Let AB be the tower standing vertically on the ground and O be the position of the observer we now have:

OA=20m angle OAB=90⁰ and AOB=60⁰

Let

AB=hm

Now, in the right ∆OAB, we have:

so, the height of the pole is 34.64m.

Let CE and AD be the heights of the observer and the chimney, respectively.

We have,

BD=CE=1.5m

BC=DE=30m

And ACB=60⁰

In ∆ABC,

tan 60=AB/BC

√3=AD-BD/30

AD-1.5=30√3

AD=30√3+1.5

AD=30x1.732+1.5

AD=51.96+1.5

AD=53.46m

So, the height of the chimney is 53.46 m (approx).

Let AB be the unfinished tower, AC be the raised tower and O be the point of observation

We have:

OA =75m

AOB=30⁰ and

AOC=60⁰

Let

AC=Hm such that

BC= H-h m

In AOB we have

AB/OA=tan30­­­⁰=1/√3

h/75=1/√3

h=75/√3

in AOC we have

AC/OA=tan60⁰=√3

H/75=√3

H=75x√3

Required height=(H-h)=75√3-25√3=50√3=86.6m

Let AB and CD be the equal poles; and BD be the width of the roaD.

We have,

AOB=60

COD=30

In ∆AOB, tan60=AB/BO

√3=AB/BO

BO=AB/√3

In ∆COD, tan30=CD/DO

1/√3=CD/DO

DO=√3CD

As, BD=80

BO+DO=80

AB/√3+√3CD=80

AB/√3+√3AB=80 (AB=CD)

AB(1/√3+√3)=80

AB=80x√3/4

AB=20√3

Also, BO=AB/√3=20√3/√3=20m

DO=80-20=60m

Let, Z = (1+i√3)²

= (1)² + (i√3)² + 2 √3i

= 1 -3 + 2 √3i

Z = -2 + 2 √3i

we have, i⁵⁷+ i⁷⁰+ i⁹¹+ i¹⁰¹+ i¹⁰⁴

= (i⁴)¹⁴.i + (i⁴)¹⁷.i² + (i⁴)²².i³ + (i⁴)²⁵.i + (i⁴)²⁶

We know that, i⁴ = 1

⇒ (i⁴)¹⁴.i + (i⁴)¹⁷.i² + (i⁴)²².i³ + (i⁴)²⁵.i + (i⁴)²⁶

= i + i² + i³ + i +1

= i -1 –i + i + 1

=i

To Find: People who speak at least one of these two languages

Let us consider,

People who speak Hindi = n(H) = 50

People who speak English = n(E) = 20

People who speak both Hindi and English = n(H ∩ E) = 10

People who speak at least one of the two languages = n(H ∪ E)

Now, we know that,

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

Therefore,

n(H ∪ E) = n(H) + n(E) – n(H ∩ E)

= 50 + 20 – 10

= 60

Thus, People who speak at least one of the two languages are 60.

we have, i + i²+ i³+ i⁴+……… up to 400 terms

We know that given series in GP where a = I, r = I and n = 400

To find: that the middle term in the expansion of

Formula used:

A General term t_r+1 of binomial expansion (x+y)ⁿ is given by,

T_r+1 = ⁿC_r X^n-r Y^r where

Total number of terms in the expansion is 11.

Thus, the middle term of the expansion is T₆ and is given by.