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• 8 Suppose $ \alpha, \beta>0 $. Compute: $ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $
• 7 Differentiability of $ G(x)=\int_{\mathbb R} e^{tx}f(t)dt $ on $ (0,1) $
• 6 Suppose that $G$ is a group of order $924=2^2\cdot3\cdot7\cdot 11$. Prove that $G$ has an element of order $77$.
• 6 What is the intuition behind the law of quadratic reciprocity?
• 6 Finding all the ideals of $\mathbb{Z} [x]/(2, (x^3+1))$?
• 6 Evaluate $\sum_{k=0}^n {{2n + 1}\choose {2k + 1}}$
• 6 Evaluate the limit the following series
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