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UPSC CDS 2 2021 Math Paper
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© examsnet.com
Question : 36
Total: 100
From an aeroplane flying about a river at an altitude of
1200
m
, it is observed that the angles of depression of opposite points on the two banks of a river are
30
∘
and
θ
. If the width of the river is
3000
m
, then which one of the following is correct ?
θ
<
30
∘
30
∘
<
θ
<
45
∘
45
∘
<
θ
<
60
∘
60
∘
<
θ
<
90
∘
Validate
Solution:
👈: Video Solution
Given, altitude of aeroplane is 1200 m.
and the width of the river is
3000
m
.
So, we have,
B
C
=
3000
m
,
A
D
=
1200
m
,
∠
B
=
30
∘
and
∠
C
=
θ
,
In
△
A
B
D
,
tan
30
∘
=
A
D
B
D
⇒
1
√
3
=
1200
B
D
⇒
B
D
=
1200
√
3
∴
D
C
=
B
C
−
B
D
=
3000
−
1200
√
3
ln
∆
A
C
D
,
cot
θ
=
D
C
A
D
=
3000
−
1200
√
3
1200
=
30
−
12
√
3
12
=
5
−
2
√
3
2
cot
θ
=
2.5
−
1.732
=
0.77
We know,
cot
θ
is a decreasing function in first quadrant.
cot
45
∘
=
1
and
cot
60
∘
=
1
√
3
=
√
3
3
=
1.732
3
≈
0.6
∴
cot
45
∘
>
cot
θ
>
cot
60
∘
∴
θ
∈
(
45
∘
,
60
∘
)
Hence, option (c) is correct.
© examsnet.com
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