Q2
5 marks
Complex Numbers Arithmetic
True/False or Property Verification Statements
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The four questions in this exercise are independent. For each question, a statement is proposed. Indicate whether each of them is true or false, by justifying the answer.
One point is awarded for each correct answer that is properly justified. An unjustified answer earns no points. An absence of an answer is not penalized.
In questions 1. and 2., the plane is referred to the direct orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$. We denote by $\mathbb{R}$ the set of real numbers.
Statement 1: The point with affix $(-1+i)^{10}$ is located on the imaginary axis.
Statement 2: In the set of complex numbers, the equation $$z - \bar{z} + 2 - 4\mathrm{i} = 0$$ admits a unique solution.
Statement 3: $$\ln\left(\sqrt{\mathrm{e}^{7}}\right) + \frac{\ln\left(\mathrm{e}^{9}\right)}{\ln\left(\mathrm{e}^{2}\right)} = \frac{\mathrm{e}^{\ln 2 + \ln 3}}{\mathrm{e}^{\ln 3 - \ln 4}}$$
Statement 4: $$\int_{0}^{\ln 3} \frac{\mathrm{e}^{x}}{\mathrm{e}^{x}+2}\,\mathrm{d}x = -\ln\left(\frac{3}{5}\right)$$
Statement 5: The equation $\ln(x-1) - \ln(x+2) = \ln 4$ admits a unique solution in $\mathbb{R}^{*}$.