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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Que es un ángulo? Un ángulo es la porción de plano limitada por dos semirrectas y que tienen el mismo origen.
Los ángulos se simbolizan con la letra
a, como se muestra:
Como se compone un ángulo? Los lados del ángulo son las semirrectas que lo forman el ángulo. El vértice del ángulo es el punto común que es origen de las rectas que forman el ángulo.
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Como se miden los ángulos?
SISTEMA SEXAGESIMAL
En este sistema, el ángulo que forman dos rectas perpendiculares es de 90º. Los ángulos se miden en grados (º), minutos (') y segundos (''). Un ángulo de 1º es el que resulta al dividir en 90 partes iguales un ángulo recto por lo que: 1 Angulo recto = 90 º 1 grado = 60 minutos. 1º = 60’ 1 minuto = 60 segundos. 1'= 60''
Para medir ángulos se utiliza un transportador de ángulos.
Para medir un ángulo, se coloca el centro del transportador transportad or sobre el vértice del ángulo, ángul o, y uno de los lados sobre la línea del cero. Observa que tiene dos graduaciones en orden inverso; esto es para facilitar facilitar la medida en cualquier posición posición del ángulo.
Ejercicio: Observa como el ángulo A mide 40 º, pasa exactamente por el punto marcado con el 40 en el transportador. transportador.
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Que es la bisectriz de un ángulo? es la semirrecta que divide al ángulo en dos partes iguales.
Ejemplo: La semirrecta OA es bisectriz del ángulo O y divide el ángulo dado en ángulos 1,2
Clasificación de los ángulos: Completo, es el ángulo formado por la coincidencia de dos semirrectas y su amplitud es de 360º.
Llano, es el ángulo formado por dos semirrectas opuestas. Tiene sus lados en la misma recta. Su amplitud es la mitad de un ángulo completo, es decir, de 180º.
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Ángulo Recto, Angulo de 90º de amplitud.
Agudo, es el ángulo cuya amplitud está entre 0 y 90º.
Obtuso, es el ángulo cuya amplitud es mayor que la del ángulo recto y menor que la del llano, es decir que entre el
rango de 90º y 180º.
Cóncavo, es el que vale menos que un llano. Por tal razón se dice que los ángulos cóncavos, comprenden a los agudos, rectos y a los obtusos.
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- ------------------Dos ángulos opuestos por el vértice (vértice común y lados de uno prolongación de los del otro) siempre serán iguales.
Ángulos formados por paralelas y una secante
•
Al trazar dos líneas pueden ocurrir dos situaciones: la primera, que se crucen en un punto; la segunda, que por más que se prolonguen no lleguen a unirse.
Dos rectas que se cortan o interceptan en algún punto, se llaman secantes
Dos rectas que jamás se cruzan son Paralelas
Al cortar dos dos rectas rectas con una secante secante se forman forman ocho ángulos ángulos ,
(De los cuales 4 son iguales si y los otros cuatros también son iguales entre si, por lo que se forman dos grupos de diferente Angulo).
Para que lo puedas apreciar bien pondré un ejemplo en donde los ángulos son muy diferentes a simple vista, y los llamaremos ángulos Grandes y pequeños. G= Ángulos Pequeños P=Ángulos Pequeños
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Un punto muy importante es que la suma de ambos ángulos dará 180º, recordando que ambos ángulos están formados por una línea recta.
Por lo que en el examen te podrán poner solo un Angulo y preguntarte cuánto mide el otro, para lo cual solo deberás hacer una resta simple. 180º- el ángulo conocido= ángulo desconocido (vital comprenderlo, ya que varias preguntas o ejercicios, implican tu dominio del tema).
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Más sin embargo también te podrán poner una ecuación simple, como el siguiente ejemplo:
En ejemplo anterior, de donde salió el 70º de la substitución?, pues se tomo el valor de la primer respuesta al azar, y se resolvió la ecuación.
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Otro ejemplo:
Ejemplo de la guía (IMPORTANTE) (IMPORTANTE)
Datos conocidos E y F Lo primero que se debe hacer es factorisar, ya que no sabemos cuántos grados mide ninguno E ni F Entonces Se sabe que E mas F es igual a 180º, por lo que podemos factorisar 90º- 2x + 5x = 180º Se pasan los 90º al otro lado de la ecuación quedando -2x+5x=180º-90º 3x=90º =30º X =30º
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Medición y conversión de ángulos y radianes Un Radian: es el ángulo que limita un arco de una circunferencia cuya longitud es igual al radio de la misma. •
Es decir, este indica que el ángulo formado por dos radios de una circunferencia, medido en radianes, es igual a la longitud del arco formado sobre el radio,
,
Es decir θ = s /r θ = ángulo s = longitud del arco r = radio. Por lo que el ángulo “α”, completo en radianes de una circunferencia de radio “r”, es:
Formula de la circunferencia
Por lo que para convertir una circunferencia completa a radianes, se debe multiplicar el 2 x π , Lo que equivale a 360º
EJEMPLOS A) Por lo tanto si queremos sacar el valor radial de 180º, la formula será solamente π, ya que 180 es la mitad de 360. B) Por lo tanto si queremos sacar el valor radial de 45º, la formula será solamente π/2, ya que 180 es la mitad de 180 y 4ta parte de 360. Ejemplos de los ángulos más comunes en radianes
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Clasificación de polígonos
•
Clasificación de polígonos por su número de lados Triángulo equilátero
Tiene los 3 lados y ángulos iguales.
Cuadrado Tiene 4 lados y ángulos iguales.
Pentágono regular
Tiene 5 lados y ángulos iguales.
Hexágono regular
Tiene 6 lados y ángulos iguales.
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Octágono regular
Tiene 8 lados y ángulos iguales.
Eneágono regular
Tiene los 9 lados y ángulos iguales.
Decágono regular regular
Tiene 10 lados y ángulos iguales.
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Clasificación de polígonos según sus ángulos Polígono Convexos Convexos
Es el polígono en que todos sus ángulos son menores que 180° y todas sus diagonales son interiores.
Polígono Cóncavo Cóncavo
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Propiedades generales de los triángulos 1 Un lado de un triángulo es menor que la suma de los otros dos y mayor que su diferencia. 2 La suma de los ángulos interiores i nteriores de un triángulo es igual a 180°. 180°. 3 El valor de un ángulo exterior es igual a la suma de los dos interiores no adyacentes.
Clasificación de triángulos Según sus lados lados Triángulo equilátero
Tres lados iguales.
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Clasificación de los triángulos según sus ángulos Triángulo acutángulo
Tres ángulos agudos
Triángulo rectángulo
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Rectas notables de los triángulos Recordando que la mediatriz de un segmento, es la recta perpendicular al segmento en su punto medio. Se llaman mediatrices del triángulo a las mediatrices de cada uno de sus lados. Se traza la mediatriz de cada uno de los lados. Observa que las tres mediatrices se cortan en un punto, que se denomina circuncentro
El circuncentro tiene una propiedad muy importante, si trazamos una circunferencia con centro en él, que pase por uno de los vértices del triángulo, también pasara por los otros dos vértices. El circuncentro es el centro centro de la circunferencia que pasa por por los tres vértices de un triángulo. triángulo.
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Ejemplo de la aplicación Se desea construir una escuela rural, que sirva para la enseñanza de tres pueblos, a los que les llamaremos (A, B, C) y estos no están alineados por lo que forma un triangulo. Exactamente en donde debemos construirla para asegurar la misma distancia entre los 3 pueblos?
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Alturas de un triángulo La altura de un triángulo es el segmento que une un vértice con el lado opuesto o su prolongación formando ángulo recto
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Medianas de un triángulo
Se llama mediana de un triángulo al segmento que une un vértice con el punto medio del lado opuesto.
Baricentro
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Bisectrices de un triángulo triángulo
Bisectriz es cada una de las rectas que divide a un ángulo en dos ángulos iguales.
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Cálculo de perímetros y áreas En este tema de áreas y perímetros, difícilmente te pondrán hacer los más complicados, dado los tiempos Pero si te aseguro te podrán salir de problemas de cuadrados, rectángulos, triangulo y circulo De los más difíciles como esfera, cono, pirámide, etc. Muy seguramente, te pedirán que identifiques las formulas, para medir tu conocimiento del tema pero limitados por el tiempo, no podrán poner problemas de este tipo. Así es que identifica TODAS las formulas, y aprende a solucionar las básicas.
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Cuadrado
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Ejemplo si el lado lado a=5, b=10 b=10 entonces el área será 5*10=50
Paralelogramo
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Rombo
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Trapecio
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
Polígono regular
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
6. La factorización factori zación completa de 36 es a) b) c) d)
4x6 2x18 2x3x6 2x2x3x3
7. ¿Cuál de los siguientes números es divisible de 50? a) b) c) d)
8.
25 75 100 150
¿Cuál de los siguientes números es irracional? irraci onal? 10
a)
25 16
b)
4 5
c)
5 4
d)
1
9. La representación en forma fraccionaria fracci onaria de 0.75 es 3
a)
4 10
b)
2 4
c)
3 1
d)
5
10. El resultado de
9 5 6 2 3 4 1 5 1
es
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- ------------------2 x
y
11. Es el reciproco de x 2 y
a) b)
2 xy
y
c) 2 x d) 2 xy 2a 12b 4a 24b
12. El resultado de a) - 2
es
1
b) 2 c) 2 40
d)
4
13. El resultado de a) b) c) d)
6 y
2
8 y
2
6 y
4
8 y
4
y 6
9y 6 2
y
9y
2
4 x
3
4 x
3
4 x
3
4 x
3
4 y 2
7 y
2
5y 4
es
6
2 x 8 y
2
2 x 4 y
2
10 x 8 y
2
6
4 x
14. El resultado de a) b) c) d)
y
2 x 4 y
2
10
2
2
2
10
3
6 x 6 y
2
6
4 x
2y
2
4
es
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
15. El resultado de 4 x a) b) c) d)
7
16 x y 16 x
12
y
2 4
es
6
y
8
7
6
12
y8
256 x y 256 x
3
5
2
2
2
8 x y
16. El resultado de 7 a) 4 x
2 x y
es
3
b) 4 x y 3 c) 4 x d)
7
4 x y
4
3
2
12w z 18w z 24wz 6wz
17. El resultado de a) b) c) d)
2w
2
12w
2
3w 4 z
3
2
2w z 3w z 4wz 4
2w z
2
3
3w z
2
18. El resultado de a) 4a 7 b) 4a 1 c) 4a 7 d) 4a 1
es
3
18w 24z
3
4
4 2
4w z
12a
2
5
9a 3
3a 3 es
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- ------------------3
2 x 3 19. El resultado de 3 es 3
6 x
a)
9 8 x
b) c) d)
27
3
27 8 x
27
3
27
8 x
4 x
2
3
6 x 27
3
27
4 x
2
18x 27
20. El resultado de factorizar a) b) c) d)
y y y y
2 2
2 2
15 15y 15y 15
y 4
225
es
2 2
2 2
15 15
21. Es ejemplo de numero primo e) f) g) h)
20 23 45 48
22. ¿Cuál de los siguientes números es múltiplo de 6? e) f) g) h)
3 9 15 18
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
23. La factorización completa de 36 es e) f) g) h)
4x6 2x18 2x3x6 2x2x3x3
2
21. La forma más simple en que se puede expresar el resultado de la expresión a)
b2 4b
b 10
b)
4b
2
b 22
8 3b
c) d)
2
b
2
b
2
2b 4
5b 4
b 5
22. ¿Cuál de los siguientes números es irracional? 10
e)
25 16
f)
4 5
g)
5 4
h)
1
b b
2
2
b 4
2b 4
es
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- ------------------2a 12b
23. El resultado de a) - 2
4a 24b
es
1
b) 2 c) 2 40
d)
4
24. El resultado de e) f) g) h)
6 y
2
8 y
2
6 y
4
8 y
4
y 6
9y 6
y
2
9y
2
y
e) f) g) h)
16 x y 16 x
12
6
3
y
y
2 4
es
8
7
6
12
y8
5
2
2
2
8 x y
26. El resultado de 7
4 y 2
6
256 x y 256 x
6
25. El resultado de 4 x 7
2
2 x y
es
7 y
2
5y 4
es
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- ------------------3
2
12w z 18w z 24wz 6wz
27. El resultado de e) f) g) h)
2w
2
12w
3w 4 z
2
3
2
2w z 3w z 4wz 4
2w z
2
3
3w z
2
28. El resultado de
4 2
4w z
12a
2
5
9a 3
e) 4a 7 f) 4a 1 g) 4a 7 h) 4a 1
3
2 x 3 29. El resultado de 3 es 3
6 x
e)
9 8 x
f) g) h)
27
3
27 8 x
27
3
27
8 x
4 x
2
3
6 x 27
3
27
4 x
2
es
3
18w 24z
3
4
18x 27
3a 3 es
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
31. La representación decimal de
A)
5.4
5
es :
B) 4.5
32. El resultado de ( x 2
A) 6x2 + x + 6
C) 1.25
4 x 2) (7 x
2
B) 8x2 + 9x + 6
3
33. El resultado de
A) 2p2 – 3p – 4q3 C) 2p3q - 3p2q – 4pq4
34.
4
5x 4) es :
C) 6x 4 + x2 + 6
2
12 p q 18 p q 24 pq 6 pq
B) (x4 + 12) (x 2 – 12)
D) 8x4 + 9x2 + 6
4
es : B) 12p2 – 18p – 24q3 D) 2p4q2 – 3p3q2 – 4p2q5
El resultado de factorizar la expresión
A) (x4 + 12) (x 4 – 12)
D) 0.8
X 8 - 144 C) (x4 – 12)2
es : D) (x6 – 12)2
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
36.
¿Cuál es el sueldo de un empleado si después de descontarle el 15% por concepto de impuestos y prestaciones cobra $ 15 525.25 ?
A) $ 22 178.93
37.
B) $18 265.00
C) $17 854.04
D) $13 196.46
Observe el siguiente plano :
Y M
X ¿Cuáles son las coordenadas del punto M ? A) (4 , 3)
B) (4 , - 3)
C) (- 4 , 3)
38.
¿Cómo se define número positivo?
a) b) c) d)
Aquel número que está a la derecha de otro número Aquel número número que esta después del origen origen Aquel número número que está a la izquierda del origen Aquel número número que es mayor que otro
D) (- 4, - 3)
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- ------------------39.
{2,
Del siguiente conjunto de números ¿Cuáles son números racionales?
1 3 6 5 1 , , 2 , ,2, ,3 ,0,0.2323..., ,3.1416,0.3} 2 3 2 3 4 1 3 6
5
1
a) {2, , , ,2, ,3 ,0,0.2323...,3.1416,0.3} 2 3 2 3 4 1 3 6 5 1 b) { , , , ,3 ,0.2323...,3.1416,0.3} 2 3 2 3 4 1 3 6 5 1 c) { , , , ,3 } 2 3 2 3 4 1 3 5 13 d) {2, , ,3,2, , ,0.2323...,3.1416,0.3} 2 3 3 4
40.
{3,2,
Del siguiente conjunto de números. ¿Cuáles son números racionales? 3 3 7 1 , , , 2 , ,0, } 4 2 12 3 3 3 7
1 , } 4 2 12 3 3 3 7 1 b) {3,2, , , ,0, } 4 2 12 3 3 3 7 1 c) { , , ,0, } 4 2 12 3 3 3 7 1 d) { , , , } 4 2 12 3
a) {3,2, , ,
41.
3 3 7
¿Cuál es el orden del siguiente conjunto de números de menor a mayor? {3,2, , ,
1 , 2 , ,0, , 7}
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
42.
¿Cuál es el orden del siguiente conjunto de números de mayor a menor? {8,8,4,4,
3
,
2 3
, , ,
4 7 12 , , } 4 3 4
4 7 2 , , ,0.666} 4 4 3 3 4 7 2 12 b) {8,8, ,4, , , ,0.666, ,4, } 4 3 3 4 4 7 2 12 c) {8, ,4, , , ,0.666, ,4, ,8} 4 3 3 4 7 4 2 12 d) {8,4, , , ,0.666, , , ,4,8} 3 4 3 4
a) {8, 16 , ,
43.
a)
Cuál es la raíz 2 2
12
2
72 2
,8, ,4,
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
45.
Dada la ecuación de primer grado encontrar la incógnita 15 + 2x = 24 es:
a) 4.2 b) c)
9 2
9 4
d) 3.8
48.- En qué consiste el sistema coordenado cartesiano a) Establece la correspondencia entre el conjunto de los pares ordenados (x,y) y el conjunto de todos los puntos de un plano b) Correspondencia de los elementos de un conjunto con otro c) Combinación de los elementos de 2 conjuntos d) Correspondencia de pares ordenados ordenados con los elementos de un tercer conjunto
49.- Cuáles son los ejes del sistema cartesiano
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
50. ¿Cuál es la orientación positiva o negativa que tienen los ejes del sistema coordenado? a) Eje de la abscisa
– a la derecha + a la izquierda Eje de las ordenadas – hacia a bajo + hacia arriba
b) Eje de la abscisa
i zquierda – a la izquierda + a la derecha Eje de las ordenadas – hacia arriba + hacia abajo
c) Eje de las abscisas – del origen hacia la izquierda + del origen hacia la derecha Eje de las ordenadas – del origen hacia abajo + del origen hacia arriba d) Eje de las abscisas
– hacia la derecha + hacia la izquierda Eje de las ordenadas – hacia abalo
+ hacia la izquierda
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
52. Observe las gráficas De acuerdo con el conjunto de pares ordenados {(-5, 2), (-2, 2), (-3, 2), (-3, -2)} ¿Cuál es la opción correcta? a)
c)
b)
d)
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
54. Observe la gráfica
De acuerdo con ella, las coordenadas del conjunto de puntos se indican en la opción a) b) c) d)
{(5,3),(3,5),(-3,-2),(7,-5)} {(5,3),(3,5),(-3,2),(7,-5)} {(-5,3),(3,5),(-4,-2),(7,-5)} {(-5,3),(3,5),(-4,2),(7,-5)}
55. ¿Cuál de las siguientes gráficas corresponde a una función? a)
b)
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
56. ¿Cuál de las siguientes gráficas correspondientes a una función? a)
c)
b)
d)
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
58. Ordenar los siguientes siguientes números reales de menor a mayor :
0.0001 , -0.2 , 0.1 , -2 , 1
A) -0.2 -2 0.0001 0.1 1
B) -2 -0.2 0.0001 0.1 1
C) -2 -0.2 0.1 0.0001 1
D) 1 0.1 0.0001 -2 -0.2
59. ¿Cuáles son las coordenadas del siguiente plano cartesiano?
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
61.-¿Cuál es una de las soluciones soluciones de la ecuación 4x + 5y + 3 = 0?
a) x = - 4, y = -12
b) x = 6, y = - 3 c) x = 5, y = - 23/5 d) x = 11/3, y = 5/7
62. -¿Cuál es la pendiente de la recta 3x + 5y – 2 = 0? a) b) c) d)
3/5 - 3/5 2/5 - 2/5
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------------------------- ----------------------------------- GUIA PARA EXAMEN CENEVAL DE EXANI ----------------------------- -------------------
66. – Observa la siguiente gráfica
Según ella, ¿cuál es la ordenada al origen? a) b) c) d)
-2 0 - 5/2 2