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CBSE Questions for Class 10 Maths Some Applications Of Trigonometry Quiz 12 - MCQExams.com
CBSE
Class 10 Maths
Some Applications Of Trigonometry
Quiz 12
A
,
B
,
C
are three collinear points on the ground such that
B
lies between
A
and
C
and
A
B
=
10
m
.
If the angles of elevation of the top of a vertical tower standing at
C
are respectively
30
∘
and
60
∘
as seen from
A
and
B
,
then the height of the tower is:
Report Question
0%
5
√
3
m
0%
5
m
0%
10
√
3
3
m
0%
20
√
3
3
m
The angle of elevation of the top of a tower from a point A on the ground is
30
0
. On moving a distance of 20 metres towards the foot of the tower to a point B, the angle of elevation increases to
60
0
. The height of the tower is :
Report Question
0%
√
3
m
0%
5
√
3
m
0%
10
√
3
m
0%
20
√
3
m
Explanation
Let the height of the tower be 'h' m.
In right
△
D
C
B
tan
60
=
D
C
C
B
⇒
√
3
=
h
x
∴
x
=
h
√
3
In right
△
D
C
A
tan
30
=
D
C
C
A
⇒
1
√
3
=
h
x
+
20
⇒
h
√
3
+
20
=
√
3
h
∴
h
=
10
√
3
Hence the height of the tower is
10
√
3
m
.
The angles of elevation of the top of a tower from two points A and B lying on the horizontal plane through the foot of the tower are respectively
45
0
and
30
0
. If A and B are on the same side of the tower and
A
B
=
48
m
e
t
r
e
, then the height of the tower is:
Report Question
0%
24
(
√
3
+
1
)
metre
0%
24 metre
0%
14
√
2
metre
0%
96 metre
Explanation
Let
Q
A
=
x
m
Q
B
=
A
B
+
Q
A
=
(
48
+
x
)
m
In
Δ
P
Q
A
,
∠
P
Q
A
=
90
°
,
∠
P
A
Q
=
45
°
tan
45
°
=
P
Q
Q
A
=
P
Q
x
∴
P
Q
=
x
m
→
(
1
)
In
Δ
P
Q
B
,
∠
P
Q
B
=
90
°
,
∠
P
B
Q
=
30
°
tan
30
°
=
P
Q
Q
B
=
P
Q
(
48
+
x
)
⇒
P
Q
=
(
48
+
x
)
√
3
m
→
(
2
)
From
(
1
)
&
(
2
)
x
=
(
48
+
x
)
√
3
⇒
x
=
48
√
3
−
1
m
∴
h
e
i
g
h
t
o
f
t
o
w
e
r
(
P
Q
)
=
x
=
48
√
3
−
1
=
24
(
√
3
+
1
)
m
Option A is correct
At a point A, the angle of elevation of a tower of a lower is such that its tangent is
5
12
;
on walking 120 meters nearer the tower the tangent of the angle of elevation is
3
4
.
The height of the tower is :
Report Question
0%
225 meters
0%
200 meters
0%
230 metres
0%
None of these
Explanation
Here AB is the tower, it is given that
tan
ϕ
=
5
12
……….(i)
On walling
240
m near to tower, it reaches point C, then
tan
θ
=
3
4
………(ii)
Let
B
C
=
x
m,
A
B
=
y
m
Now in
Δ
ABC,
tan
θ
=
y
x
=
3
4
⇒
x
=
4
3
y
Now
tan
θ
=
y
240
+
x
=
5
12
=
y
240
+
4
3
y
=
5
12
⇒
12
y
=
1200
+
20
3
y
⇒
36
y
−
20
y
=
3600
⇒
16
y
=
3600
⇒
y
=
225
m.
A man is walking towards a vertical pillar in a straight path, at uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is
30
o
. After walking for
10
minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is
60
o
. Then the time taken(in minutes) by him, from B to reach the pillar, is?
Report Question
0%
6
0%
10
0%
20
0%
5
Explanation
R.E.F image
let
A
B
=
x
,
Q
B
=
y
,
P
Q
=
z
In
△
P
A
Q
tan
30
∘
=
z
x
+
y
⇒
1
√
3
=
z
x
+
y
⇒
z
=
x
+
y
√
3
In
△
P
B
Q
tan
60
∘
=
z
y
⇒
z
=
√
3
y
∴
x
+
y
√
3
=
√
3
y
⇒
x
+
y
=
3
y
⇒
x
=
3
y
−
y
⇒
x
=
2
y
⇒
y
=
x
2
So to go with x, it takes around 10 minutes.
with y, it takes around 5 minutes.
To cross a river a person covers a straight forward distance of 325 m along a bridge over the river. If bridge subtends 30
o
angle with edge of the river, find the width of the river?
Report Question
0%
152.5
0%
155
0%
165
0%
162.5
In a storm, a tree got broken by the wind whose top meets the ground at an angle of 30
o
, at a distance of 30 meters from the root. What was the height of the tree before breaking?
Report Question
0%
30
m
0%
30
√
3
m
0%
60
m
0%
60
√
3
m
Explanation
Based on the given information, we can draw the figure shown above.
Here, line segment
A
B
represents the height of the tree.
The tree breaks at point
D
.
∴
Segment
A
D
is the broken part of tree which then takes the position of
D
C
.
∴
A
D
=
D
C
Given
m
∠
D
C
B
=
30
∘
,
B
C
=
30
m
In right angled
Δ
D
B
C
,
t
a
n
30
∘
=
B
D
B
C
⇒
1
√
3
=
B
D
30
⇒
B
D
=
30
√
3
∴
B
D
=
10
√
3
m
Also
c
o
s
30
∘
=
B
C
D
C
⇒
√
3
2
=
30
D
C
⇒
D
C
=
60
√
3
∴
D
C
=
20
√
3
m
Hence
A
D
=
D
C
=
20
√
3
m
From the figure,
A
B
=
A
D
+
D
B
∴
A
B
=
20
√
3
+
10
√
3
⇒
A
B
=
30
√
3
m
Hence, the height of tree is
30
√
3
m
.
The angles of elevation of the top of a tower from two points A and B lying on the horizontal through the foot of the tower are respectively
15
∘
and
30
∘
. If A and B are on the same side of the tower and AB = 48 m, Then the height of the tower is:
Report Question
0%
24
√
3
m
0%
24 m
0%
24
√
2
m
0%
96 m
Explanation
A
B
is the tower
In
△
A
B
C
,
tan
15
∘
=
A
B
A
C
⇒
2
−
√
3
=
h
d
+
48
.....
(
1
)
⇒
tan
30
∘
=
A
B
B
D
=
h
d
⇒
1
√
3
=
h
d
or
h
=
d
√
3
Eqn
(
1
)
⇒
2
−
√
3
=
d
√
3
d
+
48
⇒
2
−
√
3
=
d
√
3
(
d
+
48
)
⇒
2
√
3
−
3
=
d
(
d
+
48
)
⇒
(
2
√
3
−
3
)
(
d
+
48
)
=
d
⇒
d
(
2
√
3
−
3
)
+
48
(
2
√
3
−
3
)
=
d
⇒
d
(
2
√
3
−
3
−
1
)
=
−
48
(
2
√
3
−
3
)
⇒
d
(
2
√
3
−
4
)
=
48
(
3
−
2
√
3
)
⇒
2
d
(
1
√
3
−
2
)
=
48
(
3
−
2
√
3
)
⇒
d
=
24
(
3
−
2
√
3
)
(
√
3
−
2
)
We have
h
=
d
√
3
⇒
√
3
h
=
d
⇒
√
3
h
=
24
(
3
−
2
√
3
)
(
√
3
−
2
)
where
d
=
24
(
3
−
2
√
3
)
(
1
√
3
−
2
)
from above
⇒
h
=
24
(
3
−
2
√
3
)
√
3
(
√
3
−
2
)
⇒
h
=
24
(
3
−
2
√
3
)
(
3
−
2
√
3
)
∴
h
=
24
m
The shadow of a tree is
17
√
3
m
. If the height of the tree is
17
m
, then the sun's altitude is :
Report Question
0%
30
∘
0%
45
∘
0%
60
∘
0%
90
∘
Explanation
Length of the tree
=
17
√
3
m
Height of the tree
=
17
m
then , the sun's altitude is
tan
θ
=
Height of tree
Length of shadow
tan
θ
=
17
17
√
3
tan
θ
=
1
√
3
θ
=
30
∘
A tower substends an angle of
30
o
at a point on the same level as the foot of the tower. At a second point,
h
meter above first, point the depression of the foot of the tower is
60
o
, the horizontal distance of the tower from the point is _________________.
Report Question
0%
h
cos
60
o
0%
(
h
/
3
)
cot
30
o
0%
(
h
/
3
)
cot
60
o
0%
h
cot
30
o
On the level ground, the angle of elevation of the top of angle nearer to it the angle of elevation becomes
60
∘
.
The height is _________________.
Report Question
0%
10
m
0%
15
m
0%
20
m
0%
none
If a tower
30
m
high, casts a shadow
10
√
3
m
long on the ground, then what is the angle of elevation of the sun?
Report Question
0%
60
o
0%
30
o
0%
45
o
0%
90
o
Explanation
Given, Length = 30 m (Tower)
Base =
10
√
3
m (Shadow)
We have to find the angle, lets take that A
So,
tan A =
P
e
r
p
e
n
d
i
c
u
l
a
r
B
a
s
e
tan A =
30
10
√
3
tan A =
3
√
3
tan A =
√
3
So, tan A = tan
60
o
So, A =
60
o
An airplane flying horizontally at a height of
2500
√
3
m
above that ground, is observed to be at an angle of elevation
60
∘
from the ground. After a flight of
25
sec
the angle of elevation is
30
∘
. Find the speed of the plane in
m/sec
.
Report Question
0%
100
m/sec
0%
200
m/sec
0%
300
m/sec
0%
400
m/sec
Explanation
Let
P
and
Q
be the two positions of the airplane.
Q
B
=
2500
√
3
m
Let the speed of the aeroplane be
x
m/sec
.
In
△
A
B
Q
,
tan
60
∘
=
Q
B
A
B
√
3
=
2500
√
3
x
x
=
2500
m
In
△
A
P
C
,
tan
30
∘
=
2500
√
3
x
+
y
1
√
3
=
2500
√
3
x
+
y
x
+
y
=
3
×
2500
2500
+
y
=
7500
y
=
5000
m
To travel
5000
m
the aeroplane takes $$25
\text{ sec}
$$.
So, the speed of the plane will be,
5000
25
=
200
m/sec
A chinny of
20
m
t
height, standing on the top of a building subtends an angle whose tangent is
1
6
at a distance of
70
m
t
from the foot of the building. The height of the building is
Report Question
0%
50
m
t
0%
100
m
t
0%
13.749
m
t
0%
80
√
3
m
t
The angles of elevation of the top of a mountain from two points A and B on a horizontal line are
15
0
and
75
0
.If AB=650 mt,then the height of the mountain is
Report Question
0%
325
√
3
3
m
t
0%
325
√
3
2
m
t
0%
324
√
3
3
m
t
0%
324
√
3
2
m
t
The angles of elevation of the top of a building from the top and bottom of a tree are
x
and
y
respectively. If the height of the tree is
h
meter then, in meter, the height of the building is:
Report Question
0%
h
c
o
t
x
c
o
t
x
+
c
o
t
y
0%
h
c
o
t
y
c
o
t
x
+
c
o
t
y
0%
h
c
o
t
x
c
o
t
x
−
c
o
t
y
0%
h
c
o
t
y
c
o
t
x
−
c
o
t
y
Explanation
In
△
A
E
D
tan
x
=
D
E
A
E
∴
tan
x
=
H
−
h
A
E
------- (1)
A
E
=
H
−
h
tan
x
------(A)
In
△
D
B
C
tan
y
=
D
C
B
C
∴
tan
y
=
H
A
E
------- (2)
A
E
=
H
tan
y
------ (B)
Comparing
A
and
B
H
−
h
tan
x
=
H
tan
y
(
H
−
h
)
tan
y
=
H
.
tan
x
H
tan
y
−
H
tan
x
=
h
tan
y
H
=
h
tan
y
tan
y
−
tan
x
=
h
cot
y
1
cot
y
−
1
cot
x
=
h
cot
y
cot
x
−
cot
y
cot
x
.
cot
y
=
h
cot
x
cot
x
−
cot
y
The shadow of the tower becomes 60 meters longer when the altitude of the sun changes from
45
∘
t
o
30
∘
. Then the height of the tower is:
Report Question
0%
20
(
√
3
+
1
)
m
0%
24
(
√
3
+
1
)
m
0%
30
(
√
3
+
1
)
m
0%
30
(
√
3
−
1
)
m
Explanation
In
Δ
T
O
W
,
tan
45
o
=
h
x
∴
x
=
h
In
Δ
T
X
W
,
tan
30
o
=
h
60
+
x
1
√
3
=
h
60
+
h
60
+
h
=
√
3
h
60
=
h
(
√
3
−
1
)
h
=
60
√
3
−
1
=
60
(
√
3
+
1
)
3
−
1
=
30
(
√
3
+
1
)
m
The upper part of a tree broken over by the wind makes an angle of
30
o
with the ground, and the distance from the
root to the point where the top of the tree meets the ground is
15
m
. The present height of the tree is
Report Question
0%
15
m
0%
10
√
3
m
0%
20
m
0%
N
o
n
e
o
f
t
h
e
s
e
Explanation
Height of the tree is =x
tan
θ
=
P
B
tan
30
0
=
x
15
∴
x
=
15
×
1
√
3
=
8.66
A ladder is inclined to a well at an angle
α
to the horizontal. Its foot is drawn up to a distance and as such, it slides a distance
b
along the well and makes angle
β
with the horizontal, then
a
is equal to:
Report Question
0%
b
2
tan
(
α
+
β
)
0%
b
tan
(
α
+
β
)
0%
b
tan
(
α
+
β
2
)
0%
b
2
tan
(
α
+
β
2
)
Explanation
R.E.F image
t
a
n
α
=
a
√
l
2
−
a
2
t
a
n
β
=
b
+
a
√
l
2
−
(
b
+
a
)
2
on solving,
a = b tan
(
α
+
β
)
A flagstaff
5
m high is placed on a building
25
m high. If the flag and building be subtend equal angels on the observer at a height
30
m, then the distance between observer and the top of flag is
Report Question
0%
5
√
3
2
0%
5
√
2
3
0%
5
√
3
2
0%
5
√
2
3
Explanation
Given,
∠
D
A
E
=
∠
D
A
C
=
θ
(Let's assume)
Thus,
∠
E
A
C
=
2
θ
Let's assume the distance between the building and the observer is
′
x
′
m
.
Thus, for the triangle
ΔACE
;
EC = ED + DC = 5+25 = 30m
Opposite/adjacent
= \dfrac{EC}{EA} = \dfrac{30}{x} = tan2\theta
For the
ΔEDA;
Opposite/adjacent
= \dfrac{ED}{EA} = \dfrac{5}{x} = tan\theta
Now,
tan2\theta = \dfrac{2tan\theta }{ (1-tan^2\theta)}
or,
\dfrac{30}{x} = \dfrac{2(\dfrac{5}{x}) }{ (1 - \dfrac{25}{x^2})}
[As
tan\theta = \dfrac{5}{x}
and
tan2\theta = \dfrac{30}{x}]
or,
3 =\dfrac{ 1 }{ (1-\dfrac{25}{x^2}}
or,
3x^2 - 75 = x^2
or,
2x^2 = 75
or,
x = 5\sqrt{(\dfrac{3}{2})} = 6.124m
Thus the distance between the building and the observer is
5\sqrt{(\dfrac{3}{2})} m
The length of the shadow of a vertical tower on level ground increase by 10 meters when the altitude of the sun changes from
45^o
to
30^o
. Then the height of the tower is:
Report Question
0%
5\sqrt{3}
0%
10(\sqrt{3} + 1) meter
0%
5(\sqrt{3} + 1)
meter
0%
10\sqrt{3} meter
Explanation
Let\text{ the distance when the sun elevation is 45 be x}
\text{whereas when sun elevation is 30 the distance is 10+x}
\text{let h be the height of the tower}
\text{when the sun is at elevation of 45}
\text{tan45=h/x}
\text{1=h/x}
\text{h=x}\,\,\,\,\,\,\,.........\left( i \right)
when\,the\,sun\,elevation\,is\,{{30}^{0}}
\tan {{30}^{0}}=h/\left( x+10 \right)
\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{x+10}
\Rightarrow h=\frac{x+10}{\sqrt{3}}\,\,\,\,\,\,......\left( ii \right)
from\,\left( i \right)and\,\left( ii \right)
x=\frac{x+10}{\sqrt{3}}
\Rightarrow \sqrt{3}x=x+10
\Rightarrow x\left( \sqrt{3}-1 \right)=10
\Rightarrow x=\frac{10}{\sqrt{3}-1}
\,\,\,\,\,\,\,\,\,\,\,=\frac{10}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}
\,\,\,\,\,\,\,\,\,\,=5\left( \sqrt{3}+1 \right)
hence\,height=5\left( \sqrt{3}+1 \right)m
A tower is
50\sqrt{3}
m high. The angle of elevation of its top from a point 50 m away from its foot has measure............degree
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0%
65
0%
60
0%
30
0%
15
A ladder leans against a wall making an angle of measure
60^\circ
with the ground. If the foot of the ladder is
3
metre away from the wall, then the length of the ladder is ..... metre.
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0%
\dfrac{3}{2}
0%
3\sqrt{3}
0%
6
0%
\dfrac{3\sqrt{3}}{2}
Explanation
Given distance between the foot ofthe ladder and wall is
3\ m
Let the length of ladder be
l
\implies \cos 60^{\circ}=\dfrac{3}{l}
\implies l=3\text{sec} 60^{\circ}
\implies l=6
So the length of the ladder is
6\ m
The tops of two towers of height x and y standing on a level round subtends angle of 30 and 60 respectively at the line joining their feet then x:y is.
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0%
1 : 2
0%
2 : 1
0%
1 : 3
0%
3 : 1
Explanation
\tan 30^o=\dfrac{x}{a}
\tan 60^o=\dfrac{y}{a}
x=a\tan 30^o
y=a\tan 60^o
\dfrac{x}{y}=\dfrac{\tan 30^o}{\tan 60^o}=\dfrac{1}{3}
x : y =1 : 3
.
The measure of the angle of elevation of sun increases from
30\ to\ 45
, then the length of the shadow of a
100\ meter
high building reduces by
x\ meter
. Then
x
is..... meter.
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0%
100
0%
100\sqrt {3}
0%
100(\sqrt {3}-1)
0%
\dfrac {100}{\sqrt {3}}
A balloon is connected to a meteorological ground station by a cable of length 215 m inclined at
60^o
to the horizontal. Determine the height of the balloon from the ground. Assume that there is no stack in the cable.
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0%
175 m
0%
186 m
0%
190 m
0%
204 m
Explanation
Let
A
be the position of the balloon.
In right triangle
ABC,
\sin 60^o=\dfrac{AB}{AC}
AB=215 \sin 60^o
AB=215 \times \dfrac{\sqrt 3}{2}
AB =186 \ m
The upper
\dfrac 34 ^{th}
portion of a vertical pole subtends an angle
\tan^{-1} \bigg(\dfrac 35\bigg)
at a point in the horizontal plane through its foot and at a distance
40\ m
from the foot. A possible height of the vertical pole is:
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0%
40 m
0%
60 m
0%
80 m
0%
20 m
Explanation
Here,
\alpha =A- \beta
\therefore \beta = A - \alpha
Hence
\tan \beta = \dfrac{{\tan A + \tan \alpha }}{{1 - \tan A\tan \alpha }}
\Rightarrow \dfrac{3}{5} = \dfrac{{\dfrac{h}{{40}} + \left(- {\dfrac{{ h}}{{160}}} \right)}}{{1 - \left( {\dfrac{h}{{40}}} \right)\left( -{\dfrac{{ h}}{{160}}} \right)}}
\Rightarrow {h^2} - 200h + 6400=0
\Rightarrow \left( {h - 40} \right)\left( {h - 160} \right) = 0
\Rightarrow h = 40\,\,or\,\,h = 160
A 80 m long ladder is leaning on a wall. If the ladder makes an angle of
45^o
with the ground, find the distance of the ladder from the wall.
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0%
40\sqrt2 \ m
0%
20 \sqrt 3 \ m
0%
50 \sqrt 3 \ m
0%
None of these
Explanation
Here,
\cos \theta = \dfrac{Base}{Hypotenuse}
\cos 45^o=\dfrac{Base}{80}
Base = 80 \cos 45^o=\dfrac{80}{\sqrt 2}=40\sqrt 2
Therefore, Distance of the ladder from the wall
=40\sqrt 2 \text{ m}
A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min for the angle of depression to change from
30^{\circ}
to
45^{\circ}
: then after this , the time taken (in min) by the car to reach the foot of the tower, is
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0%
9(1+\sqrt{3})
0%
18(1+\sqrt{3})
0%
27(1+\sqrt{3})
0%
18(1+\sqrt{9})
A flag staff of height
'h'
stands in the centre of a rectangular field and subtends angle of
15^{o}
and
45^{o}
at the mid point of its sides. Find the distance between them
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0%
h\sqrt{2+\sqrt{3}}
0%
2h\sqrt{2+\sqrt{3}}
0%
4h\sqrt{2+\sqrt{3}}
0%
3h\sqrt{2+\sqrt{3}}
Explanation
\begin{array}{l} Q\tan { 15^{ \circ } } =\dfrac { h }{ { OQ } } \\ h\cot { 15^{ \circ } } =OQ \\ \dfrac { h }{ { 2-\sqrt { 3 } } } \, \, \, \left[ { \tan { { 15 }^{ \circ } } =2-\sqrt { 3 } } \right] \\ OP=\dfrac { h }{ { \tan { { 45 }^{ \circ } } } } =h \\ PQ=\sqrt { O{ P^{ 2 } }+O{ Q^{ 2 } } } \\ =h\sqrt { { { \left( { \dfrac { 1 }{ { 2-\sqrt { 3 } } } } \right) }^{ 2 } }+1 } \\ l\sqrt { { { \left( { \sqrt { 3 } +2 } \right) }^{ 2 } }+1 } \\ l\sqrt { 3+4+4\sqrt { 3 } +1 } \\ h\sqrt { 8+4\sqrt { 3 } } \\ 2h\sqrt { \sqrt { 3 } +2 } \\ 2h\sqrt { 2+\sqrt { 3 } } \\ Hence,\, option\, \, A\, is\, the\, \, correct\, answer. \end{array}
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Practice Class 10 Maths Quiz Questions and Answers
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