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Grade: 11
        
answer this question with proper explanation Complex numbers 
10 months ago

Answers : (1)

Rajdeep
231 Points
							
HELLO THERE!
 
First, let’s look at the equation closely:
 
u2 – 2u + 2 = 0
Alpha and Beta are the roots of the equation.
 
Sum of roots and product of roots are respectively:
 
\alpha + \beta = 2\\ \alpha \beta = 2 \\ \\ (\alpha-\beta)^{2} = (\alpha+\beta)^{2} - 4\alpha \beta \\ \implies \alpha-\beta = \sqrt{-4} = 2i
 
Now on solving the two equations:
\alpha +\beta =2\\ \alpha -\beta = 2i
 
We get the roots as:
 
\alpha = 1+i\\ \beta = 1-i
 
Now, we are given that cot theta = x + 1,
so 
x = cot \theta - 1
 
Now in the given equation, put the value of x, alpha and beta:
 
\frac{(x+\alpha)^{n} -(x+\beta)^{n}}{\alpha-\beta} \\= \frac{(cot\theta - 1+1+i)^{n}-(cot\theta-1+1-i)^{n}}{2i} \\= \frac{(cot\theta+i)^{n}-(cot\theta-i)^{n}}{2i}
 
Now, put:
 
cot\theta = \frac{cos\theta}{sin\theta}
 
The equation now becomes:
 
\frac{(e^{i\theta})^{n} -(e^{-i\theta})^{n}}{2i\times sin^{n}\theta} \\\\=\frac{cosn\theta+isinn\theta-cosn\theta+isinn\theta}{2i\times sin^{n}\theta} \\\\= \frac{2i\times sinn\theta}{2i\times sin^{n}\theta} \\\\= \frac{sinn\theta}{sin^{n}\theta}
 
So, your answer is option (A).
The explanation is above.
 
THANKS!
 
9 months ago
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