Guia 2 -Algebra y Funciones
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Guía Nº 2 Álgebra 1.
2.
Si A)
13 5
B) C)
13 3
D)
3 5
E)
5
8(x 5) 2(x 4) , x = 3
A) B) C) D) E) 3.
1 2 32 16 8 6
En la ecuación en x, 5x - (2 - k)x = 3, ¿cuál debe ser el valor de k para que la solución sea x = -3? A) B) C) D) E)
4.
3 x - 5 = 2 x + 8, entonces x =
4 3 -7 -4 6
Si el número 386 se divide en dos partes tales que, si la mayor se divide divide entre entre 4 y la menor se disminuye en 36, los resultados son iguales, entonces la mayor de las partes es A) B) C) D) E)
70 106 144 200 280
1
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5.
Si
A) B) C) D) E) 6.
7.
x3
5 11 11 , entonces 2x – 6 = 2x 2x 6 2
2 12 6 8 4
“La tercera parte de un número q resulta ser igual a la mitad de su cuadrado”. Este enunciado se traduce como A)
3q = 2q2
B)
1 1 q2 3q 2
C)
1 1 q 3 2q2
D)
1 1 2 3q 2q
E)
1 1 q q2 3 2
Pedro tiene b + 3 bolitas, si Antonio tiene 8 bolitas menos que Pedro, ¿cuántas tiene Antonio? A) B) C) D) E)
8.
3
b b b b 5
+ 11 +5 -5 -b
La tabla de la figura figura 1, muestra muestra el número número de monedas extranjeras y el valor valor correspondiente en pesos, pesos, (A + C) – (B + D) = Nº de monedas 1 5 10 50
Dólar $ 600 C $ 6.000 $ 30.000 fig. 1
Euro D $ 3.500 B A
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9.
Una señora va a la feria con $ 10.000, en el primer local compra 3 kilos de papas a $ 200 cada kilo, en el segundo local un paquete de cebollas a $ 300, compra mas allá media docena de huevos a $ 800 la docena, si la señora hasta aquí quiere saber cuánto ha gastado debe hacer la siguiente secuencia de operaciones( S: suma, R : resta, M: multiplicación y D: división) A) B) C) D) E)
10.
¿Cuál es al valor de x en la ecuación A) B) C) D) E)
11.
B) C) D) E)
3x 1 x 3? 2
8 4 7 2 3 6
Al despejar n en la ecuación S A)
12.
3 M $200 S $300 S $800 D 2 3 S $200 S $300 S $800 R 2 2 $200 D 2 S $300 S $800 M 2 3 M $200 S $300 S $800 M 2 Ninguna de las anteriores
11( 11(n 2) , se obtiene 5
5(S 2) 11
5S 2 11 5S 2 11 11S 11S 2 5 5S 2 11
Al dividir el número número 576 en dos partes tales que el triple de la menor exceda en 103 unidades a la cuarta parte del mayor. ¿Cuál es la parte menor? A) B) C) C) D)
500 412 247 76 ninguna de las anteriores
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13.
Si 0,2x - 1 = 5, entonces el valor de x es
A) B) C) D) E) 14.
Si a(1+x) = 2a , entonces x = A) B) C) D) E)
15.
a 2a 1 2 -a
Si al triple de un número número x se le suma 20 resulta resulta ser ser igual igual al exceso del número número sobre 5, entonces la ecuación que resuelve el problema es A) B) C) D) E)
16.
30 20 15 3 1,2
3x + 20 = x – 5 3x + 20 = 5 – x 3 + x + 20 = x – 5 3·(x + 20) = x – 5 3x + 20 + x = 5
Para diluir una solución de sal al 5% se deben agregar x litros de agua , a los 20 litros originales de solución. ¿Cuál es la ecuación que permite obtener x si se quiere una solución al 1% de sal? A)
B)
C) D) E)
5 1 x 20 100 100 10 0 5 ·20 1 10 0 20 x 10 0 5 ·20 x 1 10 0 20 x 10 0 95 1 (20 x) 10 0 10 0
otra ecuación
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17.
El perímetro de un cuadrado es 64 cm, si cada lado mide (3x – 2) cm, entonces la ecuación que permite encontrar x es A)
(3x – 2)2 = 64
B)
3x 2 64 4
C) D) E) 18.
64
2 -2 -4 7 -7
La solución de la ecuación A) B) C) D) E)
20.
4
Si 5·(r + 4) = -2(r – 3), entonces 3,5r = A) B) C) D) E)
19.
3x – 2 = 4 · 64 12x – 8 = 64 3x 2 1
6x 18 2x 6 , es 3
0 1 2 Todos los reales positivos Todos los reales
La expresión que permite saber a cuentos grados Celsius equivalen los grados Fahrenheit es; C
5 (F 32) , donde C son grados Celsius y F es grados Fahrenheit. ¿Cuál sería la 9
expresión para obtener los grados Fahrenheit equivalentes a grados Celsius?
A) B) C) D) E) 21.
9 C 5 9 F = C + 32 5 9 F = (C 32) 5 5 F = (C 32) 9
F=
F = 5C + 160
La crema contiene, aproximadamente, 22% de grasa. ¿Cuántos litros de crema se deben mezclar con leche al 2% de grasa, para obtener 20 l itros de leche con 4% de grasa?
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22.
Un alumno alumno tenía el 70% 70% de puntos en un examen examen de 30 preguntas. preguntas. Para Para mejorar mejorar su calificación, el profesor estuvo de acuerdo con poner 15 preguntas más. ¿Cuántas preguntas de las 15 adicionales debe contestar correctamente para elevar su calificación a un 80%? A) B) C) D) E)
23.
1 5 10 12 Todas
Un árabe árabe que tenía tenía tres hijos dispuso dispuso en su su testamento testamento que el el hijo mayor heredara la mitad de la herencia, al del medio la tercera parte y al menor la novena parte. Cuando el árabe murió dejo de herencia 17 camellos. Los hijos no podían repartirse los camellos, ya que el mayor le correspondía 8,5 camellos, al del medio 5
8 2 y al tercero 1 , por tanto 9 3
decidieron llevar el caso a un juez. El juez que sabía matemáticas decidió regalarle un camello, ante los cual el mayor recibió 9 camellos (medió más de la repartición inicial), el del medio recibió 6 camellos y el tercero 2 camellos, ante lo cual todos los hermanos quedaron felices ya que recibieron más. Ante esto el juez sumó los camellos repartidos; 9 + 6 + 2 = 17, ante lo cual el juez tomó su camello y todos quedaron satisfechos. ¿De qué se dio cuenta el juez para donar su camello y saber que no lo perdería?
24.
A) B) C) D)
Que Que Que Que
los los los los
camellos iniciales a camellos iniciales a camellos iniciales a camellos iniciales a
E)
Que el árabe que murió, solo repartió
m
repartir no repartir no repartir no repartir no
era múltiplo era múltiplo era múltiplo era múltiplo
de de de de
2. 3. 9. 18.
17 1 de su herencia quedando sin repartir. 18 18 m
Sea E la parte entera de la l a fracción impropia , luego es siempre verdadero que n n A)
n m E E m n
B)
m n E + E = 0 n m
C)
E
D)
m n m n E + E = E n m n m
m n - E = 0 n m
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25.
La solución de la ecuación A) B) C) D) E)
26.
x x x x x
+a – b – ab + ab +b
Para que el polinomio –a2x + x3 sea igual a cero, x debe ser igual a A) B) C) D) E)
28.
0 1 3 -3 no tiene solución
Uno de los factores del polinomio x2 +(b – a)x – ab es A) B) C) D) E)
27.
3x 12 x 4 es 3
1 2a a 1+a 1 – a
¿Cuál de las siguientes funciones no representa a cantidades directamente proporcionales? A)
f(x) = 2x
B)
f(x) =
C) D) E)
f(x) = 0,04 · x f(x) = 3x + 1 f(x) = 7x
3 x 4
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30.
Dado el siguiente sistema
A) B) C) D) E) 31.
Si A) B) C) D) E)
32.
40 35 30 25 50 3x y 38 2x 3y 100
, entonces y – x =
32 30 28 14 2
Dos números suman 50 y uno de de ellos excede al otro en 6 unidades, luego el menor menor de ellos es A) B) C) D) E)
33.
x y 55 , 2y = x y 15
47 28 25 22 20
En el sistema de ecuaciones 2? A) B) C)
2 3 4
nx y 10 x 3y 20
, ¿cuál debe ser el valor de n para que x sea igual a
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35.
1 1 1 Si x y 3 , entonces x + y = xy 9
A) B) C)
36.
3
1 9
E)
otro valor
Un avión pequeño puede cargar 950 kilos de equipaje en dos compartimientos de carga. En un vuelo, el avión va totalmente cargado con 150 kilos más en un compartimiento que en el otro. ¿Cuánto equipaje hay en cada compartimiento? 475 k. y 475 k. 400 k. y 550 k. 450 k. y 600 k. 350 k. y 600 k. 405 k. y 555 k.
x y 4 2 3 En el sistema y = x y 0 2 9
A) B) C) D) E) 38.
1
D)
A) B) C) D) E)
37.
3 6
1 2 3 -2 9
En un triángulo rectángulo, uno de sus ángulos agudos mide 16º 16º más más que el otro otro ángulo
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39.
En el sistema
A) B) C) D) E) 40.
1 -1 -2 -4 No tiene solución
Un automóvil y un camión camión salen de Santiago al mismo tiempo en direcciones opuestas. Al cabo de dos horas están a 350 km. de distancia y el automóvil ha recorrido 70 km. más que el camión, ¿cuál es la rapidez del automóvil? A) B) C) D) E)
41.
70 K/h 80 K/h 90 K/h 105 K/h 110 K/h
Se tienen dos soluciones de alcohol alcohol una al 8% y la otra al 15%, 15%, ¿cuántos ¿cuántos litros de cada una se deben considerar para obtener 100 litros de una solución al 12,2%? 8% A) B) C) D) E)
42.
3 y5 x = 2 2x 3y 8 x
15%
35 55 55 35 60 40 40 60 Ninguna de las anteriores
Dado el siguiente sistema de ecuaciones
3x 4y 9 x 2y 2y 8
luego 6x + 2y =
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Dada la tabla siguiente referida a la producción de pinturas mediante dos procesos. Si la pintura se venderá a $ 1.800 el galón, entonces responde las siguientes dos preguntas.
Proceso Costos fijos Costo por galón A $ 3.250.000 $ 1.300 B $ 8.060.000 $ 500 43.
¿Cuántos galones debe producir producir solo solo mediante el procesa procesa A, para no perder perder ni ganar ganar dinero? A) B) C) D) E)
44.
Si se producen 13.000 galones mediante el proceso A, entonces, ¿cuánto galones se deberán producir mediante el proceso B para igualar los costos de ambos procesos? A) B) C) D) E)
45.
650 5.500 6.500 7.000 7.500
13.000 8.060 3.700 1.300 ninguna de las anteriores
Un vendedor vendedor puede elegir entre entre dos opciones de salario; salario; 1) sólo sólo comisión comisión del 7% 7% de las ventas, 2) un sueldo fijo de $ 150.000 más una comisión por las ventas del 2%. ¿Desde qué monto total de las ventas es más conveniente el plan 1)? A) B) C)
$ 150.000 $ 1.000.001 $ 1.500.001
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47.
Dado
A) B) C)
48.
2 3 30 x y
, luego x =
0 1 6 1 5
D)
E)
6
1 6
En el sistema de ecuaciones a -b, A) B) C) D) E)
49.
3 2 30 x y
ax by 0 bx ay 1
, luego (x + y)-1 =
1 a – b a+b 1 a
1 b
ab
Maritza (M) tiene tiene 4 años más que el 75% de la edad de Carlos Carlos (C). (C). Si hace 5 años años Carlos Carlos
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51.
Un señor señor tiene cuarenta cuarenta y dos años y su su hijo diez. ¿Dentro ¿Dentro de cuántos años más la edad del padre será el triple que la de su hijo? A) B) C) D) E)
52.
Dos enteros positivos son pares consecutivos, si el el cuociente cuociente entre el el mayor mayor y el menor menor es es 1,090909…, entonces el menor de ellos es A) B) C) D) E)
53.
18 12 8 6 4
11 14 22 24 26
La sexta sexta parte parte de la suma de dos números es 16, y la cuarta parte de su diferencia es es 10. ¿Cuáles son los números? A) B) C) D)
28 40 56 40
y y y y
60 96 136 68
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56.
En una una fila de pago pago 2 de cada cada 5 personas personas son hombres, hombres, la última persona tiene el número 96 y la primera el 42, si todas las personas tienen un número correlativo, entonces ¿cuántas mujeres hay en la fila? A) B) C) D) E)
57.
La última vez vez que Antonia y Andrés Andrés se encontraron en Santiago Santiago fue fue este este lunes. lunes. Si Si Andrés Andrés viaja cada 8 días a Santiago y Antonia cada 15 días, ¿cuál será el día en que ambos estén en Santiago? A) B) C) D) E)
58.
11 22 33 44 55
Lunes Martes Miércoles Jueves Viernes
Si tuviera tuviera $ 80 más de los los que que tengo podría comprar exactamente 4 bebidas de $ 240 cada cada una. ¿Cuánto dinero me falta si quiero comprar 6 pasteles de $ 180 cada uno? A) B) C)
$ 100 $ 120 $ 200
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61.
62.
63.
Si al al triple de un número se le resta resta su mitad más más uno, resulta 9, luego el número número es A)
16 5
B) C) D) E)
2 4 20 25
(x 1)2(x2 x 1)2 (x 1)2 (x2 x 1)2 (x3 1)2 (x3 1)2 A)
(x 1)4
B)
(x3 1)4
C)
1
D)
(x3 1)(x3 1)
E)
(x3 1)4
2
A tiene 15 años años más más que B. Dentro Dentro de 4 años más sus edades edades sumarán sumarán 57 años, años, ¿cuál ¿cuál es la edad del mayor?
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66.
Si A es un número de tres cifras diferentes y B es otro número de tres cifras formado por las mismas de A pero en orden inverso, i nverso, luego A – B puede ser A) B) C) D) E)
67.
96 personas deben ser transportados al aeropuerto, solo se disponen de minibuses con capacidad para 7 personas, ¿cuál es el menor número de viajes que deben hacerse para llevarlas a todas? A) B) C) D) E)
68.
165 297 360 561 683
12 13 14 15 18
¿Cuántos números de dos dos cifras cifras son tales que al sumar el número número con otro formado por las mismas cifras, pero invertidas resulta un cuadrado perfecto? A) B) C)
4 5 6
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71.
Un ciclista ciclista recorre recorre una cierta cierta distancia distancia a 30 km/h, km/h, la vuelta la hace hace por por el mismo camino a 40 km/h. ¿Cuál es la rapidez promedio aproximada en la i da y vuelta? A) B) C) D) E)
72.
33 34 35 36 37
km/h km/h km/h km/h km/h
A una reunión asistieron n personas, si se detectó que hubo 28 saludos de manos y todos se saludaron, entonces ¿cuántas personas asistieron a la reunión? A) B) C) D)
7 8 14 28
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76.
En una encuesta de mercado mercado sobre el el consumo consumo de tres marcas A, B y C de un producto arrojó los siguientes resultados; A lo consumen el 48%, B el 45%, C el 50%, A y B el 18%, B y C el 25%, A y C el 15% y ninguno de los tres el 5%. ¿Qué porcentaje de los encuestados consumen solo producto C? A) B) C) D) E)
77.
10 % 20 % 30 % 8% 5%
Si un número es dividido por 10 da resto resto 9, por 9 da resto resto 8, por 8 da resto resto 7 y así hasta que se divide por 2 dando resto 1, entonces el menor número que cumple con estas condiciones es A)
59
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81.
¿Cuál es el valor de a en la ecuación 4a + 17 = 7(a + 2)? A) B) C) D) E)
82.
3 1 0 5 14
Al resolver el sistema de ecuaciones
A) B) C) D) E)
1 –1 0 2 7
2x 3y 12 x 4y 5
el valor que se obtiene para x – y es
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87.
Si xy = 6 y x2y + xy2 +x + y = 63, entonces x 2 + y2 = A) B) C) D) E)
88.
13 55 69 81 42
Si b y c son enteros y (x +2)(x + b) = x 2 + cx + 6, entonces c = A) B) C) D) E)
-5 -1 3 5 1 b
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91.
Si
A) B) C) D) E) 92.
xz xy 10 , entonces el valor de es yz zy
11 -11 9 -9 10
8wxy – 12yxw = A) B) C) D) E)
4 xyw 4 ywx -4 wxy -4 -4x – 4y – 4w
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96.
a6 + a6 + a6 + a6 + a6 + a6 = A) B) C) D)
a7 6a7 (a6)2 6a6
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100.
3
A)
3 3 3 s
=
4s 3 s 1
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103.
Si x e y son enteros positivos de dos cifras tal tal que xy = 555, entonces x + y =
A) B) C)
45 52 66
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107.
Si a es es el doble de b y b es el el doble de c, entonces entonces la octava octava parte parte del producto producto a·b·c es A) B) C)
1 3 c 2 2 c3 4 c3
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112.
Al desarrollar desarrollar el siguiente siguiente producto producto (x (x + m)(x – n) = x 2 + (……..)x – – mn, se borró el coeficiente de x, ¿cuál es? A) B) C)
1 m+n m – n
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277.
(5)2 =
A) B) C)
no es un número real -5 5
D)
1
E) 278.
5
Se dice que un número está escrito en “Notación Abreviada” si se expresa como el producto del menor entero posible y una potencia de base 10, luego 0,0000012 escrito en notación abreviada sería equivalente a 1,2 · 10-6 12 · 10-7 120 · 10-5 0,12 · 10-4 3 · 10-7
A) B) C) D) E) 4
279.
280.
1
3 3 33 2 33
=
A)
3
3
B)
3
9
C)
3 3
D) E)
3
3
Al simplificar
A) B) C) D) E) 281.
1 5
213 216 215
se obtiene
2
1,5 2,25 27 1
Dados los números; x = 3,5 · 10 -5 · 4 · 10-5, y = 1,1 · 10-9 y z = 17 · 10 -9, entonces es correcto afirmar que
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282.
283.
La solución de la ecuación (0,25) 2x = A)
5 8
B)
5 4
C)
5 8
D)
5 4
E)
1 2
¿Cuál es el valor de x en la ecuación A) B) C) D) E)
284.
2x 4 5 6 ?
0 1 2 2,5 no tiene solución en los reales
Dado; log2 = 0,3 y log3 = 0,48, luego al resolver 3x 3x 1 8 , x = A) B) C) D) E)
285.
32 , es
1 0,78 1,78 0,625 otro valor
Si log4 25
A) B) C) D) E)
x , entonces log5 = 3
x 3 x x3
5x 3x x+3
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286.
287.
Si log n = 8, entonces
A)
1 16
B)
1 8
C)
-
D)
E)
- 16
logn2
1 8 1 16
Si log( log(10 100x 0x)) log0,00 log0,001 1 6 , entonces x =
A) B) C) D) E) 288.
1
0 1 5 2 10
Dado que 3x-2 – 2 = 0, entonces x = A)
log3 18
B)
2 log3 2
C)
log3 11
D) E)
2 + log 2 – log 3 ninguna de las anteriores
289.
1 El resultado de la potencia , ¿entre qué valores esta? 2
A) B) C) D) E) 290.
entre entre entre entre entre
-8 y 0 0y8 8 y 16 16 y 27 24 y 32
El número de bacterias se reproduce de manera que duplica su cantidad cada hora. Si inicialmente hay 8 de ellas, al cabo de 10 horas habrá una cantidad de
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291.
292.
293.
Para qué
3k sea un número racional, k debe ser
(1) (2)
un número par un múltiplo de 4
A) B) C) D) E)
(1) por sí sola (2) por sí sola Ambas juntas, (1) y (2) Cada una por sí sola, (1) ó (2) Se requiere información adicional
Para determinar entre que enteros está log2 a , es necesario (1) (2)
a>2 a
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