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Questão 6824

ESCOLA NAVAL

(ESCOLA NAVAL - 2014) Considere as seguintes matrizes

$R = egin{bmatrix} 4 & (16)^y & -1 \ 9^x & a & 0 end{bmatrix}$ ; $S = egin{bmatrix} 1 & (4)^{(2y-1)} & 2^{-1} \ 3^x & b & 1 end{bmatrix}$ e $T = egin{bmatrix} b & (2)^{(2y-1)}-10 & c \ 27 & 13 & -6 end{bmatrix}$

A soma dos quadrados das constantes reais x, y, a, b, c que satisfazem à equação matricial é

Gabarito:

Resolução:

Temos então que:

$egin{pmatrix}:4:&:left(16 ight)^y:&:-1:\ ::::9^x:&:a:&:0:end{pmatrix}-6egin{pmatrix}:1:&:left(4 ight)^{left(2y-1 ight)}:&:2^{-1}:\ ::::3^x:&:b:&:1:end{pmatrix}=egin{pmatrix}:b:&:left(2 ight)^{left(2y-1 ight)}-10:&:c:\ ::::27:&:13:&:-6:end{pmatrix}$

$egin{pmatrix}:4:&:left(16 ight)^y:&:-1:\ ::::9^x:&:a:&:0:end{pmatrix}+egin{pmatrix}:-6:&:-6*left(4 ight)^{left(2y-1 ight)}:&:-6*2^{-1}:\ ::::-6*3^x:&:-6b:&:-6:end{pmatrix}=egin{pmatrix}:b:&:left(2 ight)^{left(2y-1 ight)}-10:&:c:\ ::::27:&:13:&:-6:end{pmatrix}$

Com isso temos o sistema de equações:

$left{egin{matrix} 4-6=b \ 16^y-6*(4)^{2y-1}=2^{2y-1}-10 \ -1 -6*2^{-1}=c \ 9^x-6*3^x=27 \ a-6b=13 \ 0-6=-6 end{matrix} ight.$

Resolvendo temos que:

$left{egin{matrix} b=-2 \ 16^y-6*(4)^{2y-1}=2^{2y-1}-10 \ c = -4 \ 9^x-6*3^x=27 \ a=13+6b=13-12=1end{matrix} ight.$

a = 1, b = -2, c = -4.

Falta descobrir os valores de x e de y.

$left{egin{matrix}16^y-6*(4)^{2y-1}=2^{2y-1}-10 \ 9^x-6*3^x=27 end{matrix} ight.$

Organizando:

Para o y:

$left(4^2 ight)^y-6cdot :4^{2y-1}=2^{2y-1}-10$

$\ 4^{2y}-6cdot :4^{2y-1}=2^{2y-1}-10$

$left(2^2 ight)^{2y}-6left(2^2 ight)^{2y-1}=2^{2y-1}-10$

$2^{2cdot :2y}-6cdot :2^{2left(2y-1 ight)}=2^{2y-1}-10$

$2^{2cdot :2y}-6cdot :2^{4y}cdot :2^{-2}=2^{2y}cdot :2^{-1}-10$

$left(2^y ight)^4-6left(2^y ight)^4cdot :2^{-2}=left(2^y ight)^2cdot :2^{-1}-10$

$Considerando 2^y=u$

$left(u ight)^4-6left(u ight)^4cdot :2^{-2}=left(u ight)^2cdot :2^{-1}-10$

Resolvendo, encontramos que

$u=2,:u=-2$

Sendo que $u=2^y$

Encontramos que y=1

Para o x:

$left(3^2 ight)^x-6cdot :3^x=27$

$3^{2x}-6cdot :3^x=27$

$left(3^x ight)^2-6cdot :3^x=27$

$Considerando que 3^x=u$

$left(u ight)^2-6u=27$

Resolvendo, encontramos que

$u=9,:u=-3$

Sendo que $u=3^x$

Encontramos que x = 2.

Com isso, a = 1, b = -2, c = -4, y = 1 e x = 2.

A soma dos quadrados destas constantes reais é

$1^2 + (-2)^2 + (-4)^2 + 1^2 + 2^2 = 26$

Questão 3185

ESCOLA NAVAL

(Esc. Naval - 2015)

When a major international software developer needed __________ a new product quickly, the project manager assembled a team of employees from India and the United States. From the start the team members could __________ on a delivery date for the product. The Americans thought the work could be done in two weeks the Indians predicted two to three months. As time went on, the Indian team members proved reluctant __________ setbacks in the production process, which the American team members would find out about only when work was due to be passed to them.

(Adapted from https://hbr.org/2OO6/11/managing-multicultural-teams)

Which is the correct option to complete this paragraph from an article on Managing Multicultural Teams?

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Questão 3229

ESCOLA NAVAL

(Esc. Naval 2015)

Which is the correct way to complete the text below?

Many wounder as Moroccan police beat protestor

(Reuters) – Moroccan police beat protesters who defied a ban on demonstrations across the country on Sunday, leading to arrests and dozens of injuries, some of them life threatening, witnesses said.

Much of the anger __________ at the Makhzen, Morocco’s royal court. “Protest is a legal right, why is the Makhzen afraid?”, crowds in Casablanca chanted. “Makhzen get out. Down with despotism.”

A Reuters correspondent __________ seven riot police attacking one bearded man in his 30s, repeatedy hitting his head and body, causing severe bleeding.
“We __________ here to preserve order because of this unauthorized protest”, said a senior police officer on the scene who __________ to give his name.

No one was available at the Interior Ministry to comment on the protesters’ reports.

(Adapted from http://www.linkedin.com)

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Questão 3505

ESCOLA NAVAL

(Esc. Naval - 2016) Mark the correct option.

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Questão 3506

ESCOLA NAVAL

(Esc. Naval - 2016) Choose the correct option to complete the text below.

International Congress Join us _____ our 2016 International Education Conference _____ Orlando, at the Disney’s BoardWalk Inn! The BoardWalk is located within the Walt Disney World Resort and 10 minutes away _____ the Epcot Theme Park. In addition to the Education Conference, we are also hosting a Business Conference that will be held on the same days, at the same venue.

(Abridged from http://www.cluteinstitute.com/education-conferences/)

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