Which of these expressions does not simplify to a positive integer?
2^-3 * 2^5
(3^2)^-1 * 3^4
(4^0 + 1)^2
(5/2)^-1
✓
Correct Answer
(5/2)^-1
The shared property is simplifying to a positive integer. 2^-3 * 2^5 simplifies to 2^2 = 4. (3^2)^-1 * 3^4 simplifies to 3^-2 * 3^4 = 3^2 = 9. (4^0 + 1)^2 simplifies to (1 + 1)^2 = 2^2 = 4. (5/2)^-1 simplifies to 2/5, which is not a positive integer.
Question 2
Which of these polynomials is not a difference of squares?
x^2 - 9y^2
4a^2 - 25
x^2 + 4
✓
16m^2 - 1
Correct Answer
x^2 + 4
The shared property is being a difference of squares, which means it can be factored into the form (A - B)(A + B). x^2 - 9y^2 is (x - 3y)(x + 3y). 4a^2 - 25 is (2a - 5)(2a + 5). 16m^2 - 1 is (4m - 1)(4m + 1). x^2 + 4 is a sum of squares, not a difference of squares, and cannot be factored into real binomials.
Question 3
Which of these equations is NOT in slope-intercept form?
y = (1/2)x + 5
2x + 3y = 6
✓
y = -x
y = 3x - 7
Correct Answer
2x + 3y = 6
The shared property is being in slope-intercept form, which is y = mx + b. y = (1/2)x + 5, y = 3x - 7, and y = -x (which is y = -1x + 0) are all in slope-intercept form. 2x + 3y = 6 is in standard form.
Question 4
Which of these radical expressions does not simplify to an integer?
sqrt(49)
sqrt(81)
cube_root(64)
sqrt(12)
✓
Correct Answer
sqrt(12)
The shared property is simplifying to an integer. sqrt(81) simplifies to 9. cube_root(64) simplifies to 4. sqrt(49) simplifies to 7. sqrt(12) simplifies to 2*sqrt(3), which is not an integer.
Question 5
Which of these quadratic equations does not have exactly two distinct real solutions?
x^2 - 5x + 6 = 0
x^2 - 7x + 10 = 0
x^2 + 4x + 4 = 0
✓
2x^2 + x - 3 = 0
Correct Answer
x^2 + 4x + 4 = 0
The shared property is having exactly two distinct real solutions, which occurs when the discriminant (b^2 - 4ac) is greater than 0. For x^2 - 5x + 6 = 0, D = (-5)^2 - 4(1)(6) = 1 > 0. For 2x^2 + x - 3 = 0, D = 1^2 - 4(2)(-3) = 25 > 0. For x^2 - 7x + 10 = 0, D = (-7)^2 - 4(1)(10) = 9 > 0. For x^2 + 4x + 4 = 0, D = 4^2 - 4(1)(4) = 0, which means it has exactly one real solution (a repeated root).
Question 6
Which of these functions does not have a domain of all real numbers?
f(x) = 2x - 7
f(x) = |x + 4|
f(x) = sqrt(x - 2)
✓
f(x) = 3x^2 - 5x + 1
Correct Answer
f(x) = sqrt(x - 2)
The shared property is having a domain of all real numbers. Polynomial functions (f(x) = 3x^2 - 5x + 1), absolute value functions (f(x) = |x + 4|), and linear functions (f(x) = 2x - 7) 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, meaning x >= 2. Its domain is not all real numbers.
Question 7
Which of these systems of linear equations does not have a consistent solution?
y = 2x + 1; y = 2x - 3
✓
2x + y = 4; 4x + 2y = 8
x + y = 5; x - y = 1
3x - y = 7; x + 2y = 0
Correct Answer
y = 2x + 1; y = 2x - 3
The shared property is having a consistent solution, meaning the system has at least one solution (either one unique solution or infinitely many solutions). The system x + y = 5; x - y = 1 has one unique solution. The system 2x + y = 4; 4x + 2y = 8 represents the same line (the second equation is twice the first), so it has infinitely many solutions. The system 3x - y = 7; x + 2y = 0 has one unique solution. The system y = 2x + 1; y = 2x - 3 consists of two parallel lines with different y-intercepts, so it has no solution and is inconsistent.
Question 8
Which of these linear equations does not represent a line with a negative slope?
3x + y = 7
x + 2y = 6
y - 4 = 1/2(x + 1)
✓
y = -2x + 5
Correct Answer
y - 4 = 1/2(x + 1)
The shared property is representing a line with a negative slope. In y = -2x + 5, the slope is -2. In 3x + y = 7, rearranging to y = -3x + 7 shows the slope is -3. In x + 2y = 6, rearranging to 2y = -x + 6, then y = (-1/2)x + 3 shows the slope is -1/2. In y - 4 = 1/2(x + 1), the slope is 1/2, which is positive.
Question 9
Which of these expressions is NOT equal to 1?
3^-1 * 3
(1/2)^-1
✓
(5x)^0, where x != 0
(7/7)^10
Correct Answer
(1/2)^-1
The shared property is being equal to 1. (5x)^0 (where x != 0) equals 1 because any non-zero base raised to the power of 0 is 1. 3^-1 * 3 equals 3^( -1+1) = 3^0 = 1. (7/7)^10 equals 1^10 = 1. (1/2)^-1 equals 2, which is not 1.
Question 10
Which of these polynomials does not have a greatest common factor (GCF) of 2x?
6x^2y + 10x
8x^3 - 4x^2 + 2
✓
2x^3 - 8x
4x^2 + 6x
Correct Answer
8x^3 - 4x^2 + 2
The shared property is having a greatest common factor (GCF) of 2x. For 4x^2 + 6x, GCF is 2x. For 2x^3 - 8x, GCF is 2x. For 6x^2y + 10x, GCF is 2x. For 8x^3 - 4x^2 + 2, the GCF is 2, not 2x, because the constant term 2 does not have a factor of x.
Question 11
Which of these quadratic functions does not have its vertex on the y-axis?
f(x) = -2x^2 + 3
f(x) = (x - 3)^2 + 1
✓
f(x) = x^2 + 5
f(x) = 4x^2
Correct Answer
f(x) = (x - 3)^2 + 1
The shared property is having its vertex on the y-axis, which means the x-coordinate of the vertex is 0. For f(x) = x^2 + 5, the vertex is (0, 5). For f(x) = -2x^2 + 3, the vertex is (0, 3). For f(x) = 4x^2, the vertex is (0, 0). For f(x) = (x - 3)^2 + 1, which is in vertex form a(x-h)^2+k, the vertex is (3, 1), so its vertex is not on the y-axis.
Question 12
Which of these relations does NOT represent a function?
x = y^2
✓
y = x^2
y = |x|
y = 3x - 1
Correct Answer
x = y^2
The shared property is representing a function (passing the vertical line test). y = x^2, y = |x|, and y = 3x - 1 are all functions because for every x-value, there is only one y-value. The relation x = y^2 is not a function because for a single x-value (e.g., x=4), there can be two y-values (y=2 and y=-2), failing the vertical line test.
Question 13
Which of these radical expressions is NOT in simplest radical form?
sqrt(48)
✓
2*sqrt(11)
5*sqrt(3)
cube_root(10)
Correct Answer
sqrt(48)
The shared property is being in simplest radical form. 5*sqrt(3) is in simplest form as 3 has no perfect square factors. cube_root(10) is in simplest form as 10 has no perfect cube factors. 2*sqrt(11) is in simplest form as 11 has no perfect square factors. sqrt(48) is not in simplest form because 48 has a perfect square factor of 16 (sqrt(48) = sqrt(16 * 3) = 4*sqrt(3)).
Question 14
Which of these quadratic equations cannot be solved by factoring using integer coefficients?
x^2 + 2x - 1 = 0
✓
x^2 - 4 = 0
2x^2 - 3x - 2 = 0
x^2 + 5x + 6 = 0
Correct Answer
x^2 + 2x - 1 = 0
The shared property is being solvable by factoring using integer coefficients. x^2 - 4 = 0 factors to (x-2)(x+2). x^2 + 5x + 6 = 0 factors to (x+2)(x+3). 2x^2 - 3x - 2 = 0 factors to (2x+1)(x-2). For x^2 + 2x - 1 = 0, the discriminant is b^2 - 4ac = 2^2 - 4(1)(-1) = 4 + 4 = 8. Since 8 is not a perfect square, this quadratic equation cannot be factored into linear terms with integer coefficients.
Question 15
Which of these linear equations does not have a solution that is a positive integer?
3x - 5 = 10
x/4 + 1 = 2
2(x + 1) = 8
x + 7 = 3
✓
Correct Answer
x + 7 = 3
The shared property is having a solution that is a positive integer. For 3x - 5 = 10, 3x = 15, so x = 5 (positive integer). For 2(x + 1) = 8, x + 1 = 4, so x = 3 (positive integer). For x/4 + 1 = 2, x/4 = 1, so x = 4 (positive integer). For x + 7 = 3, x = -4, which is not a positive integer.