The document contains 20 multiple choice questions from an exam in Brazil (ITA 2018). The questions cover a range of math and geometry topics including: arithmetic and geometric progressions, matrices, polynomials, probability, triangles, circles, trigonometry and complex numbers.
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ITA 2018 - fechada
1. ITA 2018 - FECHADA
1
01. (Ita 2018) Uma progressão aritmética 1 2 n
(a , a , , a )
satisfaz a propriedade: para cada 𝑛𝑛 ∈ ℕ, a soma da progressão é
igual a 2
2n 5n.
+ Nessas condições, o determinante da matriz
1 2 3
4 5 6
7 8 9
a a a
a a a
a 2 a a
+
é
a) 96.
−
b) 85.
−
c) 63.
d) 99.
e) 115.
02. (Ita 2018) Considere as funções 𝑓𝑓, 𝑔𝑔: ℝ → ℝ dadas por f(x) ax b
= + e g(x) cx d,
= + com 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑 ∈ ℝ, a 0
≠ e
c 0.
≠ Se 1 1 1 1
f g g f ,
− − − −
=
então uma relação entre as constantes a, b, c e d é dada por
a) b ad d bc.
+ = +
b) d ba c db.
+ = +
c) a db b cd.
+ = +
d) b ac d ba.
+ = +
e) c da b cd.
+ = +
03. (Ita 2018) As raízes do polinômio 2 3 4 5 6 7
1 z z z z z z z ,
+ + + + + + + quando representadas no plano complexo, formam
os vértices de um polígono convexo cuja área é
a)
2 1
.
2
−
b)
2 1
.
2
+
c) 2.
d)
3 2 1
.
2
+
e) 3 2
04. (Ita 2018) Sejam 1 5
x , , x
e 1 5
y , , y
números reais arbitrários e ij
A (a )
= uma matriz 5 5
× definida por
ij i j
a x y ,1 i, j 5.
= + ≤ ≤ Se r é a característica da matriz A, então o maior valor possível de r é
a) 1.
b) 2.
c) 3.
d) 4.
e) 5.
2. ITA 2018 - FECHADA
2
05. (Ita 2018) Sejam A e B matrizes quadradas n n
× tais que A B A B
+ = ⋅ e n
I a matriz identidade n n.
× Das afirmações:
I. n
I B
− é inversível;
II. n
I A
− é inversível;
III. A B B A.
⋅ = ⋅
é (são) verdadeira(s)
a) Somente I.
b) Somente II.
c) Somente III.
d) Somente I e II.
e) Todas.
06. (Ita 2018) Se o sistema 2 4
3
x y z 0
2a y (2a a)z 0
x ay (a 1)z 0
+ + =
+ − =
+ + − =
, admite infinitas soluções, então os possíveis valores do parâmetro a
são
a)
1 3 1 3
0, 1, , .
2 2
− − − +
−
b)
1 3 1 3
0, 1, , .
2 2
− +
−
c)
1 3 1 3
0, 1, , .
2 2
− + +
−
d) 0, 1, 1 3, 1 3.
− − − − +
e) 0, 1,1 3,1 3.
− − +
07. (Ita 2018) Sobre duas retas paralelas r e s são tomados 13 pontos, m pontos em r e n pontos em s, sendo m n.
>
Com os pontos são formados todos os triângulos e quadriláteros convexos possíveis. Sabe-se que o quociente entre o
número de quadriláteros e o número de triângulos é 15 11. Então, os valores de n e m são, respectivamente,
a) 2 e 11.
b) 3 e 10.
c) 4 e 9.
d) 5 e 8.
e) 6 e 7.
3. ITA 2018 - FECHADA
3
08. (Ita 2018) Sejam a e b números inteiros positivos. Se a e b são, nessa ordem, termos consecutivos de uma
progressão geométrica de razão
1
2
e o termo independente de
12
b
ax
x
−
é igual a 7.920, então a b
+ é
a) 2.
b) 3.
c) 4.
d) 5.
e) 6.
09. (Ita 2018) São dadas duas caixas, uma delas contém três bolas brancas e duas pretas e a outra contém duas bolas
brancas e uma preta. Retira-se, ao acaso, uma bola de cada caixa. Se 1
P é a probabilidade de que pelo menos uma bola
seja preta e 2
P a probabilidade de as duas bolas serem da mesma cor, então 1 2
P P
+ vale
a)
8
.
15
b)
7
.
15
c)
6
.
15
d) 1.
e)
17
.
15
10. (Ita 2018) Considere a classificação: dois vértices de um paralelepípedo são não adjacentes quando não pertencem à
mesma aresta. Um tetraedro é formado por vértices não adjacentes de um paralelepípedo de arestas 3 cm, 4 cm e 5 cm.
Se o tetraedro tem suas arestas opostas de mesmo comprimento, então o volume do tetraedro é, em cm³
a) 10.
b) 12.
c) 15.
d) 20.
e) 30.
4. ITA 2018 - FECHADA
4
11. (Ita 2018) Os triângulos equiláteros ABC e ABD têm lado comum AB. Seja M o ponto médio de AB e N o ponto
médio de CD. Se MN CN 2 cm,
= = então a altura relativa ao lado CD. do triângulo ACD mede, em cm,
a)
60
.
3
b)
50
.
3
c)
40
.
3
d)
30
.
3
e)
2 6
.
3
12. (Ita 2018) Considere a definição: duas circunferências são ortogonais quando se interceptam em dois pontos distintos
e nesses pontos suas tangentes são perpendiculares. Com relação às circunferências 2 2
1
C : x (y 4) 7,
+ + = 2 2
2
C : x y 9
+ =
e 2 2
3
C : (x 5) y 16,
− + = podemos afirmar que
a) somente 1
C e 2
C sгo ortogonais.
b) somente 1
C e 3
C são ortogonais.
c) 2
C é ortogonal a 1
C e a 3
C .
d) 1 2
C , C e 3
C são ortogonais duas a duas.
e) não há ortogonalidade entre as circunferências.
13. (Ita 2018) Se 2
log a
π = e 5
log b,
π = então
a)
1 1 1
.
a b 2
+ ≤
b)
1 1 1
1.
2 a b
< + ≤
c)
1 1 3
1 .
a b 2
< + ≤
d)
3 1 1
2.
2 a b
< + ≤
e)
1 1
2 .
a b
< +
5. ITA 2018 - FECHADA
5
14. (Ita 2018) Para que o sistema 3 3 2
x y 1
x y c
+ =
+ =
admita apenas soluções reais, todos os valores reais de c pertencem ao
conjunto
a)
1
, .
4
−∞ −
b)
1 1
, , .
4 4
−∞ − ∪ ∞
c)
1 1
, .
2 4
− −
d)
1
, .
2
∞
e)
1 1
, , .
2 2
−∞ − ∪ ∞
15. (Ita 2018) Os lados de um triângulo de vértices A, B e C medem AB 3 cm, BC 7 cm
= = e CA 8 cm.
= A circunferência
inscrita no triângulo tangencia o lado AB no ponto N e o lado CA no ponto K. Então, o comprimento do segmento NK,
em cm, é
a) 2.
b) 2 2.
c) 3.
d) 2 3.
e)
7
.
2
16. (Ita 2018) O lugar geométrico das soluções da equação 2
x bx 1 0,
+ + = quando |𝑏𝑏| < 2, 𝑏𝑏 ∈ ℝ, é representado no
plano complexo por
a) dois pontos.
b) um segmento de reta.
c) uma circunferência menos dois pontos.
d) uma circunferência menos um ponto.
e) uma circunferência.
6. ITA 2018 - FECHADA
6
17. (Ita 2018) Considere a matriz �
1 𝑥𝑥 𝑥𝑥2
𝑥𝑥3
1 2 3 4
−1 3 4 5
−2 2 1 1
� , 𝑥𝑥 ∈ ℝ. Se o polinômio p(x) é dado por p(x) det A,
= então o produto
das raízes de p(x) é
a)
1
.
2
b)
1
.
3
c)
1
.
5
d)
1
.
7
e)
1
.
11
18. (Ita 2018) Em um triângulo de vértices A, B e C são dados ˆ
B̂ , C
2 3
π π
= = e o lado BC 1cm.
= Se o lado AB é o
diâmetro de uma circunferência, então a área da parte do triângulo ABC externa à circunferência, em 2
cm , é
a)
3 3
.
8 16
π
−
b)
5 3
.
4 2
π
−
c)
5 3 3
.
8 4
π
−
d)
5 3
.
16 8
π
−
e)
5 3 3
.
8 16
π
−
19. (Ita 2018) Com relação à equação
3
2
tg x 3tgx
1 0,
1 3tg x
−
+ =
−
podemos afirmar que
a) no intervalo ,
2 2
π π
−
a soma das soluções é igual a 0.
b) no intervalo ,
2 2
π π
−
a soma das soluções é maior que 0.
c) a equação admite apenas uma solução real.
d) existe uma única solução no intervalo 0, .
2
π
e) existem duas soluções no intervalo , 0 .
2
π
−
7. ITA 2018 - FECHADA
7
20. (Ita 2018) Se x é um número real que satisfaz 3
x x 2,
= + então 10
x é igual a
a) 2
5x 7x 9.
+ +
b) 2
3x 6x 8
+ +
c) 2
13x 16x 12.
+ +
d) 2
7x 5x 9.
+ +
e) 2
9x 3x 10.
+ +
GABARITO
1 - A 2 - A 3 - D 4 - B 5 - E
6 - B 7 - E 8 - B 9 - E 10 - D
11 - A 12 - C 13 - E 14 - E 15 - A
16 - C 17 - D 18 - D 19 - B 20 - C