De Moivre's Theorem MCQ Quiz in मल्याळम - Objective Question with Answer for De Moivre's Theorem - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Calculation

Using De Moivre's Theorem:

Comparing with -i, we have:

This implies:

⇒ 

⇒ 

⇒ 

⇒ 

For the least positive integer value of k, let n = 0:

Let n = 1:

Let n = 2:

So, m = 3.

For the greatest negative integer value of k, we can analyze the values of k for negative n.

For n = -1:

So, n = -1.

Hence option 1 is correct

Concept:

Euler's theorem:

e^iθ = cosθ + isinθ 

De Moivre's Theorem:

(cosθ + isinθ)ⁿ = cos(nθ) + isin(nθ) 

Calculation:

Given, 

Putting z = e^ir, we get:

...(i)

Now 

= , where Img(z) represents the imaginary part of z

= 

= 

= 

= 

= 

= 

=  =  (given)

Comparing both sides we get, k = 50

∴ The value of k is 50.

Concept:

• If  then by componendo and dividendo rule  

Calculation:

Given z = e^iθ and   = 

⇒    = 

We know that if  then 

⇒  = 

⇒  =   [e^iθ = cos θ + i sin θ ]

⇒  =  

We know z^r = e^irθ and in the R.H.S the value of r are 2, 3 and 5

∴ a₀ + b₀ = a₁ + b₁ = a₄ + b₄ = a₆ + b₆ = a₇ + b₇ = a₈ + b₈ = a₉ + b₉ = a₁₀ + b₁₀ = 0

∴ a₂ + b₂ = 2 and a₃ + b₃ = 3 and a₅ + b₅ = 5 .

⇒ =  = 1

Required value of the expression is 1