 192 m towards the tower in the same straight line the tangent of the angle is found to be 3/4. Find the height of the tower?

9) Two ships are sailing in the sea on the either side of the light house, the angle of depression of two ships as observed from the top of the light house are 60° and 45° respectively. If the

distance between the ships is 200 ⎟⎟⎠

⎞⎜⎜⎝

⎛ +

313

. Find the height of the light house.

10) A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building. [NCERT]

11) The angle of elevation of the top Q of the vertical tower PQ from a point X on the ground is 60°. At a point Y, 40m vertically above X, the angle of elevation is 45°. Find the height of the tower PQ and the distance XQ.

12) The angle of elevation of a jet fighter from a point A on the ground is 60°. After a flight of 15 seconds, the angle of elevation changes to 30°. If the jet is flying at a speed of 720 km/hour, find the constant height at which the jet is flying?



13) The angles of elevation of the top of a tower from two points at distances a and b metres from the base and in the same straight line with it are complementary. Prove that the height of the tower is ab metres.

14) A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag-staff are α

and β respectively. Prove that the height of the tower is α−β

αtantan

tanh . [CBSE 08]

15) A round balloon of radius r subtends an angle α at the eye of the observer while the angle of elevation of its centre is β. Prove that the height of the centre of the balloon is r sin β cosec α/2.

16) If the angle of elevation of a cloud from a point h metres above a lake is α and the angle of

depression of its reflection in the lake is β, prove that the height of the cloud is α−βα+β

tantan)tan(tanh .

17) From the top of a hill, the angles of depression of two consecutive kilometer stones due east are found to be 30° and 45°. Find the height of the hill.

18) Determine the height of a mountain if the elevation of its top at an unknown distance from the base is 30° and at a distance of 10km further off from the mountain, along the same line, the

angles of elevation is 15°. (Take tan15° = 1313

+

− )

19) The angles of depression of the top and bottom of a tower as seen from the top of a 60 3 m high cliff are 45° and 60° respectively. Find the height of the tower. [CBSE 2012]

20) The angles of elevation and depression of the top and bottom of a light house from the top of a 60 m high building are 30° and 60°. Find (i) the difference between the heights of the light house and the building (ii) the distance between the light–house and the building. [CBSE 2012]

21) A kite is flying at a height of 30 m from the ground. The length of string from the kite of the ground is 60 m. Assuming that there is no slack in the string, the angle of elevation of the kite at the ground is: [2012]

a) 45° b) 30° c) 60° d) 90° 22) At a point A, 20 metres above the level of water in a lake, the angle of elevation of a cloud is 30°.

The angle of depression of the reflection of the cloud in the lake, at A is 60°. Find the distance of the cloud from A. [2014]

23) In fig. a tower AB is 20 m high and BC, its shadow on the ground, is 20 3 m long. Find the Sun’s altitude. [2014] 24) The angle of depression of a car, standing on the ground, from the top of a 75 m high tower, is

30°. The distance of the car from the base of the tower (in m) is: [2013] a) 25 3 b) 50 3 c) 75 3 d) 150 25) The horizontal distance between two poles is 15 m. The angle of depression of the top of first

pole as seen from the top of second pole is 30°. If the height of the second pole is 24 m, find the height of the first pole. [Use 3 = 1.732] [2014]

26) The angle of elevations of the top of a 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°. If the tower is 60 m high, find the height of the building. [2014]

27) The angle of elevation of an aeroplane from a point A on the ground is 60°. After a flight of 15 seconds, the angle of elevation changes to 30°. If the aeroplane is flying at a constant height of 1500 3 m, find the speed of the plane in km/hr. [2014]

A

B C θ

6


HW Exercise: 1) From a point 20 m away from the foot of a tower, the angle of elevation of the top of the tower is

30°. Find the height of the tower. 2) If the shadow of a tower is 30 m long when the sun’s elevation is 30°, what is the length of its

shadow when the sun’s elevation is 60°? 3) A vertically straight tree, 15 m high, is broken by the wind in such a way that its top just touches

the ground and makes an angle of 60° with the ground. At what height from the ground did the tree break? (Use 3 = 1.73)

4) A flagstaff stands on the top of a 20m high tower. From a point on the ground, the angles of elevation of the top and bottom of the flagstaff are 60° and 45° respectively. Find the height of the flag staff.

5) At a point on level ground, the angle of elevation of a vertical tower is found to be such that its tangent is (7/12). On walking 64m towards the tower, the tangent of the angle of elevation is found to be (3/4). Find the height of the tower.

6) A flag–staff stands on the top of a 5 m high tower. From a point on the ground, the angle of elevation of the top of the flag–staff is 60° and from the same point, the angle of elevation of the bottom of the tower is 45°. Find the height of the flag–staff.

7) There is a small island in the middle of 100m wide river and a tall tree standing on the island. P and Q are points directly opposite to each other on the two banks and in line with the tree. If the angle of elevation of the top of the tree from P and Q are respectively 30° and 45°. Find the height of the tree.

8) Two ships are sailing in the sea on the two sides of a lighthouse. The angles of elevation of the top of the lighthouse as observed form the two ships are 30° and 45° respectively. If the lighthouse is 100m high, find the distance between the two ships.

9) A man standing on the bank of a river observes that angle of elevation of a tree on the opposite bank is 60°, when he moves 50m away from the bank, he finds the angle of elevation to be 30°, calculate: (a) the width of the river (b) the height of the tree.

10 A boy is standing on the ground and flying a kite with 120m of string at an elevation of 30°. Another boy is standing on the roof of a 14m high building and is flying his kite at an elevation of 45°. Both the boys are on opposite sides of the kites. Find the length of the string that the second boy must have so that the two kites meet.

11) From a point P on the ground, the angles of elevation of the top of a 10 m tall building and of a helicopter, hovering at some height over the top of the building, are 30° and 60°, respectively. Find the height of the helicopter above the ground.

12) Find the angle of elevation of the Sun (Sun’s altitude) when the length of shadow of a vertical pole is equal to its height.

13) Two men standing on either side of a tower 30m high observe the angles of elevation of the top of the tower to be 30°and 60° respectively. Find the distance between the two men.

14) A parachutist is descending vertically. At a certain height, the angle of elevation from a point on the ground is 60° and when he has descended 300 m further, the angle of elevation becomes 45° from the same point of observation. Find the distance of the point of observation from the place where the parachutist lands.

15) A vertical wall and a 60m high tower are in the same horizontal plane. From the top of the tower the angles of depression of the top and bottom of the wall are 30° and 45° respectively. Find the height of wall.

16) As observed from the top of a 150m tall light house, the angles of depression of two ships approaching it are 30° and 45°. If one ship is directly behind the other, find the distance between the ships.

17) The angles of depression of top and bottom of a 7 m tall building from the top of a tower are 45° and 60° respectively. Find the height of the tower.



18) A straight highway leads to the foot of a tower of height 50 m. From the top of the tower, the angles of depression of two cars standing on the highway are 30° and 60°. What is the distance between the two cars and how far is each car from the tower?

19) From the top of a cliff 50 m high, the angles of depression of the top and bottom of a tower are observed to be 30° and 45°, respectively. Find the height of the tower. [CBSE 08]

20) A man on the top of a vertical tower observes a car moving at a uniform speed coming directly towards it. If it takes 12 minutes for the angle of depression to change from 30° to 45° how soon after this, will the car reach the tower? Give your answer to the nearest second.

21) There are two temples, one on each bank of a river, just opposite to each other. One temple is 50 m high. From the top of this temple, the angles of depression of the top and the foot of the other temple are 30° and 60° respectively. Find the width of the river and the height of the other temple.

22) From an aeroplane vertically above a straight horizontal road, the angles of depression of two consecutive mile stones on opposite sides of the aero plane are observed to be α and β. Show

that the height in miles of aero plane above the road is given by β+α

βαtantan

tantan .

23) From the top of a light house, the angles of depression of two ships on the opposite sides of it are observed to be α and β. If the height of the light house be h metres and the line joining the ships passes through the foot of the light house, show that the distance between the ships is

βαβ+α

tantan)tan(tanh metres.

24) An aero plane at an altitude of 900m finds that two ships are sailing towards it in the same direction. The angles of depression of the ships as observed from the plane are 60° and 30° respectively. Find the distance between the ships.

MISCELLANEOUS EXERCISE:

1) An aero plane when flying at a height of 4000 m from the ground passes vertically above another aero plane at an instant when the angles of the elevation of the two planes from the same point on the ground are 60° and 45° respectively. Find the vertical distance between the aero planes at that instant. [CBSE 08–09]

2) The angle of elevation of the top of a 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°. If the tower is 50 m high, find the height of the building. [CBSE 09]

3) A man in a boat rowing away from a lighthouse 150m high takes 2 minutes to change the angle of elevation of the top of the lighthouse from 45° to 30°. Find the speed of the boat.

4) A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point. [NCERT, CBSE 08–09]

5) The angle of elevation of the top of a tower as observed from a point on the ground is ‘α’ and on moving ‘x’ metres towards the tower, the angle of elevation is ‘β’. Prove that the height of the

tower is α−ββα

tantantantanx .

6) The length of the shadow of a tower standing on level plane is found to be 2x metres longer when the sun’s altitude is 30° than when it was 45°. Prove that the height of tower is x ( 3 + 1) metres.

7) The angle of elevation of an aero plane from a point A on the ground is 60°. After a flight of 30 seconds, the angle of elevation changes to 30°. If the plane is flying at a constant height of 3600

3 m, find the speed, in km/hour, of the plane. [CBSE 08]

8


8) A statue 1.46 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point, the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal (use 3 = 1.73). [CBSE 08]

9) The shadow of a tower, when the angle of elevation of the sun is 45° is found to be 10 m longer than when it is 60°. Find the height of the tower.

10) The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower. [NCERT]

11) Find the angle of elevation of the sun when the length of the shadow of a 48 metre long pole, is 48 3 m.

12) The horizontal distance between two towers is 140m. The angle of elevation of the top of the first tower when seen from the top of the second tower is 30°. If the height of the second tower is 60m, find the height of the first tower.

13) A circus artist is climbing from the ground along a rope stretched from the top of vertical pole and tied at the ground. The height of the pole is 12 cm and the angle made by the rope with ground level is 30°. Calculate the distance covered by the artist in climbing to the top of the pole.

14) An observer 1.6m tall is 20 3 m away from a tower. The angle of elevation from his eye to the top of the tower is 30°. Determine the height of the tower.

15) A ladder is placed along a wall of a house such that its upper end is touching the top of the wall. The foot of the ladder is 2 m away from the wall and the ladder is making an angle of 60° with the level of the ground. Determine the height of the wall.

16) A balloon is connected to a meteorological ground station by a cable of length 215 m inclined at 60° to the horizontal. Determine the height of the balloon from the ground. Assume that there is no slack in the cable.

17) A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30°. [NCERT]

18) A person standing on the bank of a river observes that the angle of elevation of the top of a tree

standing on the opposite bank is 60°. When he moves 40 m away from the bank, he finds the angle of elevation to be 30°. Find the height of tree and the width of the river. (use 3 =1.732)

19) A bridge across a river makes an angle of 45° with the river. If the length of the bridge across the river is 150m. What is the width of the river? 20) The shadow of a tower, standing on level ground, is found to be 45 m longer when Sun’s altitude

is 30° than when it was at 60°. Find the height of the tower. 21) A tower stands vertically on the ground. From a point on the ground, 20 m away from the foot of

the tower, the angle of elevation of the top of the tower is 60°. What is the height of the tower? 22) A vertical tower stands on a horizontal plane and is surmounted by a flag–staff of height 7m.

From a point on the plane, the angle of elevation of the bottom of the flag–staff is 30° and that of the top of the flag–staff is 45°. Find the height of the tower.

23) A 10m high pole is fixed on the top of a tower. The angles of elevation of the top and bottom of the pole from a point on the ground are 45° and 30° respectively. Find the height of the tower.

150m

45° RiverRiver

C B

A

30°

20m



24) From a point on the ground 40 m away from the foot of a tower, the angle of elevation of the top of the tower is 30°. The angle of elevation to the top of a water tank (on the top of the tower) is 45°. Find the (i) height of the tower (ii) the depth of the tank.

25) Two poles of equal heights are standing opposite to each other on either side of a road, which is

100 metres wide. From a point between them on the road, the angles of elevation of their tops are 30° and 60°. Find the position of the point and also the heights of the poles. [NCERT]

26) A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is 60°. From another point 20 m away from this point on the line joining this point to the foot of the tower; the angle of elevation of the top of the tower is 30°. Find the height of the tower and the width of the canal. [NCERT]

27) A man standing on the deck of a ship, which is 10 m above the water level, observes the angle

of elevation of the top of a hill as 60° and the angle of depression of the base of the hill at 30°. Calculate the distance of the hill from the ship and the height of the hill.

28) A tower subtends an angle α at a point on the same level as the foot of the tower and at a second point h metres above the first, the depression of the foot of the tower is β. Show that the height of the tower is h (tan α.tan β + 1).

29) A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case? [NCERT]

30) From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower. [NCERT]

31) As observed from the top of a 75 m high lighthouse from 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 lighthouse, find the distance between the two ships. [NCERT]

32) The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 6 m. [NCERT]

33) If the angle of elevation of a cloud from a point h metres above a lake is α and the angle of depression of its reflection in the lake be β, prove that the distance of the cloud from the point of

observation is α−β

αtantan

sec2h .

MULTIPLE CHOICE QUESTIONS:

CW Exercise:

1) The ratio of the length of a rod and its shadow is 1: 3 . The angle of elevation of the sun is

a) 30° b) 45° c) 60° d) 90°

30° 45A

B

h

C

D

40 m

D C B

A

60°30° 20 m

10


2) If the altitude of the sun is at 60°, then the height of the vertical tower that will cast a shadow of length 30m is

a) 30 3 m b) 15 m c) 3

30 m d) 15 2 m

3) If the angles of elevation of the top of tower from two points distant a and b from the base and in the same straight line with it are complementary, then the height of the tower

a) ab b) ab c) ba d)

ba .

4) The angle of elevation of the top of a tower standing on a horizontal plane from a point A is α.After walking a distance d towards the foot of the tower the angle of elevation is found to be β. The height of the tower is

a) β+α cotcot

d b) β−α cotcot

d c) α−β tantan

d d) α+β tantan

d .

5) From the top of a cliff 25m high the angle of elevation of a tower is found to be equal to the angle of depression of the foot of the tower. The height of the tower is

a) 25 m b) 50 m c) 75 m d) 100 m 6) If the angle of elevation of a cloud from a point 200 m above a lake is 30° and the angle of

depression of its reflection in the lake is 60°, then the height of the cloud above the lake is a) 200 m b) 500 m c) 30 m d) 400 m 7) Two posts are a metres apart and the height of one is double that of the other. If from the

middle point of the line joining their feet, an observer finds the angular elevation of their tops to be complementary, then the height of the shorter post is

a) 4a b)

2a c) a 2 d)

22a

8) A flagstaff 6 metres high throws a shadow 2 3 metres long on the ground. The angle of elevation is:

a) 30° b) 45° c) 90° d) 60° 9) An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of

the chimney from her eyes is 45°. The height of the chimney is: a) 30 m b) 27 m c) 28.5 m d) none of these 10) A kite is flying at a height of 60 m above the ground. The string attached to the kite is

temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. The length of the string is :

a) 40 3 m b) 30 m c) 20 3 m d) 60 3 m. 11) From a point on a bridge across a river, the angle of depression of the banks on opposite

sides of the river are 30° and 45° respectively. If the bridge is at a height of 3m from the bank then the width of the river is :

a) 3 ( 3 – 1) m b) 3 ( 3 + 1) m c) (3 + 3 ) m d) (3 – 3 ) m 12) A 1.5m tall boy is standing at some distance from a 30 m tall building. The angle of

elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. The distance he walked toward the building is :

a) 19 3 m b) 57 3 m c) 38 3 m d) 18 3 m 13) The shadow of a tower standing on a level ground is found to be 40 m longer when the

sun’s altitude is 30° than when it is 60°. The height of the tower is: a) 30 3 m b) 20 3 m c) 40 3 m d) 10 3 m.



HW Exercise: 1) If the angle of elevation of a tower from a distance of 100 meters from its foot is 60°, then

the height of the tower is

a) 100 3 m b) 3

100 m c) 50 3 m d) 3

200 m

2) If the angles of elevations of a tower from two points distant a and b (a > b) from its foot and in the same straight line from it are 30° and 60°, then the height of the tower is

a) ba + b) ab c) ba − d) ba

3) From a light house the angles of depression of two ships on opposite sides of the light house are observed to be 30° and 45°. If the height of the light house is h metres, the distant between the ships is

a) ( )13 + h metres b) ( )13 − h metres c) 3 h metres d) 1+ ⎟⎟⎠

⎞⎜⎜⎝

⎛+

311 h metres

4) The tops of two poles of height 20m and 14 m are connected by a wire. If the wire makes an angles of 30° with horizontal, then the length of the wire is

a) 12 m b) 10 m c) 8 m d) 6 m 5) The angles of depression of two ships from the top of a light house are 45° and 30°

towards east. If the ships are 100 m apart, the height of the light house is

a) 13

50+

m b) 13

50−

m c) 50( 13 − ) m d) 50( 13 + ) m

6) The height of a tower is 100 m. When the angle of elevation of the sun changes from 30° to 45°, the shadow of the tower becomes x meters less. The value of x is

a) 100 m b) 100 3 m c) 100( 13 − )m d) 3

100 m

7) A tower subtends an angle of 30° at a point on the same level as its foot. At a second point h metres above the first, the depression of the foot of the tower is 60°. The height of the tower is

a) 2h m b) 3 hm c)

3h m d)

3h m

8) An observer 3 m tall is 3 m away from the pole 2 3 high. The angle of elevation of the top of the pole from the eye of the observer is,

a) 45° b) 30° c) 60° d) 15° 9) The angle of elevation of the top of a tower from a distance 100 m from its foot is 30°. The

height of the tower is:

a) 100 3 m b) m3

200 c) 50 3 d) m3

100

10) A tree is broken by the wind, its top stuck the ground at an angle 30° at a distance of 30 m from its foot. The whole height of the tree is:

a) 10 3 m b) 20 3 m c) 40 3 m d) 30 3 m 11) The Angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m

from the base of the tower and in the same straight line with it are complementary. The height of the tower is:

a) 5 m b) 13 m c) 6 m d) 2.25 m

12


12) As observed from the top of a 75 m high lighthouse from the sea–level, the angles depression of two ships are 30° and 60°. If one ship is exactly behind the other one the same side of the light–house, then the distance between the two ships is.

a) 25 3 m b) 75 3 m c) 50 3 m d) none of these

PREVIOUS YEAR BOARD QUESTIONS:

1) From the top of a tower 100 m high, a man observes two cars on the opposite sides of the tower with angles of depression 30º and 45º respectively. Find the distance between the cars. [Use

3 = 1.732] [Delhi 2011] 2) a ladder of length 6 m makes an angle of 45º with the floor while leaning against one half of a

room. If the foot of the ladder is kept fixed on the floor and it is made to lean against the opposite wall of the room, it makes an angle of 60º with the floor. Find the distance between these two walls of the room. [Foreign 2011]

3) The angle of depressions of two ships from the top of a light house and on the same side of it are found to be 45º and 30º. If the ships are 200 m apart, then find the height of the light house. [Delhi 2012, NCERT]

4) The horizontal distance between two poles is 15 m. The angle of depression of the top of first pole as seen from the top of second pole is 30º. If the height of the second pole is 24 m, find the height of the first pole. (Use 3 = 1.732) [Delhi 2013]

5) The angle of elevation of the top of a tower at a distance of 120 m from a point A on the ground is 45º. If the angle of elevation of the top of a flagstaff fixed at the top of the tower, from A is 60º, then find the height of the flagstaff. (Use 3 = 1.73) [2014]

6) Two ships are there in the sea on either side of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of two ships as observed from the top of the light house are 60° and 45°. If the height of the light house is 200 rn, find the distance between the two ships. [Use 3 = 1.73] [Delhi 2014]

7) The angle of elevation of an aeroplane from a point on the ground is 60°. After a flight of 30 seconds the angle of elevation becomes 30°. If the aeroplane is flying at a constant height of 3000 3 m, find the speed of the aeroplane. [2014]

8) At a point A, 20 meters above the level of water in a lake, the angle of elevation of a cloud is 30º. The angle of depression of the reflection of the cloud in the lake, at A is 60º. Find the distance of the cloud from A. [2015]

9) The angle of elevation of the top of a 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 45°. If the tower is 30 m high, find the height of the building. [Delhi 2015]

10) The angle of elevation of an aeroplane from a point A on the ground is 60°. After a flight of 15 seconds, the angle of elevation changes to 30°. If the aeroplane is flying at a constant height of 1500 3 m, find the speed of the plane in km/hr. [2015]

11) A bird is sitting on the top of a 80 m high tree. From a point on the ground, the angle of elevation of the bird is 45º. The bird flies away horizontally in such a way that it remained at a constant height from the ground. After 2 seconds, the angle of elevation of the bird from the same point is 30º. Find the speed of flying of the bird. (Take 3 = 1.732) [Delhi 2016]

12) From the top of a tower 50 m high the angles of depression of the top and bottom of a pole are observed to be 45º and 60º respectively. Find the height of the pole. [Foreign 2016]

13) As observed from the top of light–house, 100 m high above sea level, the angle of depression of a ship, sailing directly towards it, changes from 30º to 60º. Determine the distance travelled by the ship during the period of observation. (Use 3 = 1.732) [Foreign 2016]



ANSWERS TO UNSOLVED EXERCISES:

CW Exercise 9.1: 1) 2.59 m 2) 60 m 3) 1.75 m 4) 8 3 m

5) 40 3 m 6) 20 m 7) 14.1 m 8) 180 m

9) 200 m 10) 19 3 m 11) 94.64, 109.3 12) 1500 3 m

17) 2

13 + or 1.366 km 18) 5 km 19) 60( 3 – 1) 20) (i) 20 m (ii) 20 3

21) b 22) 20 3 m 23) 30° 24) c 25) 15.35 m 26) 20 m 27) 720 km/hr HW Exercise 9.1: 1) 11.53 m 2) 10 m 3) 6.9 m 4) 14.64 m

5) 168 m 6) 3.65 m 7) 36.6 m 8) 100 ( 13 + )

9) 25, 25 3 10) 46 2 m 11) 30 m 12) 45°

13) 40 3 14) 409.8 m 15) 14.6 3 16) 109.5 m 17) 16.56 m 18) 57.7, 28.9, 86.6 19) 21.2 m 20) 16 min 23 sec 21) 28.83 m, 33.33 m 24) 1039.2 m Miscellaneous Exercise:

1) 1690.53 m 2) 1632 m 3) 3.294 km/hr 4) 3 sec

7) 864 km/hr 8) 2 m 9) 23.66 m 10) 10 3 m 11) 30° 12) 140.83 m 13) 24 m 14) 21.6

15) 3.46 or 2 3 m 16) 186 m 17) 10 m 18) 34.64 m, 20 m

19) 105.75 m 20) 38.93 m 21) 20 3 22) 9.56 m

23) 13.65 m 24) 23.1 m, 16.9 m 25) 43.3 m, 75 m 26) 10 3 m, 10 m

27) 17.3 m, 40m 29) 3m, 2 3 m 30) 7 ( 3 + 1) m 31) 75 ( 3 – 1) m Multiple Choice Questions: CW Exercise:

1) a 2) a 3) b 4) b 5) b 6) d 7) d 8) c 9) a 10) a 11) b 12) a 13) b

HW Exercise: 1) a 2) b 3) a 4) a 5) d 6) c 7) c 8) b 9) d 10) d

11) c 12) c Previous Year Board Question: 1) 273 m 2) 7.23 m 3) 100 ( 3 +1) 4) 15.34 m 5) 87.6 m 6) 315.33 m 7) 720 km/hr. 8) 40 m

9) 10 3 m 10) 720 km/hr 11) 29.28 m/sec.

12) ( )3

1350 − m 13) 115.466 m

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