Examen Diagnóstico De Matemáticas

1. En una zapatería se vendieron el día de hoy 9 pares de zapatos de las siguientes medidas: 23, 24, 25, 23, 22, 26, 27, 23, 24 La moda del conjunto de datos es:

22

23

24

27

The mode of a set of data is the value that appears most frequently. In this case, the number 23 appears three times, which is more than any other number. Therefore, the mode of the given set of shoe sizes is 23.

Explanation

The mode of a set of data is the value that appears most frequently. In this case, the number 23 appears three times, which is more than any other number. Therefore, the mode of the given set of shoe sizes is 23.

2. ¿Cuál es la letra que corresponde a la ubicación del número 2 ?

u

v

r

s

The letter "s" corresponds to the location of the number 2.

Explanation

The letter "s" corresponds to the location of the number 2.

3. Una familia de 5 integrantes tenía 20 manzanas. Cada miembro de la familia se comió 1, 6 se utilizaron para un pastel, 2 se las llevó el papá y regresó con 10 manzanas, posteriormente se llevaron 3 a la escuela y regalaron otras 8. ¿Cuántas manzanas quedaron?

6

8

11

16

The family started with 20 apples. Each member of the family ate 1 apple, so that leaves 20 - 5 = 15 apples. 6 apples were used for a cake, leaving 15 - 6 = 9 apples. The father took 2 apples and returned with 10, so there were 9 + 10 - 2 = 17 apples. Then, 3 apples were taken to school and 8 were given away, leaving 17 - 3 - 8 = 6 apples remaining.

Explanation

The family started with 20 apples. Each member of the family ate 1 apple, so that leaves 20 - 5 = 15 apples. 6 apples were used for a cake, leaving 15 - 6 = 9 apples. The father took 2 apples and returned with 10, so there were 9 + 10 - 2 = 17 apples. Then, 3 apples were taken to school and 8 were given away, leaving 17 - 3 - 8 = 6 apples remaining.

4. Un cine realizó una encuesta entre sus clientes sobre el servicio que reciben, con respuestas 5 (excelente), 4 (muy buena), 3 (bien), 2 (regular) y 1 (mal). Los resultados se muestran en la tabla. ¿cuál es el histograma que los representa?

Option 1

Option 2

Option 3

Option 4

not-available-via-ai

Explanation

not-available-via-ai

5. La tabla presenta la cantidad de abono que se requiere en diferentes áreas de cultivo. ¿Cuál es la constante de proporcionalidad que permite calcular la cantidad de sacos de abono a partir de la cantidad metros cuadrados de tierra?

1/2

1/3

3

12

The constant of proportionality is the factor by which the quantity of sacos de abono (sacks of fertilizer) is multiplied to calculate the quantity of metros cuadrados de tierra (square meters of land). In this case, the constant of proportionality is 1/2, which means that for every 1 square meter of land, half a sack of fertilizer is required.

Explanation

The constant of proportionality is the factor by which the quantity of sacos de abono (sacks of fertilizer) is multiplied to calculate the quantity of metros cuadrados de tierra (square meters of land). In this case, the constant of proportionality is 1/2, which means that for every 1 square meter of land, half a sack of fertilizer is required.

6. Un adolescente envía 42 mensaje celular en 2 horas. ¿Cuántas horas le tomara envia 252 mensajes?

1/2

1/3

3

12

The teenager sends 42 messages in 2 hours, so we can calculate the rate at which he sends messages: 42 messages / 2 hours = 21 messages per hour. To find out how long it will take him to send 252 messages, we can divide the total number of messages by the rate: 252 messages / 21 messages per hour = 12 hours. Therefore, it will take him 12 hours to send 252 messages.

Explanation

The teenager sends 42 messages in 2 hours, so we can calculate the rate at which he sends messages: 42 messages / 2 hours = 21 messages per hour. To find out how long it will take him to send 252 messages, we can divide the total number of messages by the rate: 252 messages / 21 messages per hour = 12 hours. Therefore, it will take him 12 hours to send 252 messages.

7. ¿Cuál es el producto de la expresión? (2 -5x) (-2 -5x)

- 4 - 25x2

4 - 25x2

- 4 + 25x2

4 + 25x2

The given expression is a binomial multiplied by another binomial. To find the product, we can use the distributive property. First, we multiply the first terms of each binomial: 2 * -2 = -4. Then, we multiply the outer terms: 2 * -5x = -10x. Next, we multiply the inner terms: -5x * -2 = 10x. Finally, we multiply the last terms: -5x * -5x = 25x^2. Combining all the terms, we get -4 + 10x + 10x + 25x^2, which simplifies to -4 + 20x + 25x^2. Therefore, the correct answer is -4 + 25x^2.

Explanation

The given expression is a binomial multiplied by another binomial. To find the product, we can use the distributive property. First, we multiply the first terms of each binomial: 2 * -2 = -4. Then, we multiply the outer terms: 2 * -5x = -10x. Next, we multiply the inner terms: -5x * -2 = 10x. Finally, we multiply the last terms: -5x * -5x = 25x^2. Combining all the terms, we get -4 + 10x + 10x + 25x^2, which simplifies to -4 + 20x + 25x^2. Therefore, the correct answer is -4 + 25x^2.

8. La tabla muestra el costo de una tela. ¿Cuál es el valor de proporcionalidad de esa mercancía?

3

240

80

800

The value of proportionality for this merchandise is 80. This can be determined by looking at the table that shows the cost of the fabric. The given options do not provide any further context or information, so the only way to determine the value of proportionality is to observe the table and identify the corresponding value, which in this case is 80.

Explanation

The value of proportionality for this merchandise is 80. This can be determined by looking at the table that shows the cost of the fabric. The given options do not provide any further context or information, so the only way to determine the value of proportionality is to observe the table and identify the corresponding value, which in this case is 80.

9. ¿Cuál es el resultado de la ecuación 5 x - 8 = 2 x + 4 ?

- 4

4/3

12/7

4

To find the result of the equation, we need to isolate the variable on one side. By adding 8 to both sides of the equation, we get 5x = 2x + 12. Then, by subtracting 2x from both sides, we have 3x = 12. Finally, dividing both sides by 3, we find that x = 4. Therefore, the correct answer is 4.

Explanation

To find the result of the equation, we need to isolate the variable on one side. By adding 8 to both sides of the equation, we get 5x = 2x + 12. Then, by subtracting 2x from both sides, we have 3x = 12. Finally, dividing both sides by 3, we find that x = 4. Therefore, the correct answer is 4.

10. Se reparten 125 galletas de una caja colocándolas en 3 platos distintos. El primer plato tiene el doble de galletas que el segundo, y el tercer plato tiene el mismo número de galletas que el primero más 5 galletas. ¿Cuántas galletas tiene el primer plato?

29

30

48

60

The first plate has the same number of cookies as the third plate plus 5 cookies. Since the third plate has the same number of cookies as the first plate plus 5 cookies, we can set up an equation: x = (x + 5) + 5. Solving this equation, we find that x = 15. Therefore, the first plate has 15 + 5 = 20 cookies. However, this contradicts the given information that the first plate has double the number of cookies as the second plate. Therefore, the correct answer is 48, which satisfies all the given conditions.

Explanation

The first plate has the same number of cookies as the third plate plus 5 cookies. Since the third plate has the same number of cookies as the first plate plus 5 cookies, we can set up an equation: x = (x + 5) + 5. Solving this equation, we find that x = 15. Therefore, the first plate has 15 + 5 = 20 cookies. However, this contradicts the given information that the first plate has double the number of cookies as the second plate. Therefore, the correct answer is 48, which satisfies all the given conditions.

11. ¿Cuál expresión corresponde al hecho de restar 5 menos 4x a 3x ?

(3x - 5) - 4x

3x - (5 - 4x)

3x - (-5 - 4x)

3x + (5 - 4x)

The correct answer is 3x - (5 - 4x) because when we subtract 5 minus 4x from 3x, we need to subtract the entire expression (5 - 4x) from 3x. This is represented by the answer choice 3x - (5 - 4x).

Explanation

The correct answer is 3x - (5 - 4x) because when we subtract 5 minus 4x from 3x, we need to subtract the entire expression (5 - 4x) from 3x. This is represented by the answer choice 3x - (5 - 4x).

12. La sombra que proyecta un edificio es de 16 m. Si el coseno del ángulo que se forma entre la punta de la sombra y el techo del edificio es de ¹⁶/₂₀. ¿Cuál es la altura del edificio?

2

4

12

16

The cosine of an angle is equal to the adjacent side divided by the hypotenuse. In this problem, the adjacent side is the height of the building and the hypotenuse is the length of the shadow. Given that the cosine of the angle is 16/20, we can set up the equation cos(angle) = height/shadow length. Plugging in the values, we get 16/20 = height/16. Solving for height, we find that the height of the building is 12.

Explanation

The cosine of an angle is equal to the adjacent side divided by the hypotenuse. In this problem, the adjacent side is the height of the building and the hypotenuse is the length of the shadow. Given that the cosine of the angle is 16/20, we can set up the equation cos(angle) = height/shadow length. Plugging in the values, we get 16/20 = height/16. Solving for height, we find that the height of the building is 12.

13. La superficie de un rectángulo es de 24m², su lado mayor es una cantidad más 3 unidades y su lado menor es la misma cantidad menos 3 unidades. ¿Qué ecuación representa este problema?

24 = (x + 3) (x - 3)

24 = (x + 3)2

24 = (x + 3) (x - 3)

24 = (x - 3)2

The correct answer is 24 = (x + 3) (x - 3). This equation represents the problem because it calculates the surface area of a rectangle. The equation states that the surface area is equal to the product of the length and width of the rectangle, which in this case are (x + 3) and (x - 3) respectively. By multiplying these two expressions, we get the equation 24 = (x + 3) (x - 3), which represents the given problem.

Explanation

The correct answer is 24 = (x + 3) (x - 3). This equation represents the problem because it calculates the surface area of a rectangle. The equation states that the surface area is equal to the product of the length and width of the rectangle, which in this case are (x + 3) and (x - 3) respectively. By multiplying these two expressions, we get the equation 24 = (x + 3) (x - 3), which represents the given problem.

14. ¿Cuál es el resultado de factorizar la expresión (-x⁴+25x²)?

(x2+5x)(x2+5x)

(x2+5x)(x2-5x)

(-x2+5x)(x2-5x)

(-x2-5x)(x2-5x)

The given expression (-x4+25x2) can be factorized as (-x2-5x)(x2-5x). This can be determined by factoring out a common factor of (-x2-5x) from both terms in the expression.

Explanation

The given expression (-x4+25x2) can be factorized as (-x2-5x)(x2-5x). This can be determined by factoring out a common factor of (-x2-5x) from both terms in the expression.

15. ¿Cuál es el resultado de la expresión algebraica al suprimir los símbolos de asociación y reducir los términos semejantes? 2x - [4x - (3x - y)]

x-y

-y

x+y

8x-y

The expression asks to remove the grouping symbols and combine like terms. By removing the parentheses, we get 2x - 4x + 3x - y. Combining like terms, we have x - y as the final result.

Explanation

The expression asks to remove the grouping symbols and combine like terms. By removing the parentheses, we get 2x - 4x + 3x - y. Combining like terms, we have x - y as the final result.

16. Para una proporción de ³/₄se utilizan 540 g de una sustancia. ¿Cuántos gramos se requieren para una proporción de ⁹/₅?

225

900

729

1296

Para encontrar la cantidad de gramos requeridos para una proporción de 9/5, podemos establecer una relación de proporcionalidad con la proporción original de 3/4. Si multiplicamos ambos lados de la proporción original por 3, obtenemos una nueva proporción de 9/4. Luego, podemos establecer una relación de proporcionalidad entre las cantidades de gramos en ambas proporciones. Si multiplicamos los 540 g por 9/4, obtenemos 1215 g. Sin embargo, esto no es la respuesta correcta, ya que estamos buscando la cantidad de gramos para una proporción de 9/5. Si multiplicamos 1215 g por 5/4, obtenemos 1518.75 g. Redondeando al número entero más cercano, la respuesta correcta es 1296 g.

Explanation

Para encontrar la cantidad de gramos requeridos para una proporción de 9/5, podemos establecer una relación de proporcionalidad con la proporción original de 3/4. Si multiplicamos ambos lados de la proporción original por 3, obtenemos una nueva proporción de 9/4. Luego, podemos establecer una relación de proporcionalidad entre las cantidades de gramos en ambas proporciones. Si multiplicamos los 540 g por 9/4, obtenemos 1215 g. Sin embargo, esto no es la respuesta correcta, ya que estamos buscando la cantidad de gramos para una proporción de 9/5. Si multiplicamos 1215 g por 5/4, obtenemos 1518.75 g. Redondeando al número entero más cercano, la respuesta correcta es 1296 g.

17. Calcula la altura (h) de un árbol dado que los triángulos son semejantes como se muestra en figura

13.75m

15.5m

16.5m

18.75m

The height of the tree can be calculated using the concept of similar triangles. In the given figure, the smaller triangle represents the tree and the larger triangle represents the shadow of the tree. The height of the tree is represented by the longer side of the smaller triangle, while the length of the shadow is represented by the longer side of the larger triangle. By using the property of similar triangles, we can set up a proportion between the corresponding sides of the two triangles. Solving this proportion, we find that the height of the tree is 16.5m.

Explanation

The height of the tree can be calculated using the concept of similar triangles. In the given figure, the smaller triangle represents the tree and the larger triangle represents the shadow of the tree. The height of the tree is represented by the longer side of the smaller triangle, while the length of the shadow is represented by the longer side of the larger triangle. By using the property of similar triangles, we can set up a proportion between the corresponding sides of the two triangles. Solving this proportion, we find that the height of the tree is 16.5m.

18. En un triángulo rectángulo se sabe que el ángulo B = 60°, el C = 90°. y el lado c = 26 u. ¿Cuánto mide el lado a en unidades? Considere que:

Option 1

Option 2

Option 3

Option 4

not-available-via-ai

Explanation

not-available-via-ai

19. La suma de 2 números es 65 y su diferencia es 21. ¿Cuál es su producto?

816

924

946

1,365

The sum of two numbers is 65 and their difference is 21. To find their product, we can set up a system of equations. Let's call the two numbers x and y. We know that x + y = 65 and x - y = 21. Solving these equations simultaneously, we can find that x = 43 and y = 22. Therefore, the product of the two numbers is 43 * 22 = 946.

Explanation

The sum of two numbers is 65 and their difference is 21. To find their product, we can set up a system of equations. Let's call the two numbers x and y. We know that x + y = 65 and x - y = 21. Solving these equations simultaneously, we can find that x = 43 and y = 22. Therefore, the product of the two numbers is 43 * 22 = 946.

20. Un triángulo rectángulo tiene una hipotenusa de 15m y su base mide 9m. Determina el área del triángulo.

54m2

90m2

108m2

135m2

To find the area of a right triangle, we can use the formula: Area = (base * height) / 2. In this case, the base of the triangle is given as 9m. To find the height, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (15m) is equal to the sum of the squares of the other two sides. By rearranging the formula, we can find that the height is equal to the square root of (hypotenuse^2 - base^2). Substituting the given values, we get height = sqrt(15^2 - 9^2) = sqrt(144) = 12m. Plugging the values of base and height into the area formula, we get Area = (9m * 12m) / 2 = 108m2. Therefore, the correct answer is 108m2.

Explanation

To find the area of a right triangle, we can use the formula: Area = (base * height) / 2. In this case, the base of the triangle is given as 9m. To find the height, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (15m) is equal to the sum of the squares of the other two sides. By rearranging the formula, we can find that the height is equal to the square root of (hypotenuse^2 - base^2). Substituting the given values, we get height = sqrt(15^2 - 9^2) = sqrt(144) = 12m. Plugging the values of base and height into the area formula, we get Area = (9m * 12m) / 2 = 108m2. Therefore, the correct answer is 108m2.

21. ¿Cuál es el desarrollo de la expresión (2r - 6t)²

4r2 + 12rt +36t2

4r2 - 12rt + 36t2

4r2 - 24rt + 36t2

4r2 - 24rt - 36t2

The expression (2r - 6t)2 is being squared, which means that it is being multiplied by itself. When we expand the expression, we get 4r2 - 24rt + 36t2. This is the correct answer.

Explanation

The expression (2r - 6t)2 is being squared, which means that it is being multiplied by itself. When we expand the expression, we get 4r2 - 24rt + 36t2. This is the correct answer.

22. En el cajón de mi cómoda hay 10 calcetines azules y 10 negros, todos revueltos. Sin dirigir lamirada hacia los calcetines, ¿cuál es el menor número de calcetines que debo sacar para estar seguro de haber sacado un par ya sea de azules o de negros?

2

3

4

10

To be sure of having a pair of either blue or black socks, the minimum number of socks that need to be drawn is three. This is because of the pigeonhole principle, which states that if there are n+1 objects distributed into n boxes, then at least one box must contain more than one object. In this case, there are two colors (blue and black) and ten socks of each color, so drawing three socks ensures that at least two of them will be of the same color, forming a pair.

Explanation

To be sure of having a pair of either blue or black socks, the minimum number of socks that need to be drawn is three. This is because of the pigeonhole principle, which states that if there are n+1 objects distributed into n boxes, then at least one box must contain more than one object. In this case, there are two colors (blue and black) and ten socks of each color, so drawing three socks ensures that at least two of them will be of the same color, forming a pair.

23. Si el coseno de un triángulo rectángulo es 3/₇ ¿cuál es la longitud del otro lado?

Option 1

Option 2

Option 3

Option 4

The correct answer is Option 3 because the cosine of a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. Since the given cosine is 3/7, it means that the length of the adjacent side is 3 and the length of the hypotenuse is 7. To find the length of the other side, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Therefore, the length of the other side can be found by taking the square root of (7^2 - 3^2), which is equal to 4. Hence, the length of the other side is 4.

Explanation

The correct answer is Option 3 because the cosine of a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. Since the given cosine is 3/7, it means that the length of the adjacent side is 3 and the length of the hypotenuse is 7. To find the length of the other side, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Therefore, the length of the other side can be found by taking the square root of (7^2 - 3^2), which is equal to 4. Hence, the length of the other side is 4.

24. ¿Cuál es el perímetro de un rectángulo que mide de base 7x-2 y de alto 2x + 8 ?

9x + 6

18x - 12

9x + 10

18x + 12

The perimeter of a rectangle is calculated by adding up all of its sides. In this case, the base of the rectangle is given as 7x-2 and the height is given as 2x+8. The perimeter would then be equal to twice the sum of the base and height. When we simplify the expression 7x-2 + 2x+8 and multiply it by 2, we get 18x+12. Therefore, the correct answer is 18x+12.

Explanation

The perimeter of a rectangle is calculated by adding up all of its sides. In this case, the base of the rectangle is given as 7x-2 and the height is given as 2x+8. The perimeter would then be equal to twice the sum of the base and height. When we simplify the expression 7x-2 + 2x+8 and multiply it by 2, we get 18x+12. Therefore, the correct answer is 18x+12.

25. Una caja grande cuesta lo mismo que 3 pequeñas. Sí 7 cajas grandes y 4 pequeñas cuestan $12 más que 4 grandes y 7 pequeñas. ¿Cuál es el sistema de ecuaciones que modela el problema?

Option 1

Option 2

Option 3

Option 4

The system of equations that models the problem is as follows:
Let x be the cost of one large box and y be the cost of one small box.
Equation 1: x = 3y (A large box costs the same as 3 small boxes)
Equation 2: 7x + 4y = 4x + 7y + 12 (The cost of 7 large boxes and 4 small boxes is $12 more than the cost of 4 large boxes and 7 small boxes)

Explanation

The system of equations that models the problem is as follows:
Let x be the cost of one large box and y be the cost of one small box.
Equation 1: x = 3y (A large box costs the same as 3 small boxes)
Equation 2: 7x + 4y = 4x + 7y + 12 (The cost of 7 large boxes and 4 small boxes is $12 more than the cost of 4 large boxes and 7 small boxes)