Preguntas propuestas
5
2015
• Aptitud Académica • Matemática • Cultura General • Ciencias Naturales
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Álgebra Valor Valor absoluto
Además y está entre 0 y z. Si se sabe que | x|+| y|=18 | x|+| z|=20 | y|+| z|=22
NIVEL BÁSICO
1.
Si x ∈ 〈7; 10〉, entonces halle el valor de la expresión k. − 16 −
x
k =
x
−
5
+
A) – 1 D) – 3
3 x x
−
−
2
y calcule x +
−2 z
6
B) – 2
B) 2 – 1
A) 2 D) 22
C) 3 E) 1
C) 2 – 2 E) 23
NIVEL INTERMEDIO 2.
Resuelva la ecuación 4 x
1 −
+
2x
−1 =
3x
e indique la suma de soluciones.
Determine el conjunto A={ x ∈ R /| x – 1|= x2 – x – 1} por extensión.
A) 1/3 D) 4/3
A) {0; 2; − 2; 2 }
2
B) 2/3
7.
C) 1 E) 5/3
B) f 3.
Sea la igualdad | x – a+b|=| x+a – b| (*) entonces, la proposición verdadera es
A) (*) si y solo si x=0 ∨ a2= b2 B) (*) si y solo si x=a=b C) (*) si y solo si x=0 ∧ a=b D) (*) si y solo si x=0 ∨ a=b E) (*) si y solo si x=a= – b
C) {0; 2} D) {− 2; 2} E) {0; 2; − 2 } 8.
x
Indique el número de soluciones de la ecuación x2+7+| x – 3|=6 x A) 0 D) 3
B) 1
2
−
2x
+
3
A) 3 D) – 4 9.
=
−12
x 2
−
C) 2 E) 4
B) – 3
−
{ } D) { }
Si x0 es una solución de la ecuación
3 2
1
| x2 – 4|+| x+2|+| x|=| – x|
7
determine el valor de x03. 10.
C) 4 E) 8
Resuelva la siguiente ecuación x 1 x x 1
A) 5.
+
y determine la mayor solución. UNI 2009 - I
4.
Resuelva la siguiente ecuación
;1
=
−
B)
{ } 1
2
;2
;2
Resuelva la ecuación
C) {2} E) {2; – 2}
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Álgebra 11.
I. No presenta solución negativa. II. Presenta Presenta solución racional. III. Presenta Presenta una solución irracional.
Determine el área de la región triangular ABC . B
A) VFV D) VFF
| x+1| 15.
|3 – x| A
x
x
A) 45 u 2 D) 30 u2 12.
B) 16 u2
x
C
−
x
2 −
−
3 =
1
A) 6 D) – 3
C) 24 u2 E) 12 u2
16.
−
B) 1
B) – 6
C) 9 E) 27
Halle el conjunto solución de la ecuación |3 x+2| – | x – 1|=2 x+3
A) [1; + ∞〉
1 x
A) 0 D) 3
Si x1; x2; ...; x n son las soluciones de la ecuación calcule el valor de x1 · x2 · ... · x n.
Indique la cantidad de soluciones de la ecuación x
C) FVV E) VVF
2 2 2 x ( x – 1) =| x – x|+6
Considere x el mayor entero posible.
B) VVV
3 B) − ; + ∞ 2
{ } D) {− } ∪
C) 2 E) más de tres
C) −
3 2
3
1; + ∞
2
NIVEL AVANZADO
{} 3
E) 1; + ∞ − 2 13.
Si | x|= – x, indique la variación de f ( x )
=
1
−
17.
2 1
−
x
Calcule la suma de las soluciones de la ecuación siguiente. x x
A) f ( x) ∈ [ – 1; 1〉 B) f ( x) ∈ [ – 1; +∞〉
2
2
+
x
−
x
+1 −
3
x +
x
A) – 2 D) 1
C) f ( x) ∈ 〈 – 1; 1〉
2 2
−
x
−
+
x
+1
B) – 3
3 =
2
C) 2 E) 0
D) f ( x) ∈ [0; 1〉 E) f ( x) ∈ R –
18.
Si a y b son las soluciones de la ecuación 4
4
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Álgebra Valor Valor absoluto II 6.
NIVEL BÁSICO
1.
Resuelva el siguiente sistema x < 5 − x > 2 e indique el número de soluciones enteras.
A) 0 D) 4 2.
B) 2
A) [2; + ∞〉 B) 〈3; +∞〉 C) 〈 – 2; 2] D) 〈 – ∞; – 2〉 ∪ [2; 3〉 4 E) −2; − ∪ [2; 3 3
C) 5 E) 3
NIVEL INTERMEDIO
Si A={ x ∈ R / 2 ≤ | x+1| < 5} determine la longitud de A.
7.
A) 7 D) 5 3.
B) 4
C) 3 E) 6
Resuelva la inecuación x
2
+
x
+1 +
3
<
x
−1
B) − 5 ; 5 C) 〈 – ∞; 1〉 ∪ 〈3; +∞〉 D) R E) f
5.
8.
∨ x < 3} ∧ x < 3} ∧ x < 4} ∨ x < 4} ∧ x < 4}
Determine el conjunto T por por extensión 2 1 3 ∈ ; T = x ∈Z x − 2 2 2
B) 5
9.
C) 6 E) 10
Sabiendo que la desigualdad | x – a|+5 x < 8 se verifica para todo x ∈ 〈 – ∞; 1〉. Determine un valor de a. A) – 2 D) 3
Resuelva la siguiente inecuación 2 2 | x – 1| x – 2 x – 2 < 2|
A) CS={ x ∈ R / – 2 < x B) CS={ x ∈ R / – 2 < x C) CS={ x ∈ R / – 1 < x D) CS={ x ∈ R / – 2 < x E) CS={ x ∈ R / – 2 < x
Luego de resolver el sistema x 2 − 4 x < 3 x − 2 2 x < x + 2 se obtiene S=〈a; b〉. Halle el valor de | a|+| b|.
A) 4 D) 8
A) 〈1; 3〉
4.
Dados los conjuntos 2 A={ x ∈ R /| x – x| < 6} B={ x ∈ R /|3 x – 1| ≥ 5} halle A ∩ B.
B) 0
C) 2 E) – 4
Resuelva la siguiente ecuación. | x2 – 3|+|5 – x2|=2
A) CS= f B) CS=R C) CS =
−
D) CS =
−∞ −
E) CS =
−∞ −
5; − 3
3; 5
∪
;
5
∪
;
5
∪
−
3; 3
5; + ∞
∪
5; + ∞
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Álgebra 11.
Determine el conjunto solución de la siguiente inecuación. 2 x
−1 −
x
x − 5
<
D) 2 − 5; 2 + 5 E) f
2 15.
A) 〈 – ∞; 5〉 ∪ 〈9; +∞〉 B) 〈 – ∞; 4〉 ∪ 〈8; +∞〉 C) 〈 – 5; 6〉 D) 〈 – 5; 7〉 E) 〈 – 3; 4〉 12.
1 + x − 2 , Si E = x ∈ R 2 < x − 2 halle el complemento de E . A) Z – D) {1; 3}
B) Z+
C) f E) {1; 2; 3}
A) VVV D) FVV 16.
Resuelva la siguiente inecuación. |3 x – 1| < |5 x – p|+|2 x+1 – p|
A) CS =
1 3
;
π−
17.
E) CS =
π
5
;
π
5
π−
B) 8
Si B=〈 – a; b〉 es el conjunto solución de la si guiente inecuación: 2 7 ≥ ( x − 5) ( x + 1)
∪
A) 4 D) – 2
π −1
2
;+∞
1
2
18.
B) 6
Determine el conjunto solución de la inecuación 1+
1
x
+
1
x 2
≤
x
−4 +
9−
x
≥
A) −∞; 2 − 5 ∪ 2 + 5; + ∞
−1
x +1 −7
Determine su conjunto solución.
1 2
C) 8 E) 0
Sea la inecuación x + 1 − 7
14.
C) 1 E) 4
determine el valor de a+b.
2
−∞;
C) VFV E) FFF
Si y=| x – 1|+| x – 2|+| x – 3|+| x – 4|; x ∈ R ¿cuál es el mínimo valor de y?
2 − x +
1
B) CS=f C) CS=R D) CS =
B) VVF
A) 6 D) 2
NIVEL AVANZADO
13.
Luego de resolver la ecuación (| x| – 1)(| x| – 2) ≤ 0 determine la veracidad (V) o falsedad (F) d las siguientes proposiciones. I. La longitud del conjunto solución es 2. II. Posee solo 4 soluciones enteras. III. No posee soluciones irracionales.
A) [– 4; 9] – {– 8; 6} B) R – {– 8; 6} C) [– 9; – 4] ∪ [4; 9]
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Álgebra Funciones
NIVEL BÁSICO
1.
A)
A) f ={(3; ={(3; 9), (3; – 3), ( – 1; – 1), (5; 1)} B) f ={(3; ={(3; 9), (5; 9)} C) f ={(3; ={(3; 9), (3; 7), (5; 25), (5; 1)} D) f ={(3; ={(3; 9), (– 3; 7), (5; – 1)} E) f ={(3; ={(3; 9), (– 3; 7), (1; – 1), (5; 1)}
2 x − 1; 0 ≤ x < 3 2; 3 ≤ x ≤ 10
m2 − f(2 m+ 5) 2
calcule f ( m).
7.
A) 0 D) 3
B) 1
C) 2 E) 5
Determine el dominio de la función f ( x )
9 =
x
2
−
funci ones Dado el conjunto A={ – 2; 2; 3; 4} y las funciones f=A → A y g, tales que ={(2; 3), (a; 2), (3; 4), (4; – 2)} y f ={(2; g( x)= mx+n. Además f (2)= g(3) y f (3)= g(2). Evalúe g(a). A) – 1 D) 8
1
A) [– 3; 3] B) [– 3; 3] – {0} C) 〈– ∞; – 3] 3] ∪ [3; +∞〉 D) [– 2; 2] E) [– 2; 2] – {0}
8.
x
→ →
C) 5 E) 10
x + f(1) ; x < 0 = x − f(0) ; x ≥ 0 5
calcule
Sea la función h: 〈a; b〉 b〉
B) 3
una función, tal que Sea f una f ( x )
4.
Sea f : S → R una función definida por ={(2 t+1; 4 t2 – 2 t+1) ∈ R×R / t ∈ R} f ={(2 Halle la regla de correspondencia de f .
NIVEL INTERMEDIO
= f(1); − 1 < m ≤ 2
3.
15 E) ; 2 42
además f
C) 〈 – 1; – 1/2 1/2〉
A) f ( x)= x2 – x+1 B) f ( x)= x2 – 2 x+3 C) f ( x)= x2+3 x+3 D) f ( x)= x2+3 x – 3 E) f ( x)= x2 – 3 x+3
Se sabe que f es es una función, tal que f ( x ) =
B) [2; 15]
15 2 D) ; 42 3
Si el conjunto ={(3; 9), (a; 7), (3; a2), ( b; – 1), (5; b2), (5; 4 b – 3)} f ={(3; representa una función, indique f . 6.
2.
15 2 ; 42 3
∑ f ( k) k= −5
R
A) 0 D) – 5
2 x+5
cuyo rango es 〈3; 9〉. Determine el valor de a+b. 9.
B) 10
C) 5 E) – 1
x3+2 x2= x} Sea el conjunto A={ x ∈R /
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Álgebra 10.
II. f (a x+b y)=a f ( x)+b f ( y)
III. f ( x – y)= f ( x) – f ( y)
IV. f ( – x)= – f ( x) A) VVVF D) VVVV
11.
B) VFVF
15.
=
12 + x − x
Indique la suma de los valores enteros del do minio de f ( x).
16.
2 x − 2 f( x ); x ≥ 1 f ( x ) = 1 4 xf( x ) − 4 x + 1; x < 1; x ≠ 4 2 B) ; + ∞ 3
}
)
1 − sen 2 t ; 1 − cos2 t t ∈ R
4 C) ; + ∞ 3 7 E) ; + ∞ 3
2x
Si f ( x )
=
2 x
2 2
−
2x
+
3
−
2x
+
2
,
tal que | f ( x)| ≤ k, ∀ x ∈ Dom( f ) determine el menor valor de k.
C) 3 E) 5
A) 2/3 D) 8/3
Determine el rango de f ( x).
1 A) ; + ∞ 3 5 D) ; + ∞ 3
{(
D) Ran f = 0; 2 E) Ran f =[0; =[0; 1]
2
B) 2
=
A) Ran f =[ =[ – 1; 1] B) Ran f =[0; =[0; 2] C) Ran f =[ =[ – 2; 0]
C) VVFF E) VFVV
2 x − 5
A) 1 D) 4
Halle el rango de la función f
Sea f ( x )
12.
A) [– 1; +∞〉 B) [– 1; 1] – {0} C) {– 1; 1} D) [– 1; +∞〉 – [0; 1] E) [– 1; +∞〉 – [0; 1〉
Dada la función f : R → R de modo que f ( x)= mx; ∀ x ∈ R Indique el valor de verdad de las proposiciones. I. f (a x)=a f ( x)
17.
B) 5/3
C) 7/3 E) 3
Dada la función f ( x )
x =
2
−
3x
x
−1
+
2
y sean x0 ∉ Dom f ∧ y0 ∉ Ran f . calcule el valor de ( x0+y 0)
A) – 1 D) 5
B) 0
C) 2 E) 7
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Álgebra Funciones especiales I
C)
NIVEL BÁSICO
Y
1
1.
X
Grafica la función 2 2 f ( x)=(sen x+cos) +(sen x – cos x)
A)
B)
Y
–1
Y
D) 2
1
1
E)
Y
Y
2 X
X
2
X
4 X
–2
C)
Y
4.
2
Del gráfico, calcule el valor de a+ b y ab. Y
X
12
D)
E)
Y
Y
b 1 2
2
a
X
2.
X
Si f ( x)=| x – a|+ b
A) 10; 10
La tabla adjunta muestra parte del dominio y rango de una función lineal f . x
2
5
f ( x)
10
a
8
b
28 28
37
a=2 b
B) 10; 16
D) 12; 32 5.
∧
X
C) 6; 24 E) 7; 32
La gráfica adjunta corresponde a f
2 a x +a x+a
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Álgebra 6.
Determine el área de la región que se forma al graficar las funciones
A)
B) 16
–3
C) 50 E) 36
f ( x )
A)
=
–3
(2 x2 + 7 x + 3) ( x − 2) x
2
+
–2
3
0
X
–2
x−6
D) B)
Y
E)
Y
–3 X
2
–3
3 0
Y
X
E)
Y
–2
La empresa telefónica Comunícate proporci na una nueva oferta. I. Un pago fijo de S/.30 mensual y de regal 150 minutos de llegada. II. Se cobrará 20 céntimos céntimos por cada minut adicional a los 150 minutos. Determine la gráfica de la función que repr senta el costo (en soles) por cada minuto d llamada. A)
Y
Y
30 X
0
X
X
9.
D)
Y
2
Y
X
C)
X
Y
NIVEL INTERMEDIO
Grafique la función f .
0
3 0
X
C)
7.
Y
–3
2
f ( x)=–2| x+2|+2 y g( x)=– 6
A) 64 D) 32
B)
Y
X
B)
Y
30
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Álgebra 0.
Si alcanzó un desplazamiento horizontal d 20 cm, determine la máxima altura que logr alcanzar.
Trace el gráfico de la siguiente función. f ( x)=2|2 x – 1|+1
A)
Y
A) 100 cm
3
B) 80 cm
D) 60 cm
E) 40 cm
X
1 2
B)
12.
Esboce la gráfica de f ( x)=| x|+ x+1 x ∈ [–2; 2〉
Y
3
A)
Y
1 X
1 X
2
C)
Y
B) 0
Y
X
1 X
–2
D)
Y
1 0
E)
Y
2
C) 50 cm
C)
Y
X
X
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Álgebra 15.
NIVEL AVANZADO
Dada la gráfica de P( x)=ax + bx+c Y
13.
Dadas las funciones f y g cuyas reglas de co-
rrespondencias son f ( x)=sgn( x2); g( x)=sgn( x – 1)
determine la gráfica de h( x)= f ( x) – g( x).
α
A)
B)
Y
Y
1
β X
2
2
y sea A = k
1 2
C)
X
X
indique Inf( A).
Y
A) 1 D) 4
2 1
16. 1
D)
X
E)
Y
Y
α α2 α 3 + 2 + 3 + ... k= 3+ β β β
B) 3
C) 2 E) 6
En la figura adjunta se muestran las gráficas de las funciones f y y g definidas por 2 f ( x)=ax + bx+c g( x)= mx2+ nx+ p Y
X
14.
f
X
g
Sea la función f ( x)=–| x – 3|+8 cuya gráfica se muestra a continuación. 0 Y
X
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Álgebra 17.
Sea f : R → R, tal que f (1 – x) – 2 f ( x)= x2+2 x – 2. Esboce la gráfica de f .
A)
D)
Y
Y X X Y
E) B)
Y
X
X
18.
Dada la función f( x )
C)
Y
X
=
4+
x
4
−
x
2
−
x+2
halle n; donde n es el número de puntos de intersección con el eje X .
A) 1 D) 3
B) 2
C) 0 E) 4
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Álgebra Funciones especiales II 4.
NIVEL BÁSICO
Dadas las gráficas de f ( x) y g( x). Y
1.
x−2
=
+
2 X
3
esboce la gráfica de h( x )
3
X
A)
Y
Y
1
3
2
Y
X
X
B) D)
3
X
Y
B)
Y
2
C)
1 x+b
3
–1
A)
g( x)=
f ( x)= 1 x+a
Esboce la gráfica de la siguiente función. f( x )
Y
E)
Y
Y
X
=
1 x + a + b
X
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Álgebra 5.
Esboce la gráfica de g( x)=| f ( x)| a partir de f( x )
=
2
−
NIVEL INTERMEDIO
x 7.
A)
B)
Y
Y
Se sabe que f( x ) = a − x + b es una función, tal que f (0)=1 y f (3)=0. Esboce su gráfica. Y
A) 4
4
X
B)
X
–1
C)
3
X
–1
X
3
Y
C) 4
Y
X
X
–1
D)
Y
E)
Y
Y
D)
4
X
3
4
X
X
–1
Y
E)
Y
3
X
–1
3
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Álgebra 9.
Determine la gráfica de la siguiente función. g( x )
x
−
3
x
−
4
=
11.
En la figura se muestra la gráfica del polinomio cúbico P( x). Y
A)
B)
Y
Y
1 1
– 2a 4
C)
4
X
O
2a
X
X
Y
sabiendo que P(a)=20, halle 1 4
A) 4 D) 10
X
B) 5
P(−3 a) .
C) 8 E) 12 UNI 2009 - I
D)
Y
E)
4
X
Y
12.
X
Resuelva la inecuación x 2 + 1 ≤ e indique un intervalo solución.
A) 〈– 1; 1; 1〉 B) 〈– 1; 1; +∞〉 C) 〈 〉
x+5
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Álgebra 4.
Se muestra la gráfica de la función F .
16.
Grafique f( x )
Y
A)
=
máx { x + 1; x }
X
–2
0
A) 20 D) 16 5.
B) 10
Y
X
X
4
Halle la suma de los valores absolutos de las soluciones de la ecuación g( x)=0, donde (4 – | x|). g( x)= F (4
B)
Y
C)
Y
C) 8 E) 4
X
Dada la siguiente gráfica
D)
Y
E)
Y
Y f ( x)
1
X
X
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Álgebra 18.
Indique la gráfica de g( x)= f ( x+| x|), si la grá fica de f es es
C) Y
g
Y 1 1
2
X
D)
Y 1
A) Y
X
2
f
1
1
g X
1
1
g
2 2
X
1
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Anual UNI VALOR
VALOR
ABSOLUTO I
ABSOLUTO II