JEE Main 31-Aug-2021 Shift 1 Solved Paper

Given,
Height of transmitting anteena (HT)+ height of receiving anteena (HR)=160m
Radius of earth, Re=6400‌km
Let, range = d
If height of transmitting antenna be x
then, height of receiving anteena = 160 - x
We know that, d=√2RhT+√2RhR=√2R[√hT+√hR]
=√2×R[√x+√160−x]
=√2R[√x+√160−x]
For maximum range

d

dx

(d)=

d

dx

√2R[√x+√160−x]=0
⇒

d

dx

[√x+√160−x]=0
⇒

1

2

x−

1

2

−

1

2

(160−x)−

1

2

=0
⇒

1

2

1

√x

=

1

2

1

√160−x

⇒x=160−x
⇒2x=160
⇒ x = 80
∴ Maximum range
(dmax)=√2R[√x+√160−x]
=√2×6400[√80+√160−80]
=80√2[√

80

1000

+√

80

1000

]
=80√2×2√

80

1000

=80√2×

2√80

10√10

=16√

2×80

10

=16×4=64‌km