matematica 2015
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◆❖❚❆➬Õ❊❙R
✿ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s
C
✿ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s
i
✿ ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛✿ i2 = −1
|z|
✿ ♠ó❞✉❧♦ ❞♦ ♥ú♠❡r♦ z ∈ C
❘❡(z)
✿ ♣❛rt❡ r❡❛❧ ❞♦ ♥ú♠❡r♦ z ∈ C
■♠(z)
✿ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞♦ ♥ú♠❡r♦ z ∈ C
det A
✿ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ A
tr A
✿ tr❛ç♦ ❞❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A, q✉❡ é ❞❡✜♥✐❞♦ ❝♦♠♦ ❛ s♦♠❛ ❞♦s ❡❧❡♠❡♥t♦s ❞❛ ❞✐❛❣♦♥❛❧
♣r✐♥❝✐♣❛❧ ❞❡ A.
P♦tê♥❝✐❛ ❞❡ ♠❛tr✐③ : A1 = A, A2 = A · A, . . . , Ak = Ak−1 · A, s❡♥❞♦ A ♠❛tr✐③ q✉❛❞r❛❞❛ ❡ k
✐♥t❡✐r♦ ♣♦s✐t✐✈♦.
❞(P, r)
✿ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦ P à r❡t❛ r
AB
✿ s❡❣♠❡♥t♦ ❞❡ ❡①tr❡♠✐❞❛❞❡s ♥♦s ♣♦♥t♦s A ❡ B
[a, b]
= {x ∈ R : a ≤ x ≤ b}
[a, b[
= {x ∈ R : a ≤ x < b}
]a, b]
= {x ∈ R : a < x ≤ b}
]a, b[
= {x ∈ R : a < x < b}
X \Y
= {x ∈ X ❡ x ∈ Y }
n
ak
= a0 + a1 + a2 + · · · + an , s❡♥❞♦ n ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦
k=0
❖❜s❡r✈❛çã♦✿ ❖s s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝♦♥s✐❞❡r❛❞♦s sã♦ ♦s ❝❛rt❡s✐❛♥♦s r❡t❛♥❣✉❧❛r❡s✳
◗✉❡stã♦ ✶✳
■✳
❈♦♥s✐❞❡r❡ ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s s♦❜r❡ ♥ú♠❡r♦s r❡❛✐s✿
❙❡ ❛ ❡①♣❛♥sã♦ ❞❡❝✐♠❛❧ ❞❡ x é ✐♥✜♥✐t❛ ❡ ♣❡r✐ó❞✐❝❛✱ ❡♥tã♦ x é ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✳
√
1
2
√
√ =
√ .
■■✳
n
( 2 − 1) 2
1−2 2 n=0 √
■■■✳ ln 3 e2 + (log3 2)(log4 9) é ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✳
∞
➱ ✭sã♦✮ ✈❡r❞❛❞❡✐r❛✭s✮✿
❆ ✭ ✮ ♥❡♥❤✉♠❛✳
❉ ✭ ✮ ❛♣❡♥❛s ■ ❡ ■■■✳
❇ ✭ ✮ ❛♣❡♥❛s ■■✳
❊ ✭ ✮ ■✱ ■■ ❡ ■■■✳
❈ ✭ ✮ ❛♣❡♥❛s ■ ❡ ■■✳
√
❙❡❥❛♠ A✱ B ❡ C ♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ C ❞❡✜♥✐❞♦s ♣♦r A = {z ∈ C : |z + 2 − 3i| < 19}✱
B = {z ∈ C : |z + i| < 7/2} ❡ C = {z ∈ C : z 2 + 6z + 10 = 0}✳ ❊♥tã♦✱ (A \ B) ∩ C é ♦ ❝♦♥❥✉♥t♦
◗✉❡stã♦ ✷✳
❆ ✭ ✮ {−1 − 3i, −1 + 3i}✳
❉ ✭ ✮ {−3 − i}✳
◗✉❡stã♦ ✸✳
❆✭ ✮−
2π
✳
3
❇ ✭ ✮ {−3 − i, −3 + i}✳
❊ ✭ ✮ {−1 + 3i}✳
√
1 + 3i
√
1 − 3i
❙❡ z =
❈ ✭ ✮ {−3 + i}✳
10
π
3
❇✭ ✮− ✳
✱ ❡♥tã♦ ♦ ✈❛❧♦r ❞❡ 2 ❛r❝s❡♥(❘❡(z)) + 5 ❛r❝t❣(2 ■♠(z)) é ✐❣✉❛❧ ❛
❈✭ ✮
2π
✳
3
❉✭ ✮
4π
✳
3
❊✭ ✮
5π
✳
3
❙❡❥❛ C ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ t❛♥❣❡♥t❡ s✐♠✉❧t❛♥❡❛♠❡♥t❡ às r❡t❛s r : 3x + 4y − 4 = 0 ❡ s : 3x + 4y − 19 = 0✳ ❆ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦r C é ✐❣✉❛❧ ❛
◗✉❡stã♦ ✹✳
❆✭ ✮
5π
✳
7
❇✭ ✮
4π
✳
5
❈✭ ✮
3π
✳
2
❉✭ ✮
8π
✳
3
❊✭ ✮
9π
✳
4
❙❡❥❛ (a1 , a2 , a3 , . . .) ❛ s❡q✉ê♥❝✐❛