Difference between revisions of "MIE 97/98"

(Created page with "===Problem 1=== Find the solution of <math>\sin x+\sqrt3\cos x=1</math> with <math>x\in\mathbb{R}</math>. ===Problem 2=== Solve the following matrix in terms of <math>\alpha<...")
 
(Problem 4)
 
(One intermediate revision by one other user not shown)
Line 13: Line 13:
 
Translation needed
 
Translation needed
  
Determine os parâmetros <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math> e <math>\delta</math> da transformação complexa <math>w=\frac{\alpha z+\beta}{\gamma z+\delta}</math> que leva pontos <math>z=0;-i;1</math> para <math>w=i;1;0</math>, respectivamente, bem como <math>z</math> para <math>w=-2-i</math>, onde <math>i=\sqrt{-1}</math>.
+
Determine the parameters <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math> and <math>\delta</math> of the complex transformation <math>w=\frac{\alpha z+\beta}{\gamma z+\delta}</math> which takes points <math>z =0;-i;1</math> for <math>w=i;1;0</math>, respectively, as well as <math>z</math> for <math>w=-2-i</math>, where <math>i=\sqrt{-1}</math>.
  
 
===Problem 5===
 
===Problem 5===
Line 25: Line 25:
  
 
===Problem 8===
 
===Problem 8===
A finit sum of integer consecutive numbers, odd, positives or negatives, is equal to <math>7^3</math>. Find the terms of this sum.
+
A finite sum of integer consecutive numbers, odd, positives or negatives, is equal to <math>7^3</math>. Find the terms of this sum.

Latest revision as of 21:07, 13 August 2023

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 the parameters $\alpha$, $\beta$, $\gamma$ and $\delta$ of the complex transformation $w=\frac{\alpha z+\beta}{\gamma z+\delta}$ which takes points $z =0;-i;1$ for $w=i;1;0$, respectively, as well as $z$ for $w=-2-i$, where $i=\sqrt{-1}$.

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