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• 28 $ I(r) = \int_0^{2\pi}\frac{\cos(t) - r}{1 - 2r\cos t + r^2}\,dt$ is always zero for $r\in[0,1)$. Why?
• 11 If $f : [a,b]\to\Bbb R$ is continuous, are there $x_1,x_2\in (a,b)$ such that $\tfrac{f(b)-f(a)}{b-a} = \tfrac{f(x_1)-f(x_2)}{x_1-x_2}$?
• 9 How do I prove that $|\det(A+B)|\leq2^n$ where $A,B$ are $n\times n$ unitary matrices?
• 8 How to find the limit of series? (What should I know?)
• 8 Show that $A^{-1} + B^{-1}$ is invertible when $A,B$ and $A+B$ are invertible
• 7 Given $A\in\Bbb R^{n\times n}$, is $C_A := \{SAS^{-1} : S\in GL(n,\mathbb R)\}$ connected?
• 7 Calculate $\lim_{x\to 0} {1\over x} \int_0^x \cos(t^2)\,dt$
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