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1) Dados los números complejos $\\color{blue} {z_1=3-2i}$ y $\\color{red} {z_2=5+7i}$ ¿Cuál alternativa muestra el resultado de $z_1 \\cdot z_2$.

A) $29+11i$

B) $15+14i$

C) $29-11i$

D) $11+29i$

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Para resolver la multiplicación de los complejos $z_1$ y $z_2$ debemos seguir los siguientes pasos:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

Paso 1: Multiplicar término a término los binomios que conforman cada número complejo.

$(3-2i) \\cdot (5+7i)$

$=$[[0]]$+$[[1]]$i$ $-$ [[2]]$i$ $-$[[3]]$i^2$

Paso 2: Recordar que $i^2=-1$, por lo tanto, al final se produce un cambio de signo en la expresión quedando:

$=$[[0]]$+$[[1]]$i$ $-$[[2]]$i$ $+$[[3]]

Paso 3: Se reduce la expresión del paso 2, juntando lo real con real (lo que no tiene $i$), y luego lo imaginario con imaginario (todos aquellos número con $i$)

$=$[[4]][[6]][[5]]$i$

Por lo tanto la alternativa correcta es: [[7]]

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2) La varianza de los datos $6$ ; $4$ ; $8$ y $10$, es:

A) $3$

B) $4$

C) $5$

D) $6$

Para determinar la varianza de los datos $6$ ; $4$ ; $8$ y $10$, debemos realizar los siguientes pasos:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

Paso 1: Calcular el promedio o media aritmética de los datos (recuerda que es la suma de los datos dividido por el total de datos)

[[0]]

$\\overline{x}=$________ $=$ [[2]]

[[1]]

Paso 2: A cada dato le restamos el promedio y el valor obtenido lo elevamos al cuadrado:

$($[[2]]$-6)^2=($[[3]]$)^2=$[[4]]

$($[[2]]$-4)^2=($[[5]]$)^2=$[[6]]

$($[[2]]$-8)^2=($[[7]]$)^2=$[[8]]

$($[[2]]$-10)^2=($[[9]]$)^2=$[[10]]

Paso 3: Los valores obtenidos en el paso 2 se promedian para así obtener la varianza $\\sigma^2$, es decir se suman y se divide por el total de valores obtenidos en el paso 2.

[[4]]$+$[[6]]$+$[[8]]$+$[[10]] [[11]]

$\\sigma^2=$ _______________________________________ $=$ _______ $=$ [[13]]

[[12]] [[12]]

Por lo tanto, la respuesta correcta es: [[14]]

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"Ejercicio 1 NM3 - Repaso General 2020", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Emilio Gonz\u00e1lez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9326/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

3) En una urna hay 6 bolitas blancas y 3 azules ¿Cuál es la probabilidad de sacar al azar ambas blancas sin reposición?

A) $\\Large \\frac{1}{2}$

B) $\\Large \\frac{5}{8}$

C) $\\Large \\frac{2}{3}$

D) $\\Large \\frac{5}{12}$

Para determinar la probabilidad de sacar dos bolitas blancas sin reposición, debemos seguir los siguientes pasos.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

[[0]]

$P(1°$ bolita blanca$)=$ ______

[[1]]

Paso 1: Calcular la probabilidad de que la primera bolita sea blanca.

[[2]]

$P(2°$ bolita blanca$)=$ ______

[[3]]

Paso 2: Calcular la probabilidad de que la primera bolita sea blanca. Recuerda que se debe eliminar una bolita blanca (ya que no se repone la bolita al ser sacada) y del total de bolitas también se elimina una.

[[0]] [[2]]

$P(2 $ blancas sin reponer$)= $ ______ $\\cdot$ ______ $=$

[[1]] [[3]]

[[4]]

$=$ ______

[[5]]

Paso 3: Como ambos sucesos ocurren uno después del otro, debemos multiplicar las probabilidades del paso 1 y 2.

[[6]]

$P(2$ blancas sin reponer$)=$ ______

[[7]]

Paso 4: Se simplifica el valor obtenido en el paso 3.

Por tanto, la respuesta correcta es: [[8]]

4) ¿Cuántas bacterias habrá al cabo de $5$ horas si se considera que un cultivo de $7$ bacterias se duplica por cada hora?

A) $32$

B) $35$

C) $70$

D) $224$

Para resolver este ejercicio de crecimiento exponencial, debemos realizar los siguientes pasos:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

\n $a=$[[0]] \n $b=$[[1]] \n $x=$[[2]] \n	Paso 1: Se debe determinar los valores $a$, $b$ y $c$ de la función exponencial $f(x)=a \\cdot b^x$, donde $a$ es la población inicial, $b$ es el tipo de crecimiento (doble, triple, etc) y $x$ es el tiempo transcurrido.
\n $f(x)=a \\cdot b^x$ \n [[2]] \n $f($[[2]]$)=$[[0]]$\\cdot$ [[1]] \n	Paso 2: Se debe sustuir los valores encontrados en el paso 1 en la función $f(x)=a \\cdot b^x$.
$f($[[2]]$)=$[[0]]$\\cdot$ [[3]]	\n Paso 3: Se debe calcular el valor de: \n [[2]] \n [[1]] \n \n
$f($[[2]]$)=$[[4]]	\n Paso 4: Se calcula el valor de: \n [[0]]$\\cdot$ [[3]] \n

Por tanto la respuesta correcta es: [[5]]

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5) El resultado de $\\log_2{64}+\\log_5{125}$, es:

A) $6$

B) $7$

C) $8$

D) $9$

Para determinar el valor de $\\log_2{64}+\\log_5{125}$, debemos resolver los siguientes pasos, recordando primero la definición de logaritmo:

$\\Large \\log_b{a}=c \\Leftrightarrow b^a=c$

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

Paso 1: Determinar el valor de $\\log_2{64}=x$, por definición esto es igual a:

$x$

$\\Large \\log_2{64}=x \\Leftrightarrow$ [[0]] $=$ [[1]]

Por lo tanto, debemos determinar [[0]] elevado a cuanto es igual a [[1]], obteniendo que $x=$[[2]]

Paso 2: Determinar el valor de $\\log_5{125}=y$, por definición esto es igual a:

$y$

$\\Large \\log_5{125}=x \\Leftrightarrow$ [[3]] $=$ [[4]]

Por lo tanto, debemos determinar [[3]] elevado a cuanto es igual a [[4]], obteniendo que $y=$[[5]]

Paso 3: Ahora sumamos los valores obtenidos en el paso 2 y 3, concluyendo que:

$\\log_2{64}+\\log_5{125}=$[[2]]$+$[[5]]$=$[[6]]

Por tanto, la respuesta correcta es: [[7]]