`(sqrt3+i)^100=2^99(p+iq)` then p and q are the roots of which of the following quadratic equation?

Options

A. `x^2-(sqrt3+1)x-sqrt3=0`
B. `x^2-(sqrt3-1)x-sqrt3=0`
C. `x^2+(sqrt3+1)x+sqrt3=0`
D. None

Answer

To solve the equation \((\sqrt{3} + i)^{100} = 2^{99}(p + iq)\) and find the quadratic equation for which \(p\) and \(q\) are the roots, we can follow these steps:

### Step 1: Express \(\sqrt{3} + i\) in polar form
First, we find the modulus and argument of \(\sqrt{3} + i\):
- The modulus is given by:
\[ r = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]
- The argument \(\theta\) can be calculated using:
\[ \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \]
Thus, we can express \(\sqrt{3} + i\) in polar form as:
\[ \sqrt{3} + i = 2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right) = 2e^{i\frac{\pi}{6}} \]

### Step 2: Raise to the power of 100
Now, we raise this expression to the power of 100:
\[ (\sqrt{3} + i)^{100} = (2e^{i\frac{\pi}{6}})^{100} = 2^{100}e^{i\frac{100\pi}{6}} = 2^{100}e^{i\frac{50\pi}{3}} \]

### Step 3: Simplify the argument
Next, we simplify the argument \(\frac{50\pi}{3}\):
\[ \frac{50\pi}{3} = 16\pi + \frac{2\pi}{3} = 2\cdot 8\pi + \frac{2\pi}{3} \]
Since \(e^{i(2k\pi + \theta)} = e^{i\theta}\), we have:
\[ e^{i\frac{50\pi}{3}} = e^{i\frac{2\pi}{3}} \]

### Step 4: Substitute back into the equation
Substituting back, we get:
\[ (\sqrt{3} + i)^{100} = 2^{100} e^{i\frac{2\pi}{3}} = 2^{100}\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right) \]
Calculating \(\cos\frac{2\pi}{3}\) and \(\sin\frac{2\pi}{3}\):
\[ \cos\frac{2\pi}{3} = -\frac{1}{2}, \quad \sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2} \]
Thus,
\[ (\sqrt{3} + i)^{100} = 2^{100}\left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) = 2^{99}(-1 + i\sqrt{3}) \]

### Step 5: Compare with the original equation
From the equation \((\sqrt{3} + i)^{100} = 2^{99}(p + iq)\), we can equate:
\[ p + iq = -1 + i\sqrt{3} \]
This gives us:
\[ p = -1, \quad q = \sqrt{3} \]

### Step 6: Form the quadratic equation
To find the quadratic equation whose roots are \(p\) and \(q\), we use the formula:
\[ x^2 - (p + q)x + pq = 0 \]
Calculating the sum and product of the roots:
\[ p + q = -1 + \sqrt{3}, \quad pq = (-1)(\sqrt{3}) = -\sqrt{3} \]
Thus, the quadratic equation is:
\[ x^2 - (-1 + \sqrt{3})x - \sqrt{3} = 0 \]
This can be rearranged to:
\[ x^2 - (1 - \sqrt{3})x - \sqrt{3} = 0 \]

### Final Answer
The quadratic equation for which \(p\) and \(q\) are the roots is:
\[ x^2 - (1 - \sqrt{3})x - \sqrt{3} = 0 \]