LIBRO RECOPILACIÓN PSU EJERCICIOS DEMRE
CONTENIDOS EJERCICIOS PSU RESPUESTAS ENSAYOS ÁLVARO M. SÁNCHEZ VÁSQUEZ PROF. MATEMÁTICA y FÍSICA 2010 1
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INDICE 1 2
3 4 5 6
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36
Contenido Números Enteros, operatoria, propiedades Números racionales, operatoria, propiedades Potencias, propiedades, aplicaciones aplicaciones Operatoria algebraica Simbología Razones y proporciones Tanto por ciento Raíces, propiedades, aplicaciones Ecuaciones de primer grado, lineales, sistemas de ecuaciones Desigualdades, intervalos, inecuaciones Ecuación de segundo grado Logaritmos, propiedades, aplicaciones aplicaciones Funciones, operatoria, tipos de funciones Ángulos y Triángulos, propiedades, Teorema de Pitágoras, teorema de Euclides Congruencia de triángulos Semejanza de triángulos Cuadriláteros Polígonos Ángulos en la circunferencia Relaciones métricas en la circunferencia, círculo Poliedros, volumen División interior y exterior Trigonometría Probabilidad Estadística Transformaciones isométricas Teorema de Tales Evaluación de suficiencia de datos Respuestas Resumen contenidos Primer año medio Resumen contenidos Segundo año medio Resumen tercer año medio Resumen Cuarto año medio Ensayo 1 Ensayo 2 Ensayo 3
Página 3 10 20 26
38 42 49 57 64 79 83 85 88 108 129 133 141 152 153 162 166 173 175 183 198 209 221 226 243 248 258 269 280 290
308 329
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RESUMEN PSU MATEMATICA
I. NÚMEROS NATURALES Y CARDINALES ( IN, IN0 ) Los elementos del conjunto lN = {1, 2, 3, «} se denominan ´números naturalesµ. Si a este conjunto le unimos el conjunto formado por el cero, obtenemos lN0 = {0, 1, 2, «} llamado ´conjunto de los números cardinalesµ. NÚMEROS ENTEROS (Z) Los elementos del conjunto Z = { «, -3, -2, -1, 0, 1, 2, «} se denominan ´números enterosµ Algunos subconjuntos de Z son: Z+ = {1, 2, 3, «} enteros positivos Z 0 = {0, 1, 2, « } enteros no negativos Z- = {-1, -2, -3, «} enteros negativos Z 0 = {0, -1, -2, -3, «} enteros no positivos
1. Son cuadrados perfectos los enteros: enteros: 1, 4, 9, 16, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225,
256, « 2. Son cubos perfectos los enteros: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, « y también: 1, -8, -27, -64, -125, -216, -343, « MÚLTIPLO Y DIVISOR En la expresión a = b c en que a, b y c son números enteros, a es múltiplo de b y de c o bien b y c son divisores o factores de a. REGLAS DE DIVISIBILIDAD Un número entero es divisible: Por Cuando 2 Termina en cifra par. 3 La suma de sus cifras es múltiplo de tres. 4 Las dos últimas cifras forman un número múltiplo de cuatro o bien son Ceros. 5 La última cifra es cero o cinco. 6 Es divisible por dos y por tres a la vez. 7 La diferencia entre el doble de la última cifra y el número que forman las Cifras restantes es múltiplo de siete. 8 Las tres últimas cifras forman un número múltiplo múltip lo de ocho o bien son Ceros. 9 La suma de sus cifras es múltiplo de nueve. 10 Termina en cero. 11 La diferencia entre la suma de las cifras ubicadas en los lugares pares y
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NÚMEROS PRIMOS, COMPUESTOS y DESCOMPOSICIÓN EN FACTORES Números primos rimos:: Son aquellos enteros positivos que tienen sólo dos divisores distintos. Los primeros números primos son: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, « Números compuestos uestos:: Son todos los enteros positivos mayores que uno que no son primos. Los primeros números compuestos son: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, « TEOREMA FUNDAMENTAL Todo número compuesto se puede expresar de manera única como el producto de factores de números primos MÍNIMO COMÚN MÚLTIPLO (m.c.m.) Es el menor múltiplo común positivo de dos o más enteros. MÁXIMO COMÚN DIVISOR (M.C.D.) Es el mayor divisor común entre dos o más enteros. CÁLCULO DEL m.c.m. y M.C.D MEDIANTE DESCOMPOSICIÓN EN FACTORES PRIMOS Se descomponen los números en factores primos: 1. El m.c.m. se obtiene como producto de todos los factores primos. En el caso de existir factores primos comunes se considera aquel que posea el exponente mayor. 2. El M.C.D. se obtiene como producto de los factores primos comunes considerando aquel que posea el exponente menor. OPERATORIA EN Z ADICIÓN i. Al sumar números de igual signo, se suman los valores absolutos de ellos conservando el signo común. ii. Al sumar dos números de distinto signo, al de mayor valor absoluto se le resta el de menor valor absoluto y al resultado se le agrega ag rega el signo del mayor valor absoluto. MULTIPLICACIÓN i. Si se multiplican dos números de igual signo al resultado es siempre positivo. ii. Si se multiplican dos números de distinto signo el resultado es siempre negativo. OBSERVACIÓN: La división cumple con las reglas de signos de la multiplicación. VALOR ABSOLUTO Es la distancia que existe entre un número y el 0
n, si n u 0 ® n si n 0 °
DEFINICIÓN: ¯
ALGORITMO DE LA DIVISIÓN Si D: d = c, entonces D = d c + r r //
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OBSERVACIONES: 1) 0 r < d 2) La división por cero no está definida. PRIORIDAD DE LAS OPERACIONES Al realizar distintas operaciones a la vez, se debe respetar el siguiente orden: 1. Resolver los paréntesis. 2. Realizar las potencias. 3. Realizar multiplicaciones y/o divisiones de izquierda a derecha. 4. Realizar adiciones y/o sustraccione s ustraccioness de izquierda a derecha. RELACIÓN DE ORDEN EN Z Si a y b son números enteros, entonces diremos que: i. a > b si y sólo si (a - b) es un entero positivo. ii. a < b si y sólo si (a - b) es un entero negativo. iii. a b si y sólo si (a > b) o (a = b); (no ambos a la vez). iv. a b si y sólo si (a < b) o (a = b); (no ambos a la vez). EJEMPLO PSU-1: Si al entero (² 1) le restamos el entero (² 3), resulta
A) ² 2 B) 2 C) 4 D) ² 4 E) ninguno de los valores anteriores EJEMPLO PSU-2: Si a es un número de dos dígitos, en que el dígito de las decenas es m y el de las unidades es n, entonces a + 1 =
A) m + n + 1 B) 10m + n + 1 C) 100m + n + 1 D) 100m + 10n + 1 E) 10(m + 1) + n EJEMPLO PSU-3: Si n = 2 y m = -3, ¿cuál es el valor de ²nm ²(n + m)?
A) -11 B) -5 C) 5 D) 7 E) -7
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EJEMPLO PSU-4: En una fiesta de cumpleaños hay 237 golosinas para repartir entre 31 niños invitados. invitados. ¿Cuál es el número mínimo de golosinas que se necesita agregar para que cada niño invitado reciba la misma cantidad de golosinas, sin que sobre ninguna? A) 11 B) 20 C) 21 D) 0 E) 7 EJEMPLO PSU-5: Claudia tenía en el banco $ 4p. Retiró la mitad y horas más tarde depositó el triple de lo que tenía al comienzo. ¿Cuánto dinero tiene ahora Claudia en el banco? A) $ 8p B) $ 10p C) $ 12p D) $ 16p E) $ 14p EJEMPLO PSU-6: Para completar la tabla adjunta se debe seguir la siguiente regla: el último número de cada fila es la suma de los tres números anteriores y el último número de cada columna es la suma de los tres números anteriores. ¿Cuál es el valor de x? A) 5 4 x 20 B) 7 4 9 C) 8 8 13 D) 9 16 55 24 E) 16 EJEMPLO PSU-7: Con los círculos se ha armado la siguiente secuencia de figuras:
¿Cuál(es) de las siguientes afirmaciones es(son) verdadera(s)? verdadera(s)? I) La décima figura de la secuencia está formada por 21 círculos II) De acuerdo a la formación de la secuencia cualquier figura tendrá un número impar de círculos III) La diferencia positiva en cuanto a la cantidad de círculos entre dos figuras consecutivas es 2 A) Sólo I B) Sólo I y II C) Sólo I y III D) Sólo II y III E) I, II y III
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EJEMPLO PSU-8: En un monedero hay doce monedas de $5 y nueve de $10. Estas 21 monedas representan un cuarto del total de dinero que hay en su interior. Si en el resto de dinero se tiene igual cantidad de monedas de $50 y de $100, ¿cuál(es) de las siguientes afirmaciones es(son) verdadera(s)? I) En total hay 27 monedas II) Hay 4 monedas de $50 en el monedero III) En el monedero hay $600
A) Solo I B) Solo II C) Solo III D) Solo I y III E) Solo II y III EJEMPLO PSU-9: Se define valor de (2 ¢ 5) # (-2) es:
a ¢ b ! ab b
y a # b = 2a - 4b, para a y b números enteros, el
A) 82 B) 66 C) 60 D) 38 E) 22 EJEMPLO PSU-10: Al sumar el cuarto y el quinto término de la secuencia: x - 5, 2(2x + 7), 3(3x - 9), 4(4x + 11), . . . , resulta A) 41x - 2 B) 61x + 25 C) 41x - 109 D) 41x + 109 E) 41x - 21 EJEMPLO PSU-11: ¿De cuántas formas distintas se puede pagar, en forma exacta, una
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EJEMPLO PSU-13: Si tuviera $80 más de los que tengo podría comprar exactamente 4 pasteles de $ 240 cada uno, ¿cuánto dinero me falta si quiero comprar 6 chocolates de $ 180 cada uno? A) $280 B) $200 C) $120 D) $100 E) $ 40 EJEMPLO PSU-14: El precio de los artículos M, N y T son $(n-1), $(n-2) y $(n -3), respectivamente. ¿Cuántos pesos se deben pagar por un artículo M, dos artículos N y tres artículos T? A) 6n - 14 B) 6n ² 6 C) 5n ² 14 D) 3n ² 14 E) 3n - 6 EJEMPLO PSU-15: En las siguientes igualdades los números n. p, q y r son enteros positivos. ¿Cuál de las opciones expresa la afirmación p es divisible por q?
A) p = nq + r B) q = np + r C) q = np D) p = nq E)
p 1 !1 q q
EJEMPLO PSU-16: Una prueba tiene 40 preguntas. El puntaje corregido se calcula de la siguiente manera: ´Cada 3 malas se descuenta 1 buena y 3 omitidas equivalen a 1 malaµ. ¿Cuál es el puntaje corregido si un estudiante obtuvo 15 malas y 9 omitidas? A) 8 B) 6
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EJEMPLO PSU-18: M, N y P son números enteros mayores que 1. Si ninguno de ellos tiene factores en común, salvo el 1, cuando M = 9 y N = 8, ¿cuál es el menor valor posible de P? A) 7 B) 5 C) 4 D) 3 E) 1 EJEMPLO PSU-19: En un triángulo equilátero de lado 1.000 se unen los puntos medios de cada lado y se obtiene un nuevo triángulo equilátero, como se muestra en la figura. Si repetimos el proceso 6 veces, el lado del triángulo que se obtiene es: 1.000 12 ¨ 1.000 ¸ B) 6 y © ¹ ª 2 º 1.000 C) 26 1.000 D) 6 1.000 E) 25 A)
EJEMPLO PSU-20: La suma de tres números impares consecutivos es siempre: I) divisible por 3 II) divisible por 6 III) divisible por 9 Es(son) verdadera(s): A) Solo I B) Solo II C) Solo I y III D) Solo II y III E) I, II y III
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II. NÚMEROS RACIONALES
Los números racionales son todos aquellos números de la forma
a b
con a y b números
enteros y b distinto de cero. El conjunto de los números racionales racionales se representa por la letra Q.
. IGUALDAD ENTRE NÚMEROS RACIONALES
2
ADICIÓN Y SUSTRACCIÓN DE NÚMEROS RACIONALES
Si
a c , Q, b d
entonces:
OBSERVACIONES
1. El inverso aditivo (u opuesto) de a b
o
a b
es - a , el cual se puede escribir también como b
a b
2. El número mixto A b se transforma a fracción con la siguiente fórmula: c
MULTIPLICACIÓN Y DIVISIÓN DE NÚMEROS RACIONALES
Si
a c Q, , b d
entonces:
MULTIPLICACIÓN
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RELACIÓN DE ORDEN EN Q
OBSERVACIONES 1. Para comparar números racionales, también se pueden utilizar los siguientes procedimientos: a) igualar numeradores. b) igualar denominadores. denominadores. c) convertir a número decimal. 2. Entre dos números racionales cualesquiera hay infinitos números racionales. NÚMEROS DECIMALES Al efectuar la división entre el numerador y el denominador de una fracción, se obtiene un desarrollo decimal, el cuál puede ser finito, infinito periódico o infinito semiperiódico. semiperiódico.
a) Desarrollo decimal finito: Son aquellos que tienen una cantidad limitada de cifras decimales. Ejemplo: 0,425 tiene 3 cifras decimales b) Desarrollo decimal infinito periódico: Son aquellos que están formados por la parte entera y el período. Ejemplo: 0,444.... = 0, 4 c) Desarrollo decimal infinito semiperiódico: Son aquellos que están formados por la parte entera, un anteperíodo y el período. Ejemplo: 24,42323 ... = 24,4 23 OPERATORIA CON NÚMEROS DECIMALES 1. Adición o sustracción de números decimales: Para sumar o restar números decimales se ubican las cantidades enteras bajo las enteras, las comas bajo las comas, la parte decimal
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3. División de números decimales: Para dividir números decimales, se puede transformar
el dividendo y el divisor en números enteros amplificando por una potencia en base 10. Así por ejemplo: 2,24: 1,2 se amplifica por 100 224: 120 y se dividen como números enteros enteros TRANSFORMACIÓN DE DECIMAL A FRACCIÓN finito:: Se escribe en el numerador todos los dígitos que forman el número 1. Decimal finito decimal y en el denominador una potencia de 10 con tantos ceros como cifras decimales tenga dicho número.
Por ejemplo: 3,24 =
324 100
. Decimal infinito periódico: Se escribe en el numerador la diferencia entre el número decimal completo (sin considerar la coma) y el número formado por todas las cifras que anteceden al período y en el denominador tantos tantos nueves como cifras tenga el período. 2
Por ejemplo: 2, 15 =
215 2 99
3. Decimal infinito semiperiódico: Se escribe en el numerador la diferencia entre el
número completo (sin considerar la coma) y el número formado por todas las cifras que anteceden al período y en el denominador se escriben tantos nueves como cifras tenga el período, seguido de tantos ceros como cifras tenga el anteperíodo. Por ejemplo: 5,3 4 =
534 53 90
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EJEMPLO PSU-4:
9 3 ! 8 5
A) 0,15 B) 0,5 C) 0,52 D) 0,525 E) 2 EJEMPLO PSU-5: Si a A) B) C) D) E)
5 se le resta 1 resulta: 6 3
1 2
1 2 2 3 4
3 2 9
EJEMPLO PSU-6:
1 3 0,75 8
1 3 0,25 8
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EJEMPLO PSU-8: En la igualdad
1 1 1 ! , P Q R
si P y R se reducen a la mitad, entonces
para que se mantenga el equilibrio, equilibrio, el valor de Q se debe A) duplicar. B) reducir a la mitad. C) mantener igual. D) cuadruplicar. E) reducir a la cuarta parte. EJEMPLO PSU-9: Juan dispone de $ 6.000 para gastar en entretención. Si se sabe que cobran $1.000 por jugar media hora de pool y $600 por media hora en Internet, entonces ¿cuál(es) de las siguientes afirmaciones es(son) verdadera(s)? I) Juan puede jugar a lo más 3 horas de pool II) Juan puede conectarse a lo más 5 horas en Internet III) Juan puede jugar 1,5 horas de pool y conectarse conectarse 2,5 horas a internet
A) Solo III B) Solo I y II C) Solo I y III D) Solo II y III E) I, II y III
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EJEMPLO PSU-11: Si P R
B) C)
E)
R
2P 2P R
2R P R
2P
EJEMPLO PSU-12: 5 12 2 B) 15 1 C) 9 2 D) 3 1
A)
1 RH , entonces H -1 es igual a: 2
2P
A)
D)
!
1 1 1 ! 3 6 2
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EJEMPLO PSU-14:
1 3
2
1
1
!
4
3 2 1 B) 3 11 C) 6 D) 1
A)
E) 3 50 0,5 1 00 EJEMPLO PSU-15: ! (0,5) 2
A) 10 B) 1 C) 0,1 D) 0,25 E) 0,75 EJEMPLO PSU-16: Una persona debe recorrer 12,3 kilómetros y ha caminado 7.850 metros. ¿Cuánto le falta por recorrer?
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EJEMPLO PSU-18: Se mezclan 2 litros de un licor P con 3 litros de un licor Q. Si 6 litros del licor P valen $ a y 9 litros del licor Q valen $ b, ¿cuál es el precio de los 5 litros de mezcla? ab 3 ab B) $ 5 C) $(2a 3b)
A) $
3a 2b 18 5 (3a 2b) E) $ 18
D) $
EJEMPLO PSU-19: Juan tiene un bidón de 5 litros de capacidad, llenado hasta los 2
1 3
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EJEMPLO PSU-21: Se define a b = A)
1
abc a B) bc bc C) a ab
D) E)
c c
ab
1 ab
, entonces a (b c) es igual a:
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EJEMPLO PSU-24: tres atletas corrieron los 100 metros planos, Javier cronometró 11,3 segundos, Arturo 11,02 segundo y Marcelo 11,2 segundos. ¿Cuál(es) de las siguientes afirmaciones es(son) verdadera(s)? I) Javier llegó después de Marcelo II) Entre Arturo y Marcelo hay 18 centésimas de segundo de diferencia al llegar a la meta III) Arturo llegó primero A) Solo I B) Solo I y II C) Solo I y III D) Solo II y III E) I, II y III EJEMPLO PSU-25: En una receta de un postre para 6 personas se necesitan 200 gramos de
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III. POTENCIAS EN Z DEFINICIÓN
PROPIEDADES 1. 0 n = 0, si n Z+ 2. 1 n = 1 3. Si n es par, ( 1) n = 1 4. Si n es impar, ( 1) n = -1
Positivo si a { 0 y n es par ® Signos de una potencia: p otencia: a n = ¯ Negativo si a 0 y n es impar °
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10 1 = 10 10 2 = 100
1 =0,01 100 10 3 = 1 =0,001 1000
10 2 =
10 3 = 1000 Las potencias de base 10 se utilizan para escribir un número de las siguientes formas: 1. Un número está escrito en notación científica si se escribe de la forma k 10 n , en en que 1 k < 10 y n Z. 2. Un número está escrito en forma abreviada, si se escribe de la forma p 10n, en que p es el menor entero y n Z. 3. Un número esta inscrito en notación ampliada o desarrollada si se expresa como la suma de las cantidades que resulten de multiplicar cada dígito de dicho número por la potencia de diez correspondiente a su posición (... centena, decena, unidad, décima,
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1 EJEMPLO PSU-4: ¨© a 2 ¸¹ ª2 º A ) 8a 6 B) 8a 5
3
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EJEMPLO PSU-8: ¿Cuál es la mitad de 2 6 ?
A) 25 B) 23 C) 16
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3,6 10 6 0 ,00006 EJEMPLO PSU-12: ! 20.000.000 A ) 1,08 10 4 B) 1,08 10 5
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EJEMPLO PSU-17: Si una colonia de bacterias se triplica cada 20 minutos e inicialmente hay 5.000 de ellas, el número de bacterias que hay al término de 3 horas es:
A) 5.000 33 bacterias B) 5.000 34 bacterias
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IV. ALGEBRA y FUNCIONES EVALUACIÓN DE EXPRESIONES ALGEBRAICAS Evaluar una expresión algebraica consiste en sustituir las letras por los valores numéricos dados para luego realizar las operaciones indicadas. Esta sustitución va siempre entre
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