CBSE Class 10 Mathematics Important Questions Chapter 9 ...

CBSEClass10Mathematics

ImportantQuestions

Chapter9

SomeApplicationsofTrigonometry

1MarksQuestions

1. Amanstandingonthedeckofaship,whichis10mabovethewaterlevel,observes

theangleofelevationofthetopofahillas andtheangleofdepressionofthebase

ofthehillas .Calculatethedistanceofthehillfromtheshipandtheheightofthe

hill.

Ans.LetABbetheshipandCDbethehill.

LetBD=xm

Inright

Inright

Heightofthehill=h+10

=30+10=40m

2. Aboyisstandingonthegroundandflyingakitewith100mofstringatanelevation

of .Anotherboyisstandingontheroofofa20mhighbuildingandisflyinghis

kiteat

anelevationof .Boththeboysareontheoppositesidesofboththekites.Findthe

lengthofthestringthatthesecondboymusthavesothatthetwokitesmeet.

Ans.AE=100m(Given)andDC=20m(Given)

LetEF=hm

Inright ABE,

Nowinright

3. TheangleofelevationofthetopQofaverticaltowerPQfromapointXonthe

groundis .AtapointY,40mverticallyaboveX,theangleofelevationis ,find

theheightofthetowerPQandthedistanceXQ.

Ans.Inright

Inright

Inright

4. Atthefootofamountaintheelevationofitssummitis .Afterascending1000m

towardsthemountainupaslopeof inclination,theelevationisfoundtobe .

Findtheheightofthemountain.

Ans.Inright

Inright

Inright

5. Iftheanglesofelevationofthetopofatowerfromtwopointsatdistancesaandb

fromthebaseandinthesamestraightlinewithitarecomplementary,thentheheight

ofthetoweris

(a)

(b) ab

(c)

(d)

Ans.a)

6. Iftheheightoftowerishalftheheightoftheflagstaffonitandtheangleofelevation

ofthetopofthetowerasseenfromapointonthegroundis30 ,thentheangleof

elevationofthetopoftheflagstaffasseenfromthesamepointis

(a) 30

(b)45

(c) 60

(d) 90

Ans.(b)45

7. Asobserved fromthe topofa75m tall lighthouse theanglesofdepressionof two

shipsare30 and45 . Ifoneship isexactlybehindtheotheronthesamesideof the

lighthouse,thendistancebetweenthetwoshipsis

(a)

(b)

(c) 75

(d)

Ans.(a)

8. Theangleofelevationofthetopabuildingfromthefootofthetoweris30 andthe

angleof elevationof the topof the tower from the footof thebuilding is 60 . If the

toweris50mhigh,thentheheightofthebuildingis

(a) 50

(b)

(c)

(d)

Ans.(b)

9. A1.6mtallgirlstandsatadistanceof3.2mfromalamppostandcastsashadowof

4.8montheground,thentheheightofthelamppostis

(a)

(b)

(c)

(d)

Ans.(a)

10. Atreebreaksduetostormandbrokenpartbendssothatthetopofthetreetouches

thegroundmakinganangleof30 withground.Ifthedistancebetweenthefootofthe

treetothepointwherethetoptouchesthegroundis8m,thentheheightofthetreeis

(a)

(b)

(c)

(d)

Ans.(c)

11. Theangle of elevationof the sunwhen the lengthof the shadowof apole is

times,thenheightofthepoleis

(a) 30

(b) 45

(c) 60

(d) 75

Ans.(a)30

12. Fromthefigure,theangleofelevation is

(a) 30

(b) 60

(c) 45

(d) 75

Ans.(b)60

13. A person walking 20 m towards a chimney in a horizontal line through its bare

observerthatitsangleofelevationchangesfrom30 to45 ,thenheightofchimneyis

(a)

(b)

(c)

(d)

Ans.(b)

14. Findheightofthetoweriflengthofshadow=10mandsun’saltitude=45 .

(a) 10m

(b) 12m

(c) 20m

(d) noneofthese

Ans.(a)10m

15. Iftheanglesofelevationofatowerfromtwopointsatdistancesaandbwherea>b,

fromitsfootandinthesamestraightlinefromitare30 and60 ,thenheightofthe

toweris

(a)

(b)

(c)

(d)

Ans.(b)

16. Theangleofelevationofthetopofatowerfromapointonthegroundwhichis30m

awayfromthefootofthetoweris30 ,thenheightoftoweris

(a) 10m

(b) m

(c) m

(d) m

Ans.(a)10m

17. Ifheightofthetower=shadowofthetower,thenangleofelevationis

(a) 30

(b) 45

(c) 60

(d) 90

Ans.(b)45

18. Iflengthofshadow=20mandangleofelevation=60 .Thenheightoftoweris

(a) 20m

(b)

(c)

(d)

Ans.(d)

19. Alittleboyisflyingakite.Thestringofkitemakesanangleof30 withtheground.

Iflengthofthekiteish=21m,thenlengthofstringis

(a) 63m

(b) 42m

(c) 35m

(d) 21m

Ans.(b)42m

20. Linejoininganeyeandtheobjecttobeviewediscalled

(a) Horizontal

(b) lineofsight

(c) Verticalline

(d) Noneofthese

Ans.(b)lineofsight


ImportantQuestions

Chapter9


2MarksQuestions

1. Acircusartistisclimbinga20mlongrope,whichistightlystretchedandtiedfrom

thetopofaverticalpoletotheground.Findtheheightofthepole,iftheanglemadeby

theropewiththegroundlevelis (seefigure).

Ans.InrighttriangleABC,

AB=10m

Hence,theheightofthepoleis10m.

2. A tree breaks due to storm and the broken part bends so that the top of the tree

touchesthegroundmakinganangle withit.Thedistancebetweenthefootofthe

treetothepointwherethetoptouchesthegroundis8m.Findtheheightofthetree.


m

Again,

AB= m

Heightofthetree=AB+AC

= =

= = m

3. A contractor plans to install two slides for the children to play in a park. For the

childrenbelowtheageof5years,shepreferstohaveaslidewhosetopisataheightof

1.5mandisinclinedatanangleof totheground,whereasforelderchildren,she

wants tohavea steep slideat aheightof 3mand inclinedat anangleof to the

ground.Whatshouldbethelengthoftheslideineachcase?


AC=3m

InrighttrianglePQR,

PR= m

Hence,thelengthsoftheslidesare3mand mrespectively.

4. Theangleofelevationofthetopofatowerfromapointontheground,whichis30m

awayfromthefootoftheloweris .Findtheheightofthetower.


AB= m

= m

5. Akiteisflyingataheightof60mabovetheground.Thestringattachedtothekiteis

temporarilytiedtoapointontheground.Theinclinationofthestringwiththeground

is Findthelengthofthestring,assumingthatthereisnoslackinthestring.


AC= m

Hencethelengthofthestringis m.

6. ATV towerstandsverticallyonabankofacanal.Fromapointon theotherbank

directlyoppositethetower,theangleofelevationofthetopofthetoweris From

anotherpoint20mawayfromthispointonthelinejoiningthispointtothefootofthe

tower,theangleofelevationofthetopofthetoweris (seefigure).Findtheheight

ofthetowerandthewidthofthecanal.


AB= m……….(i)

InrighttriangleABD,

AB= m……….(ii)

Fromeq.(i)and(ii),

=

3BC=BC+20

BC=10m

Fromeq.(i),AB= m

Henceheightofthetoweris mandthewidthofthecanalis10m.

7. Fromthetopofa7mhighbuilding,theangleofelevationofthetopofacabletower

is andtheangleofdepressionofitsfootis Determinetheheightofthetower.

Ans.InrighttriangleABD,

BD=7m

AE=7m

InrighttriangleAEC,

CE= m

CD=CE+ED

=CE+AB

=

= m

Henceheightofthetoweris m.

8. Aroundballoonofradiusrsubtendsanangle attheeyeoftheobserverwhilethe

angle of the elevation of its centre is . Prove that the height of the centre of the

balloonisrsin cosec

Ans.Inright

Inright

9. Anaeroplaneflyinghorizontally1kmabovethegroundisobservedatanelevation

of .After10seconds,itselevationisobservedto .Findthespeedofthe

aeroplaneinkm/hr.

Ans.Inright

Inright

Speed=

10. Fromawindow(hmhighabovetheground)ofahouseinastreet,theanglesof

elevationanddepressionofthetopandfootofanotherhouseontheoppositesideof

thestreetare and respectively,showthattheheightoftheoppositehouseis

Ans.LetDE=hm

DC=xm

Inright

Inright

HenceProved.

11. Aladderrestsagainstawallatanangle tothehorizontal.Itsfootispulledaway

fromthewallthroughadistance‘a’sothatitslidesadistance‘b’downthewallmaking

anangle‘ ’withthehorizontal.Showthat

Ans.Inright

Similarly,inright

HenceProved.

12. Averticaltowerstandsonahorizontalplaneandsurmountedbyverticalflagstaff

ofheighth.Atapointontheplane,theanglesofelevationofthebottomandthetopof

theflagstaffare and respectively,Provethattheheightofthetoweris

Ans.LetAB=Heightoftower=H

LetBC=Heightofflagstaff=h

Inright-angledtriangleOABandOAC,

And

Equatingvalueofx,weget

13. Theangleofelevationofthetopofatoweratapointonthelevelgroundis30 .

Afterwalkingadistanceof100mtowardsthefootofthetoweralongthehorizontal

linethroughthefootofthetoweronthesamelevelgroundtheangleofelevationtothe

topofthetoweris60 ,findtheheightofthetower.

Ans.In ,

In

14. Asobservedfromthetopoflighthouse100mhighabovesealeveltheangleof

depressionofashipsailingdirectlytowardsitchangesfrom30 to60 .Determinethe

distancetravelledbytheshipduringtheperiodofobservation.(Use )

Ans.LetPQbethelighthousesuchthatPQ=100m.LetAandBbethepositionsofship

whentheangleofdepressionchangesfrom30 to60 respectively.

Let andBP=ym

Fromright-angled

Fromright-angledtriangleBPQ,

From(i)and(ii),

15. TheanglesofelevationofthetopofatowerfromtwopointsPandQatdistancesof

aandbrespectivelyfromthebaseandinthesamestraightlinewithare

complementary.Provethattheheightofthetoweris wherea>b.

Ans.LetABbetowerofheighth.LetPandQbethegivenpointsinthesamestraightline

withthefootBofthetower.

LetBP=a,BQ=b

Fromrightangled

Fromright-angled

Multiplying(i)and(ii),weget

16. Anaeroplaneflyinghorizontallyataheightof1.5kmabovethegroundisobserved

atacertainpointontheearthtosubtendanangleof60 .After15seconds,itsangleof

elevationisobservedtobe30 .Calculatethespeedofaeroplaneinkm/hr.

Ans.LetObetheobservationpoint.

LetAbethepositionofaeroplanesuchthat and

LetBbethepositionofaeroplaneafter15seconds.

Inright

Inright

Distancecoveredin15seconds

Distancecoveredin1second=

Distancecoveredin3600seconds=

17. Amanisstandingonthedeckofashipwhichis25mabovewaterlevel.Heobserves

theangleofelevationofthetopofalighthouseas60 andtheangleofdepressionof

thebaseofthelighthouseas45 .Calculatetheheightofthelighthouse.

Ans.H=Heightoflighthouse=h+25…….(i)

Inright

Inright

Now

18. Anaeroplanewhenflyingataheightof3125mformthegroundpassesvertically

belowanotheraeroplaneataninstantwhentheangleofelevationofthetwoplanes

fromthesamepointonthegroundare30 and60 respectively.Findthedistance

betweenthetwoaeroplanesatthatinstant.

Ans.Let and bethepositionsofthetwoaeroplanes

Let

AndOM=y

Also

19. Aboystandingonahorizontalplanefindsabirdflyingatadistanceof100mfrom

himatanelevationof30 .Agirlstandingontheroofof20mhighbuildingfindsthe

angleofelevationofthesamebirdtobe45 .Boththeboyandthegirlareonopposite

sideofthebird.Findthedistanceofthebirdfromthegirl.

Ans.PositionsofbirdatA,boyatPandgirlatBareasshowninfigure.

In

AlsoBC=DQ=20m

In

Hence,thebirdis30 mawayfromthegirl.

20. Atapointonlevelground,theangleofaelevationofaverticaltowerisfoundtobe

suchthatitstangentis onwalking19.2mtowardsthetower,thetangentofthe

angleofelevationis .Findtheheightoftower.

Ans.Supposeheightoftowerishmeter

In

In

21. Fromthetopofabuilding60mhigh,theanglesofdepressionofthetopandbottom

ofaverticallamppostareobservedtobe30 and60 respectively.Find:

(i) thehorizontaldistancebetweenthebuildingandthelamppost

(ii) theheightofthelamppost.[Take ]

Ans.Supposeheightoflamppostishmeter.

In

Inright

Byeq.(i)and(ii)

(60–h) =20

22. Amanstandingonthedeckofashipwhichis10mabovethewaterlevelobserves

theangleofelevationofthetopofahillas60 andtheangleofdepressionofthebase

ofthehillas30 .Calculatethedistanceofthehillfromtheshipandtheheightofthe

hill.

Ans.LetH=Heightofhill

Inright

Inright

Equatingthevaluesofx,weget

FromH=10+h=10+30=40m

Andx=distanceofhillfromship=10 m

23. Theangleofelevationofajet-planefromapointPonthegroundis60 .Aftera

flightof15secondstheangleofelevation(changeto30 ).Ifthejetplaneisflyingata

constantheightof ,findthespeedofthejetplaneinkm/hour.

Ans.LetAbethepointonthegroundEisthepositionofaeroplanesuchthat

and

Cisthepositionofplaneafter15seconds

Inright

In

Distance

Speed=

24. Apole5mhighisfixedonthetopofatower.Theangleofelevationofthetopof

poleobservedfromapointAonthegroundis60 andtheangleofdepressionofthe

point A from the top of the tower is 45 . Find the height of the tower. (Take

)

Ans.In

In

25. Fromawindow15mhighabovethegroundinastreet.Theanglesofelevationand

depressionofthetopandfootofanotherhouseontheoppositesideofthestreetare30

and45 respectively.Showthattheheightoftheappositehouseis23.66m.[Take

]

Ans.Suppose and

Then,AB=

In

In

Then

Heightofanotherhouse=

26. Theangleofelevationofthetopofatowerasobservedfromapointontheground

is andonmoving‘a’metertowardsthetower.Theangleofelevationis prove

thatheightoftoweris

Ans.LetABbtowerandheightoftower=hm

In

In

27. ATVtowerstandsverticallyonabankofacanal.Fromapointontheotherbankof

acanal.Fromapointontheotherbankdirectlyoppositethetower,theangleof

elevationofthetopofthetoweris60 fromapoint20mawayfromthispointonthe

samebacktheangleofelevationofthetopofthetoweris30 .Findtheheightofthe

towerandthewidthofthecanal.

Ans.Lethbetheheightoftowerandxbethewidthoftheriver

In

In

Equating(i)and(ii),

Put in(i),


ImportantQuestions

Chapter9


3MarksQuestions

1. A1.5mtallboyisstandingatsomedistancefroma30mtallbuilding.Theangleof

elevationfromhiseyestothetopofthebuildingincreasesfrom to ashewalks

towardsthebuilding.Findthedistancehewalkedtowardsthebuilding.

Ans.AB=30mandPR=1.5m

AC=AB–BC

=AB–PR

=30–1.5

=28.5m

InrighttriangleACQ,

QC= m

InrighttriangleACP,

PQ=

PQ= = m

Hence,thewalkedtowardsthebuildingis m.

2. Fromapointontheground, theanglesofelevationof thebottomandthe topofa

transmission tower fixed at the top of a 20 m high building are and

respectively.Findtheheightofthetower.

Ans.Lettheheightofthetowerbe m.Then,

inrighttriangleCBP,

……….(i)

InrighttriangleABP,

BP=20m

Puttingthisvalueineq.(i),weget,

=

m

Theheightofthetoweris m.

3. A statue, 1.6m tall, standson the topofapostal.Fromapointon theground, the

angleofelevationofthetopofthestatueis andfromthesamepointtheangleof

elevationofthetopofthepedestalis Findtheheightofthepedestal.

Ans.Lettheheightofthepedestalbe m.

BC= m

InrighttriangleACP,

……….(i)

InrighttriangleBCP,

PC=

[Fromeq.(i)]

m

4. Asobservedfromthetopofa75mhighlighthousefromthesea-level,theanglesof

depressionoftwoshipsare and Ifoneshipisexactlybehindtheotheronthe

samesideofthelighthouse,findthedistancebetweentwoships.

Ans.InrighttriangleABQ,

BQ=75m……….(i)

InrighttriangleABP,

[Fromeq.(i)]

75+QP=

QP= m

Hencethedistancebetweenthetwoshipsis m.

5. Agirlwhois1.2mtall,spotsaballoonmovingwiththewindinahorizontallineata

heightof88.2mfromtheground.Theangleofelevationoftheballoonfromtheeyeof

thegirlatanyinstantis .Aftersometime,theangleofelevationreducesto .

Findthedistancetravelledbytheballoonduringtheinterval.

Ans.Inright

Inright

6. Fromthetopofatower96mhigh,theanglesofdepressionoftwocarsonaroadat

thesamelevelasthebaseofthetowerandonsamesideofitareθand ,where

and .Findthedistancebetweenthetwocars.

Ans.Inright

Inright


ImportantQuestions

Chapter9


4MarksQuestions

1. Theangleofelevationofthetopofabuildingfromthefootofthetoweris and

theangleofelevationofthetopofthetowerfromthefootofthebuildingis Ifthe

toweris50mhigh,findtheheightofthebuilding.

Ans.Lettheheightofthebuildingbe m.

BQ= m……….(i)

InrighttriangleABQ,

BQ= m……….(ii)

Fromeq.(i)and(ii),

m

2. Two poles of equal heights are standing opposite each other on either side of the

road, which is 80 m wide. From a point between them on the road, the angles of

elevationof thetopof thepolesare and respectively.Find theheightof the

polesandthedistancesofthepointfromthepoles.


AB= m……….(i)

InrighttriangleABD,

AB= m……….(ii)

Fromeq.(i)and(ii),

=

3BC=BC+20

BC=10m

Fromeq.(i),AB= m

Henceheightofthetoweris mandthewidthofthecanalis10m.

3. A1.2mtallgirlspotsaballoonmovingwiththewindinahorizontallineataheight

of88.2mfromtheground.Theangleofelevationof theballoonfromtheeyesofthe

girlatanydistantis Aftersometime,theangleofelevationreducesto (see

figure).Findthedistancetravelledbytheballoonduringtheinterval.


BC= m

InrighttrianglePQC,

=176.4

BQ= =

= = = m

Hencethedistancetravelledbytheballoonduringtheintervalis m.

4. Astraighthighwayleadstothefootofatower.Amanstandingatthetopofthetower

observesacaratanangleofdepressionof ,whichisapproachingthefootofthe

towerwith a uniform speed. Six seconds later, the angle of depression of the car is

foundtobe Findthetimetakenbythecartoreachthefootofthetowerfromthis

point.

Ans.InrighttriangleABP,

BP=AB ……….(i)

InrighttriangleABQ,

BQ= ……….(ii)

PQ=BP–BQ

PQ=AB = = =2BQ[Fromeq.(ii)]

BQ= PQ

TimetakenbythecartotraveladistancePQ=6seconds.

TimetakenbythecartotraveladistanceBQ,i.e. PQ= x6=3seconds.

Hence,thefurthertimetakenbythecartoreachthefootofthetoweris3seconds.

5. Theanglesofelevationofthetopofatowerfromtwopointsatadistanceof4mand

9mfromthebaseofthetowerandinthesamestraightlinewithitarecomplementary.

Provethattheheightofthetoweris6m.

Ans.Let APB=

Then, AQB=

[ APBand AQBarecomplementary]

InrighttriangleABP,

……….(i)

InrighttriangleABQ,

=

……….(ii)

Multiplyingeq.(i)andeq.(ii),

AB2=36

AB=6m

Hence,theheightofthetoweris6m.

Proved.

CBSE Class 10 Mathematics Important Questions Chapter 9 ...

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