MIE 97/98

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Problem 1

Find the solution of $\sin x+\sqrt3\cos x=1$ with $x\in\mathbb{R}$.

Problem 2

Solve the following matrix in terms of $\alpha$ and $\beta$

$\begin{vmatrix}1&-2&3\\5&-6&7\\6&8&\alpha\end{vmatrix}\begin{vmatrix}x\\y\\z\end{vmatrix}=\begin{vmatrix}-4\\-8\\\beta\end{vmatrix}$

Problem 3

Find the value of $\lambda$ that satisfies the inequation $27^{2\lambda}-\frac{4}{9}\cdot27^\lambda+27^{-1}>0$ and represent , graphically, the function $y=27^{2x}-\frac{4}{9}\cdot27^x+27^{-1}$.

Problem 4

Translation needed

Determine os parâmetros $\alpha$, $\beta$, $\gamma$ e $\delta$ da transformação complexa $w=\frac{\alpha z+\beta}{\gamma z+\delta}$ que leva pontos $z=0;-i;1$ para $w=i;1;0$, respectivamente, bem como $z$ para $w=-2-i$, onde $i=\sqrt{-1}$.

Problem 5

Translation needed

Problem 6

Translation needed

Problem 7

Find $\alpha$, $\beta$ and $\gamma$ such that the polynomial $\alpha x^{\gamma+1}+\beta x^\gamma+1$, with $x\in\mathbb{Z}$, is divisible by $(x-1)^2$ and that the numerical value of the quotient is equal to $120$ when $x=1$.

Problem 8

A finit sum of integer consecutive numbers, odd, positives or negatives, is equal to $7^3$. Find the terms of this sum.