TS EAMCET · Maths · Complex Number
For \(z \in \mathbf{C}\), if \((1+z)^n=1+{ }^n C_1 z+{ }^n C_2 z^2+\ldots{ }^n C_n z^n\) and \(\sum_{r=0}^{100} 100 c_r(\sin r x)=\left(2 \cos \frac{x}{2}\right)^{100} \sin k x\), then \(k=\)
- A 25
- B 100
- C 50
- D 75
Answer & Solution
Correct Answer
(C) 50
Step-by-step Solution
Detailed explanation
It is given that for a complex number \(z\), \((1+z)^n=1+{ }^n C_1 z+{ }^n C_2 z^2+{ }^n C_3 z^3+\ldots+{ }^n C_n z^n\) Put \(z=\cos x+i \sin x\) Then, \((1+\cos x+i \sin x)^n=1+{ }^n C_1(\cos x+i \sin x)\) \(+{ }^n C_2(\cos x+i \sin x)^2+\ldots+{ }^n C_n(\cos x+i \sin x)^n\)…
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