Which of these linear equations does not represent a line with a positive slope?
3y + x = 9
✓
y = 4x - 1
2x - 3y = 6
y - 5 = x
Correct Answer
3y + x = 9
The shared property is having a positive slope. The equations 2x - 3y = 6 (slope = 2/3), y = 4x - 1 (slope = 4), and y - 5 = x (slope = 1) all have positive slopes. The equation 3y + x = 9 can be rewritten as y = (-1/3)x + 3, which has a negative slope.
Question 2
Which of these statistical values does not indicate the spread or variability of a dataset?
Mean
✓
Standard deviation
Interquartile range
Range
Correct Answer
Mean
The shared property is being a measure of the spread or variability of a dataset. Range, standard deviation, and interquartile range are all measures of spread. The mean is a measure of central tendency, not spread.
Question 3
Which of these sets of side lengths cannot form a right-angled triangle?
6, 8, 10
5, 12, 13
3, 4, 5
4, 5, 6
✓
Correct Answer
4, 5, 6
The shared property is that the side lengths can form a right-angled triangle, meaning they satisfy the Pythagorean theorem (a^2 + b^2 = c^2). For (3, 4, 5), 3^2 + 4^2 = 9 + 16 = 25 = 5^2. For (5, 12, 13), 5^2 + 12^2 = 25 + 144 = 169 = 13^2. For (6, 8, 10), 6^2 + 8^2 = 36 + 64 = 100 = 10^2. For (4, 5, 6), 4^2 + 5^2 = 16 + 25 = 41, which is not equal to 6^2 = 36.
Question 4
Which of these systems of linear equations does not have infinitely many solutions?
x + y = 3 and 2x + 2y = 6
y - x = 0 and 2y = 2x
3x - y = 5 and 6x - 2y = 8
✓
y = 2x + 1 and 2y = 4x + 2
Correct Answer
3x - y = 5 and 6x - 2y = 8
The shared property is that the system of linear equations has infinitely many solutions, meaning the two equations represent the same line. The systems (y = 2x + 1 and 2y = 4x + 2), (x + y = 3 and 2x + 2y = 6), and (y - x = 0 and 2y = 2x) all consist of two equivalent equations, thus representing the same line and having infinitely many solutions. The system (3x - y = 5 and 6x - 2y = 8) can be rewritten as (y = 3x - 5 and y = 3x - 4), which represents two parallel lines with different y-intercepts, meaning it has no solution.
Question 5
Which of these scenarios does not represent a direct proportion?
The number of pages read in a book over time, assuming a constant reading speed.
The time it takes to complete a task as the number of workers increases, assuming equal work rate.
✓
The distance traveled by a car moving at a constant speed over time.
The total cost of apples at $2 per pound.
Correct Answer
The time it takes to complete a task as the number of workers increases, assuming equal work rate.
The shared property is representing a direct proportion (where one quantity increases or decreases at a constant rate with another). The total cost of apples, pages read, and distance traveled all increase directly with the amount of apples, time spent reading, and time traveled, respectively (Cost = 2 * pounds, Pages = speed * time, Distance = speed * time). The time it takes to complete a task as the number of workers increases is an inverse proportion; as workers increase, time decreases (Time = TotalWork / Workers).
Question 6
Which of these functions does not have a domain that includes all real numbers?
f(x) = (x+1)/(x^2 + 1)
f(x) = 3x^2 - 5x + 1
f(x) = |x + 4|
f(x) = sqrt(x-2)
✓
Correct Answer
f(x) = sqrt(x-2)
The shared property is having a domain that includes all real numbers. Polynomial functions like f(x) = 3x^2 - 5x + 1, absolute value functions like f(x) = |x + 4|, and rational functions where the denominator is never zero (like f(x) = (x+1)/(x^2 + 1) because x^2+1 is always positive) all have domains of all real numbers. The function f(x) = sqrt(x-2) requires the expression under the square root to be non-negative, so x-2 >= 0, which means x >= 2. Its domain is restricted.
Question 7
Which of these equations does not represent a circle centered at the origin?
(x-1)^2 + y^2 = 9
✓
x^2 + y^2 = 25
x^2 + y^2 - 16 = 0
2x^2 + 2y^2 = 8
Correct Answer
(x-1)^2 + y^2 = 9
The shared property is representing a circle centered at the origin (0,0). The general form of a circle centered at the origin is x^2 + y^2 = r^2. The equations x^2 + y^2 = 25, x^2 + y^2 - 16 = 0 (which simplifies to x^2 + y^2 = 16), and 2x^2 + 2y^2 = 8 (which simplifies to x^2 + y^2 = 4) all represent circles centered at the origin. The equation (x-1)^2 + y^2 = 9 represents a circle centered at (1,0).
Question 8
Which of these inequalities does not have a solution set that includes negative numbers?
x + 5 < 2
|x| < 1
x^2 + 1 < 0
✓
-2x > 6
Correct Answer
x^2 + 1 < 0
The shared property is having a solution set that includes negative numbers. For x + 5 < 2, the solution is x < -3, which includes negative numbers. For -2x > 6, dividing by -2 and reversing the inequality sign gives x < -3, which includes negative numbers. For |x| < 1, the solution is -1 < x < 1, which includes negative numbers between -1 and 0. For x^2 + 1 < 0, there are no real solutions, because x^2 is always non-negative, so x^2 + 1 is always positive and can never be less than 0.
Question 9
Which of these events does not have a probability of 1?
Drawing a red card from a standard deck of 52 cards.
✓
Flipping a coin and getting either heads or tails.
Picking a day of the week that is not a weekend day or a weekend day.
Rolling a number less than 7 on a standard six-sided die.
Correct Answer
Drawing a red card from a standard deck of 52 cards.
The shared property is having a probability of 1, meaning the event is certain to occur. Rolling a number less than 7 on a standard six-sided die (1, 2, 3, 4, 5, 6 are all less than 7), flipping a coin and getting either heads or tails, and picking a day that is either a weekday or a weekend day are all certain events with probability 1. Drawing a red card from a standard deck of 52 cards has a probability of 26/52 = 0.5, as there are 26 red cards and 26 black cards.
Question 10
Which of these polynomial expressions does not have (x - 2) as a factor?
x^2 - 5x + 6
x^2 + x + 6
✓
x^3 - 8
x^2 - 4
Correct Answer
x^2 + x + 6
The shared property is having (x - 2) as a factor, which means that x=2 is a root of the polynomial (i.e., substituting x=2 into the expression results in 0). For x^2 - 4, 2^2 - 4 = 0. For x^2 - 5x + 6, 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0. For x^3 - 8, 2^3 - 8 = 8 - 8 = 0. For x^2 + x + 6, 2^2 + 2 + 6 = 4 + 2 + 6 = 12, which is not 0, so (x - 2) is not a factor.
Question 11
Which of these trigonometric values is not positive in the second quadrant (90 to 180 degrees)?
sin(150°)
sin(120°)
cos(120°)
✓
csc(120°)
Correct Answer
cos(120°)
The shared property is being a positive trigonometric value in the second quadrant. In the second quadrant, only sine and its reciprocal, cosecant, are positive. Cosine and tangent (and their reciprocals, secant and cotangent) are negative. Sin(120°) is positive, csc(120°) (1/sin(120°)) is positive, and sin(150°) is positive. Cos(120°) is negative.
Question 12
In a linear function y = mx + b, which of these does not represent a change in the initial value or y-intercept?
Changing the value of 'b'.
Changing the value of 'm'.
✓
Adding a constant to the entire function, e.g., y = mx + b + k.
Vertically shifting the graph up or down.
Correct Answer
Changing the value of 'm'.
The shared property is representing a change to the y-intercept (the initial value) of the linear function. Changing 'b', vertically shifting the graph, and adding a constant to the function all directly alter the y-intercept. Changing the value of 'm' changes the slope of the line, causing it to rotate around the y-intercept (unless the y-intercept is also changed), but it does not inherently change the y-intercept itself if 'b' remains constant.
Question 13
Which of these calculations does not result in a unit of speed (distance per time)?
Miles per hour.
Feet per minute.
Kilometers per second.
Meters multiplied by seconds.
✓
Correct Answer
Meters multiplied by seconds.
The shared property is representing a unit of speed (distance divided by time). Miles per hour, kilometers per second, and feet per minute are all standard units of speed. Meters multiplied by seconds results in a unit of distance-time, not distance per time (speed).
Question 14
Which of these expressions is not equivalent to x^(2/3)?
(x^2)^(1/3)
sqrt(x^3)
✓
(cube root of x)^2
(x^(1/3))^2
Correct Answer
sqrt(x^3)
The shared property is being equivalent to x^(2/3). Using exponent rules, (x^2)^(1/3) = x^(2*1/3) = x^(2/3). (x^(1/3))^2 = x^((1/3)*2) = x^(2/3). (cube root of x)^2 can be written as (x^(1/3))^2 = x^(2/3). The expression sqrt(x^3) is equivalent to x^(3/2), not x^(2/3).