1.
If f(x)=3x2−2, what is f(3)?
Correct Answer
C. 25
Explanation
Here's a summary of the steps involved:
Substitute x = 3:
In the function f(x) = 3x² - 2, replace x with 3, giving you f(3) = 3(3)² - 2.
Evaluate the expression:
Calculate 3(3)², which equals 3 × 9 = 27.
Subtract 2 from 27, resulting in f(3) = 25.
Therefore, you've correctly determined that f(3) = 25 for the given function.
2.
What is the sum of the first 20 terms of an arithmetic sequence if the first term is 3 and the common difference is 2?
Correct Answer
C. 420
Explanation
To find the sum of the first 20 terms of an arithmetic sequence, we can use the formula:Sn = n/2 * (2a + (n - 1)d)where:Sn is the sum of the first n termsa is the first termn is the number of termsd is the common differenceIn this case, we have:a = 3 (the first term)n = 20 (the number of terms)d = 2 (the common difference)Plugging these values into the formula, we get:Sn = 20/2 * (2 * 3 + (20 - 1) * 2)Sn = 10 * (6 + 38)Sn = 10 * 44Sn = 440Therefore, the sum of the first 20 terms of the arithmetic sequence is 440.
3.
Solve the equation 4x−7=9:
Correct Answer
D. 4
Explanation
To solve the equation 4x - 7 = 9, we can follow these steps:Add 7 to both sides of the equation:4x - 7 + 7 = 9 + 7
4x = 16Divide both sides by 4:4x / 4 = 16 / 4
x = 4Therefore, the solution to the equation 4x - 7 = 9 is x = 4.So, the correct answer is 4.
4.
What are the roots of the quadratic equation x2−4x+4=0?
Correct Answer
A. 2
Explanation
To find the roots of the quadratic equation xÂ² - 4x + 4 = 0, we can use the quadratic formula:x = (-b Â± âˆš(bÂ² - 4ac)) / 2awhere:a = 1 (coefficient of xÂ²)b = -4 (coefficient of x)c = 4 (constant term)Plugging these values into the formula, we get:x = (-(-4) Â± âˆš((-4)Â² - 4 * 1 * 4)) / 2 * 1x = (4 Â± âˆš(16 - 16)) / 2Since the square root of 0 is 0, this simplifies to:x = (4 Â± 0) / 2So, we have two solutions:x1 = (4 + 0) / 2 = 4
x2 = (4 - 0) / 2 = 4Therefore, both roots of the quadratic equation are equal to 4.
5.
If 5 is added to three times a number, the result is 26. What is the number?
Correct Answer
C. 7
Explanation
To find the number, we can use the following steps:Translate the problem into an equation:Let x represent the unknown number.The phrase "three times a number" can be written as 3x.The phrase "5 is added to three times a number" can be written as 3x + 5.The equation becomes: 3x + 5 = 26.Solve the equation:Subtract 5 from both sides: 3x = 21.Divide both sides by 3: x = 7.Therefore, the number is 7.So, the correct answer is 7.
6.
Convert the fraction 7/87/8 to a decimal.
Correct Answer
D. 0.875
Explanation
To convert the fraction 7/8 to a decimal, we can perform long division:Divide 7 by 8: 0.875
8) 7.000
64
---
60
56
---
40
40
---
0
The division stops here because there's no remainder.Therefore, 7/8 as a decimal is 0.875.So, the correct answer is 0.875.
7.
What is the sum of the interior angles of a pentagon?
Correct Answer
A. 540°
Explanation
The sum of the interior angles of a pentagon is indeed 540Â°.Here's why:The sum of the interior angles of any polygon can be calculated with the formula: (n - 2) * 180Â°Where n is the number of sides in the polygon.For a pentagon (n = 5), applying the formula gives us: (5 - 2) * 180Â° = 3 * 180Â° = 540Â°Therefore, the sum of the interior angles of a pentagon is 540Â°. So, the correct answer is 540Â°.
8.
A car traveled 300 miles in 5 hours. What was its average speed?
Correct Answer
B. 60 mpH
Explanation
To find the average speed of the car, we can use the formula:Average speed = total distance / total timeIn this case, we have:Total distance = 300 milesTotal time = 5 hoursPlugging these values into the formula, we get:Average speed = 300 miles / 5 hoursAverage speed = 60 miles per hourTherefore, the average speed of the car was 60 mph. So, the correct answer is 60 mph.
9.
If a shirt is sold at a 15% loss and the selling price is $85, what was the cost price?
Correct Answer
D. $100
Explanation
Let's solve this step-by-step to find the cost price of the shirt:Define variables:Let C be the cost price of the shirt.We know the selling price (S) is $85 and the loss percentage is 15% (L).Relate variables:The selling price is related to the cost price and loss by the formula: S = C - C * L/100.Plugging in the known values, we get: $85 = C - C * 15/100.Solve for C:To isolate C, we need to manipulate the equation. Add C * 15/100 to both sides: C = $85 + C * 15/100.Divide both sides by (1 - 15/100): C = $85 / (1 - 15/100) = $85 / 0.85.Calculating the division, we get: C = $100.Therefore, the cost price of the shirt was $100. So, the correct answer is $100.
10.
A sum of money doubles in 5 hours at simple interest. What is the rate of interest?
Correct Answer
C. 20%
Explanation
Simple interest is given by the formula I=P⋅r⋅t, where P is the principal, r is the rate of interest, and t is the time. Since the sum doubles, I=P and r⋅t=. So, r=1/5 or 20%.
11.
If the mean of six numbers is 14, and five of them are 12, 15, 17, 10, and 11, what is the sixth number?
Correct Answer
A. 18
Explanation
To find the sixth number, we can follow these steps:Calculate the sum of the five known numbers:Add the five known numbers: 12 + 15 + 17 + 10 + 11 = 65.Recall the relationship between the mean and the sum:The mean is the sum of all values divided by the number of values. In this case, the mean is 14 and we have six numbers.Use the formula to find the sum of all six numbers:Mean * Number of numbers = Sum of all values.Therefore, 14 * 6 = 84.Find the sixth number by subtracting the sum of the five known numbers from the sum of all six numbers:Sixth number = Sum of all six numbers - Sum of five known numbers.Sixth number = 84 - 65 = 19.Therefore, the sixth number is 19.So, the correct answer is 19.
12.
A tank is filled in 5 hours by three pipes. If the first pipe fills the tank in 10 hours, the second one in 12 hours, and the third in 20 hours, what part of the tank is filled by the second pipe in one hour?
Correct Answer
D. 1/12
Explanation
To find the part of the tank filled by the second pipe in one hour, we can follow these steps:Calculate the rate of each pipe:Rate of first pipe: 1 tank / 10 hours = 1/10 tank/hourRate of second pipe: 1 tank / 12 hours = 1/12 tank/hourRate of third pipe: 1 tank / 20 hours = 1/20 tank/hourCombine the rates of all three pipes:Combined rate: 1/10 + 1/12 + 1/20 = 9/60 = 3/20 tank/hourIsolate the rate of the second pipe:Second pipe rate: 1/12 tank/hourTherefore, the second pipe fills 1/12 of the tank in one hour.So, the correct answer is 1/12.
13.
Find the value of x that satisfies the equation 3x−4=2x+5:
Correct Answer
C. 9
Explanation
To find the value of x in the equation 3x - 4 = 2x + 5, we need to solve for x by isolating it:Move all terms containing x to one side:
3x - 2x = 5 + 4Combine like terms:
x = 9Therefore, the value of x that satisfies the equation is 9.So, the correct answer is 9.
14.
If 30% of a number is 60, what is 20% of that number?
Correct Answer
D. 40
Explanation
Let's solve this step-by-step:Denote the unknown number as "n".Translate the first statement into an equation:30% of n is 60, written as: 0.3n = 60.Solve for n:Dividing both sides by 0.3, we get: n = 60 / 0.3 = 200.Find 20% of n:20% of 200 is: 0.2 * 200 = 40.Therefore, 20% of the original number is 40.So, the correct answer is 40.
15.
In a class of 40 students, 12 are girls. What percentage of the class are boys?
Correct Answer
B. 70%
Explanation
To find the percentage of boys in the class, we can follow these steps:Calculate the number of boys:Since there are 40 students in total and 12 are girls, 40 - 12 = 28 are boys.Calculate the proportion of boys to the total class:Divide the number of boys by the total number of students and express it as a decimal: 28 boys / 40 students = 0.7.Convert the decimal proportion to a percentage:Multiply the decimal by 100: 0.7 * 100 = 70%.Therefore, 70% of the class are boys.So, the correct answer is 70%.