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TIFR-2015-Maths-B-7

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A complex number $\alpha \in \mathbb{C}$ is called algebraic if there is a non-zero polynomial $P(x) \in \mathbb{Q}\left[x\right]$ with rational coefficients such that $P(\alpha)=0$. Which of the following statements is true?

  1. There are only finitely many algebraic numbers
  2. All complex numbers are algebraic
  3. $\sin (\frac{\pi}{3})+ \cos (\frac{\pi}{4}$) is algebraic
  4. None of the above
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