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SAT Math Questions.
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-
If 2x + 4 = 14, what is x?
- 3
- 4
- 5 ✓
- 6
Explanation
Subtract 4 from both sides: 2x = 10. Divide by 2: x = 5.
-
If x/3 − 2 = 4, what is x?
- 6
- 12
- 18 ✓
- 24
Explanation
Add 2 to both sides: x/3 = 6. Multiply by 3: x = 18.
-
If 5(x − 2) = 3x + 8, what is x?
- 7
- 8
- 9 ✓
- 10
Explanation
Distribute: 5x − 10 = 3x + 8. Subtract 3x: 2x − 10 = 8. Add 10: 2x = 18. Divide by 2: x = 9.
-
If (3x − 1)/2 = (x + 5)/2, what is x?
- 2
- 3 ✓
- 4
- 5
Explanation
Both sides have the same denominator, so 3x − 1 = x + 5. Subtract x: 2x − 1 = 5. Add 1: 2x = 6. Divide: x = 3.
-
If 3(x + 2) − 5(x − 1) = 2x + 7, what is x?
- 0
- 1 ✓
- 2
- 3
Explanation
Distribute: 3x + 6 − 5x + 5 = 2x + 7 → −2x + 11 = 2x + 7 → 4 = 4x → x = 1.
-
What value of x satisfies 4x = 20?
- 4
- 5 ✓
- 6
- 8
Explanation
Divide both sides by 4: x = 5.
-
If 7 − 3x = −11, what is x?
- 3
- 4
- 5
- 6 ✓
Explanation
Subtract 7 from both sides: −3x = −18. Divide by −3: x = 6.
-
For the equation kx + 6 = 3x − 2, where k is a constant, the equation has no solution. What is k?
- 1
- 2
- 3 ✓
- 6
Explanation
For no solution, the coefficients of x must be equal but the constants different. Setting k = 3 gives 3x + 6 = 3x − 2, which simplifies to 6 = −2 — a contradiction. So k = 3.
-
A line has slope 3 and passes through (0, −2). Which equation represents this line?
- y = 3x + 2
- y = −3x + 2
- y = 3x − 2 ✓
- y = −2x + 3
Explanation
Slope-intercept form: y = mx + b with m = 3 and b = −2 gives y = 3x − 2.
-
What is the slope of the line 2x − 4y = 8?
- 2
- −2
- 1/2 ✓
- −1/2
Explanation
Solve for y: −4y = −2x + 8 → y = (1/2)x − 2. The slope is 1/2.
-
A line passes through (2, 5) and (6, 13). What is the y-intercept?
- −1
- 0
- 1 ✓
- 2
Explanation
Slope = (13 − 5)/(6 − 2) = 2. Using y = 2x + b and point (2, 5): 5 = 4 + b → b = 1.
-
Which equation represents a line perpendicular to y = 3x + 5?
- y = 3x − 2
- y = −3x + 1
- y = (1/3)x + 2
- y = −(1/3)x + 4 ✓
Explanation
Perpendicular lines have slopes that are negative reciprocals. The slope of y = 3x + 5 is 3, so the perpendicular slope is −1/3. Only y = −(1/3)x + 4 has this slope.
-
Line L passes through (1, 4) and is parallel to 3x + 2y = 6. What is the equation of L?
- y = −(3/2)x + 11/2 ✓
- y = (3/2)x + 5/2
- y = −(2/3)x + 14/3
- y = −(3/2)x + 4
Explanation
3x + 2y = 6 → y = −(3/2)x + 3, so slope = −3/2. L: y − 4 = −(3/2)(x − 1) → y = −(3/2)x + 3/2 + 4 = −(3/2)x + 11/2.
-
A cab company charges a flat fee of $3.00 plus $1.50 per mile. If the total fare is $15.00, how many miles was the trip?
- 6
- 7
- 8 ✓
- 9
Explanation
3 + 1.5m = 15 → 1.5m = 12 → m = 8 miles.
-
What is the x-intercept of the line y = 2x − 6?
- 2
- 3 ✓
- 6
- −6
Explanation
Set y = 0: 0 = 2x − 6 → 2x = 6 → x = 3.
-
In the equation y = ax + b, increasing x by 4 increases y by 10. What is a?
- 2
- 5/2 ✓
- 2/5
- 5
Explanation
The slope a = Δy/Δx = 10/4 = 5/2.
-
If x + y = 8 and x − y = 2, what is x?
- 3
- 4
- 5 ✓
- 6
Explanation
Add the equations: 2x = 10 → x = 5.
-
Given 2x + y = 7 and x − y = 2, what is x + y?
- 3
- 4 ✓
- 5
- 6
Explanation
Add equations: 3x = 9 → x = 3. From x − y = 2: y = 1. So x + y = 4.
-
Given 3x + 2y = 16 and x − y = 2, what is y?
- 1
- 2 ✓
- 3
- 4
Explanation
From x − y = 2: x = y + 2. Substitute: 3(y + 2) + 2y = 16 → 5y + 6 = 16 → 5y = 10 → y = 2.
-
How many solutions does the system 5x + 3y = 11 and 10x + 6y = 22 have?
- 0
- 1
- 2
- Infinitely many ✓
Explanation
The second equation is exactly 2 times the first, so they represent the same line. Every point on the line is a solution — infinitely many.
-
The system ax + 2y = 6 and 3x + y = 4 has no solution. What is a?
- 2
- 4
- 6 ✓
- 8
Explanation
For no solution, coefficient ratios must match but constants must not: a/3 = 2/1 → a = 6. Check: 6/3 = 2 = 2/1, but 6 ≠ 2 × 4 = 8. Confirmed no solution.
-
A store sells apples for $0.50 each and bananas for $0.30 each. Maria buys 20 pieces of fruit for $8.00. How many apples did she buy?
- 8
- 9
- 10 ✓
- 12
Explanation
Let a = apples, b = bananas. a + b = 20 and 0.5a + 0.3b = 8. Substituting b = 20 − a: 0.5a + 0.3(20 − a) = 8 → 0.2a = 2 → a = 10.
-
If y = 2x + 1 and y = 5, what is x?
- 1
- 2 ✓
- 3
- 4
Explanation
Substitute y = 5: 2x + 1 = 5 → 2x = 4 → x = 2.
-
Two numbers have a sum of 24 and a difference of 8. What is their product?
- 112
- 120
- 128 ✓
- 136
Explanation
x + y = 24 and x − y = 8. Adding: 2x = 32 → x = 16, y = 8. Product = 16 × 8 = 128.
-
Which values of x satisfy 3x − 6 > 9?
- x > 2
- x > 3
- x > 5 ✓
- x > 7
Explanation
Add 6: 3x > 15. Divide by 3: x > 5.
-
Which values of x satisfy −2x + 4 ≤ 10?
- x ≤ −3
- x ≥ −3 ✓
- x ≤ 3
- x ≥ 3
Explanation
Subtract 4: −2x ≤ 6. Divide by −2 (flip the inequality): x ≥ −3.
-
A student needs an average of at least 80 to pass a two-test course. She scored 65 on the first test. What is the minimum score she needs on the second test?
- 85
- 90
- 95 ✓
- 100
Explanation
(65 + x)/2 ≥ 80 → 65 + x ≥ 160 → x ≥ 95.
-
What is the solution set for |2x − 4| < 6?
- −1 < x < 5 ✓
- −5 < x < 1
- x < −1 or x > 5
- x < 1 or x > −5
Explanation
−6 < 2x − 4 < 6 → −2 < 2x < 10 → −1 < x < 5.
-
Which of the following is NOT a solution to 4 − x > 2?
- −1
- 0
- 1
- 2 ✓
Explanation
4 − x > 2 → −x > −2 → x < 2. So x = 2 is NOT a solution (it makes the inequality false).
-
Which inequality represents "all real numbers less than or equal to 3"?
- x < 3
- x > 3
- x ≥ 3
- x ≤ 3 ✓
Explanation
"Less than or equal to 3" is written x ≤ 3.
-
The inequality 3(x + 2) ≤ 2(x + 5) is equivalent to which of the following?
- x ≤ 2
- x ≤ 4 ✓
- x ≥ 4
- x ≤ 8
Explanation
3x + 6 ≤ 2x + 10 → x ≤ 4.
-
A company's profit P (in dollars) satisfies P = 50n − 200, where n is items sold. What is the minimum number of items the company must sell to make a profit?
- 3
- 4
- 5 ✓
- 6
Explanation
50n − 200 > 0 → n > 4. The smallest integer greater than 4 is 5.
-
The function C(x) = 3x + 7 gives the cost (in dollars) of renting a bicycle for x hours. What does the 7 represent?
- The hourly rate
- The total cost
- The initial flat fee ✓
- The number of hours
Explanation
In y = mx + b, b is the y-intercept — the value when x = 0. Here, 7 is the cost before any hours are added, so it is the initial flat fee.
-
A table shows (1, 5), (2, 8), (3, 11), (4, 14). What is the rate of change?
- 2
- 3 ✓
- 4
- 5
Explanation
Rate of change = (8 − 5)/(2 − 1) = 3. This holds for every consecutive pair in the table.
-
A linear model predicts a town's population grows by 450 people per year. The population was 12,000 in 2020. What does the model predict for 2025?
- 13,500
- 14,000
- 14,250 ✓
- 15,000
Explanation
2025 is 5 years after 2020. 12,000 + 5 × 450 = 12,000 + 2,250 = 14,250.
-
Line ℓ: y = 2x passes through the origin. Line m is perpendicular to ℓ and passes through (4, 0). What are the coordinates of their intersection?
- (4/5, 8/5) ✓
- (1, 2)
- (2, 4)
- (8/5, 4/5)
Explanation
Slope of m = −1/2. m: y = −(1/2)(x − 4). Set equal to ℓ: 2x = −x/2 + 2 → 5x/2 = 2 → x = 4/5, y = 8/5.
-
If f(x) = −2x + 10, for what value of x does f(x) = 0?
- 2
- 3
- 5 ✓
- 10
Explanation
−2x + 10 = 0 → −2x = −10 → x = 5.
-
A phone plan charges $30 per month plus $0.10 per text message. Which equation gives the monthly cost C for t text messages?
- C = 0.10t
- C = 30t
- C = 30 + 0.10t ✓
- C = 30t + 0.10
Explanation
Fixed cost ($30) plus variable cost ($0.10 per text) gives C = 30 + 0.10t.
-
f and g are linear functions with the same slope. f(2) = 5, f(6) = 13, and g(0) = 7. What is g(3)?
- 11
- 12
- 13 ✓
- 14
Explanation
Slope of f = (13 − 5)/(6 − 2) = 2. g(x) = 2x + 7 (since g(0) = 7). g(3) = 6 + 7 = 13.
-
In the model y = −3x + 24, y represents remaining pages and x represents hours of reading. What is the total number of pages in the book?
- 3
- 8
- 21
- 24 ✓
Explanation
When x = 0 (no reading yet), y = 24. That is the total page count.
-
Which expression is equivalent to (3x²)(4x³)?
- 7x⁵
- 12x⁵ ✓
- 12x⁶
- 7x⁶
Explanation
Multiply coefficients: 3 × 4 = 12. Add exponents: x² · x³ = x⁵. Result: 12x⁵.
-
Which is equivalent to (x + 4)(x − 3)?
- x² + x + 12
- x² − x − 12
- x² + x − 12 ✓
- x² − 7x − 12
Explanation
FOIL: x² − 3x + 4x − 12 = x² + x − 12.
-
Which shows the complete factorization of x² − 9?
- (x + 3)²
- (x − 3)²
- (x + 3)(x − 3) ✓
- x(x − 9)
Explanation
Difference of squares: a² − b² = (a + b)(a − b). So x² − 9 = (x + 3)(x − 3).
-
Which expression is equivalent to (x² + 5x + 6)/(x + 2), where x ≠ −2?
- x + 2
- x + 3 ✓
- x − 3
- x² + 3x
Explanation
Factor the numerator: (x + 3)(x + 2). Cancel (x + 2): result is x + 3.
-
Which expression is equivalent to 4x² − 12x + 9?
- (2x + 3)²
- (2x − 3)² ✓
- (4x − 3)(x − 3)
- (4x + 3)(x − 3)
Explanation
(2x − 3)² = 4x² − 12x + 9. Check: (2x)² = 4x², 2(2x)(−3) = −12x, (−3)² = 9. ✓
-
Which is equivalent to (2x³ − 6x)/(2x), where x ≠ 0?
- x² − 6
- 2x² − 3
- x² − 3 ✓
- x³ − 3
Explanation
Divide each term: 2x³/2x − 6x/2x = x² − 3.
-
What is (3x² + 2x − 1) + (x² − 5x + 4)?
- 4x² + 7x + 3
- 4x² − 3x + 3 ✓
- 4x² − 3x − 3
- 2x² − 3x + 3
Explanation
Combine like terms: (3+1)x² + (2−5)x + (−1+4) = 4x² − 3x + 3.
-
Which is equivalent to (x + 2)²?
- x² + 4
- x² + 2x + 4
- x² + 4x + 4 ✓
- x² − 4x + 4
Explanation
(x + 2)² = x² + 2(x)(2) + 4 = x² + 4x + 4.
-
What are the solutions to x² − 5x + 6 = 0?
- x = 1 and x = 6
- x = 2 and x = 3 ✓
- x = −2 and x = −3
- x = 2 and x = −3
Explanation
Factor: (x − 2)(x − 3) = 0. So x = 2 or x = 3.
-
What are the solutions to x² + 4x − 5 = 0?
- x = 1 and x = −5 ✓
- x = −1 and x = 5
- x = 5 and x = −1
- x = 2 and x = −5
Explanation
Factor: (x + 5)(x − 1) = 0. So x = −5 or x = 1.
-
For the equation x² − 6x + 9 = 0, how many distinct real solutions are there?
- 0
- 1 ✓
- 2
- 3
Explanation
Discriminant D = 36 − 36 = 0. One repeated solution: x = 3.
-
The equation x² + bx + 16 = 0 has exactly one real solution. What is a possible value of b?
- −4
- 4
- 8 ✓
- 16
Explanation
D = b² − 64 = 0 → b² = 64 → b = ±8. Both 8 and −8 work; 8 is listed.
-
What is the vertex of y = x² − 4x + 7?
- (2, 3) ✓
- (−2, 3)
- (2, −3)
- (4, 7)
Explanation
h = −(−4)/(2·1) = 2. k = 4 − 8 + 7 = 3. Vertex: (2, 3).
-
For what value of c does x² + 6x + c = 0 have no real solutions?
- c = 8
- c = 9
- c = 10 ✓
- c = 36
Explanation
D = 36 − 4c < 0 → c > 9. The only choice greater than 9 is c = 10.
-
What are the solutions to 2x² − 8 = 0?
- x = ±1
- x = ±2 ✓
- x = ±4
- x = 2 only
Explanation
2x² = 8 → x² = 4 → x = ±2.
-
A quadratic equation has solutions x = 3 and x = −1. Which could be the equation?
- x² + 2x − 3 = 0
- x² − 2x − 3 = 0 ✓
- x² − 2x + 3 = 0
- x² + 2x + 3 = 0
Explanation
(x − 3)(x + 1) = x² + x − 3x − 3 = x² − 2x − 3 = 0.
-
What is the value of x if √(x + 5) = 4?
- 3
- 9
- 11 ✓
- 21
Explanation
Square both sides: x + 5 = 16 → x = 11.
-
What is the positive solution to (x − 2)/(x + 1) = 3/5?
- 5
- 5.5
- 6.5 ✓
- 7
Explanation
Cross-multiply: 5(x − 2) = 3(x + 1) → 5x − 10 = 3x + 3 → 2x = 13 → x = 6.5.
-
The system y = x² and y = x + 2 has how many solutions?
- 0
- 1
- 2 ✓
- 3
Explanation
x² = x + 2 → x² − x − 2 = 0 → (x − 2)(x + 1) = 0. Two solutions: x = 2 and x = −1.
-
Where does y = x² + 1 intersect y = 2x + 1? What are the x-values?
- x = 1 and x = 2
- x = 0 and x = 2 ✓
- x = −1 and x = 2
- x = 0 and x = 1
Explanation
x² + 1 = 2x + 1 → x² − 2x = 0 → x(x − 2) = 0 → x = 0 or x = 2.
-
The parabola y = x² − 4 and the line y = −x + 2 intersect at two points. What is the sum of their x-coordinates?
- −1 ✓
- 1
- 5
- −5
Explanation
x² − 4 = −x + 2 → x² + x − 6 = 0 → (x + 3)(x − 2) = 0. x-values: −3 and 2. Sum = −1.
-
Where does y = x² intersect y = 4?
- x = 2 only
- x = ±1
- x = ±2 ✓
- x = ±4
Explanation
x² = 4 → x = ±2.
-
The system y = x² + 3x and y = kx has solutions at x = 0 and x = 4. What is k?
- 4
- 5
- 7 ✓
- 9
Explanation
x² + 3x = kx → x(x + 3 − k) = 0. For x = 4: 4 + 3 − k = 0 → k = 7.
-
How many x-intercepts does the parabola y = x² − 2x − 3 have?
- 0
- 1
- 2 ✓
- 3
Explanation
D = 4 + 12 = 16 > 0, so two x-intercepts. Factor: (x − 3)(x + 1) = 0 → x = 3 and x = −1.
-
If f(x) = x² + 2, what is f(3)?
- 7
- 9
- 11 ✓
- 13
Explanation
f(3) = 3² + 2 = 9 + 2 = 11.
-
The function f(x) = 2 · (1.5)^x models a population. What is the initial population (at x = 0)?
- 1
- 1.5
- 2 ✓
- 3
Explanation
f(0) = 2 · (1.5)⁰ = 2 · 1 = 2.
-
A parabola opens downward and has vertex at (3, 5). Which equation could represent it?
- y = (x − 3)² + 5
- y = −(x + 3)² + 5
- y = −(x − 3)² + 5 ✓
- y = (x + 3)² − 5
Explanation
Vertex form with vertex (h, k) = (3, 5) and opening downward (negative a): y = −(x − 3)² + 5.
-
What is the minimum value of f(x) = 3x² − 12x + 7?
- −7
- −5 ✓
- 3
- 7
Explanation
Vertex x = 12/(2·3) = 2. f(2) = 12 − 24 + 7 = −5.
-
If f(x) = 2^x, what is f(5)?
- 10
- 16
- 25
- 32 ✓
Explanation
2⁵ = 32.
-
The function f(x) = a · 2^x passes through (0, 4) and (3, 32). What is a?
- 2
- 3
- 4 ✓
- 8
Explanation
f(0) = a · 1 = a = 4. Verify: f(3) = 4 · 8 = 32. ✓
-
For f(x) = −x² + 6x − 5, what is the maximum value?
- 3
- 4 ✓
- 5
- 6
Explanation
Vertex x = −6/(2 · −1) = 3. f(3) = −9 + 18 − 5 = 4.
-
If f(x) = x² − 4 and g(x) = 2x + 1, what is f(g(2))?
- 17
- 19
- 21 ✓
- 25
Explanation
g(2) = 2(2) + 1 = 5. f(5) = 25 − 4 = 21.
-
If f(x) = 3x − 2, what is f(4)?
- 8
- 9
- 10 ✓
- 11
Explanation
f(4) = 3(4) − 2 = 12 − 2 = 10.
-
If g(x) = x² − 3x + 2, what is g(−1)?
- 4
- 5
- 6 ✓
- 7
Explanation
g(−1) = (−1)² − 3(−1) + 2 = 1 + 3 + 2 = 6.
-
If f(x) = 2x + 1, what is f(f(2))?
- 9
- 10
- 11 ✓
- 12
Explanation
f(2) = 5. f(f(2)) = f(5) = 2(5) + 1 = 11.
-
The graph of y = f(x) is shifted 3 units up and 2 units to the left. Which equation represents the new function?
- y = f(x − 2) + 3
- y = f(x + 2) + 3 ✓
- y = f(x − 2) − 3
- y = f(x + 3) + 2
Explanation
Shifting left by 2 replaces x with (x + 2); shifting up by 3 adds 3. Result: y = f(x + 2) + 3.
-
If h(x) = f(x) + 5 and f(3) = 7, what is h(3)?
- 10
- 11
- 12 ✓
- 14
Explanation
h(3) = f(3) + 5 = 7 + 5 = 12.
-
For f(x) = x³ − x, the equation f(−x) = f(x) holds for which values of x?
- x = 0 only
- x = 0 and x = 1
- x = 0, x = 1, and x = −1 ✓
- x = ±1 only
Explanation
f(−x) = −x³ + x = −f(x). So f(−x) = f(x) only when f(x) = 0: x³ − x = 0 → x(x−1)(x+1) = 0 → x = 0, ±1.
-
If f(x) = |x − 3|, what is f(−1)?
- 2
- 3
- 4 ✓
- 5
Explanation
f(−1) = |−1 − 3| = |−4| = 4.
-
The function g(x) = 2f(x − 1) is a transformation of f(x). Which transformations are applied?
- Shift left 1 and vertical stretch by 2
- Shift right 1 and vertical stretch by 2 ✓
- Shift right 1 and vertical shrink by 2
- Shift left 1 and vertical shrink by 2
Explanation
Replacing x with (x − 1) shifts the graph right by 1. Multiplying by 2 stretches it vertically by a factor of 2.
-
What is the area of a rectangle with length 8 and width 5?
- 26
- 32
- 40 ✓
- 80
Explanation
A = length × width = 8 × 5 = 40.
-
What is the area of a triangle with base 10 and height 6?
- 16
- 30 ✓
- 60
- 80
Explanation
A = (1/2)bh = (1/2)(10)(6) = 30.
-
A cylinder has radius 3 and height 8. What is its volume?
- 24π
- 48π
- 72π ✓
- 96π
Explanation
V = πr²h = π(9)(8) = 72π.
-
A cone has radius 3 and height 12. What is its volume? (V = (1/3)πr²h)
- 12π
- 24π
- 36π ✓
- 108π
Explanation
V = (1/3)π(3²)(12) = (1/3)π(9)(12) = 36π.
-
A sphere has a surface area of 100π. What is its radius? (SA = 4πr²)
- 3
- 4
- 5 ✓
- 10
Explanation
4πr² = 100π → r² = 25 → r = 5.
-
Two parallel lines are cut by a transversal. One angle measures 65°. What is the measure of its alternate interior angle?
- 25°
- 65° ✓
- 115°
- 180°
Explanation
Alternate interior angles formed by parallel lines and a transversal are congruent. The alternate interior angle also measures 65°.
-
In a triangle, two angles measure 50° and 70°. What is the measure of the third angle?
- 50°
- 55°
- 60° ✓
- 70°
Explanation
Angle sum in a triangle = 180°. Third angle = 180 − 50 − 70 = 60°.
-
Two triangles are similar. The first has sides 3, 4, and 5. The shortest side of the second is 9. What is the longest side of the second triangle?
- 9
- 12
- 15 ✓
- 20
Explanation
Scale factor = 9/3 = 3. Longest side = 5 × 3 = 15.
-
An isosceles triangle has a vertex angle of 40°. What is the measure of each base angle?
- 40°
- 60°
- 70° ✓
- 80°
Explanation
The two base angles are equal. Each = (180 − 40)/2 = 70°.
-
Two lines intersect. One of the angles formed measures 130°. What is the measure of the adjacent supplementary angle?
- 30°
- 40°
- 50° ✓
- 60°
Explanation
Supplementary angles sum to 180°. 180 − 130 = 50°.
-
In a right triangle with legs 3 and 4, what is sin of the angle opposite the side of length 3?
- 3/4
- 3/5 ✓
- 4/5
- 4/3
Explanation
Hypotenuse = √(9 + 16) = 5. sin = opposite/hypotenuse = 3/5.
-
A 10-meter ladder leans against a wall at a 60° angle with the ground. How high on the wall does it reach?
- 5
- 5√2
- 5√3 ✓
- 10√3
Explanation
Height = 10 × sin 60° = 10 × (√3/2) = 5√3 meters.
-
In a 45-45-90 triangle, if the hypotenuse is 8√2, what is the length of each leg?
- 4
- 8 ✓
- 8√2
- 4√2
Explanation
In a 45-45-90 triangle, leg = hypotenuse/√2 = 8√2/√2 = 8.
-
In a right triangle, cos θ = 5/13. What is tan θ?
- 5/12
- 12/5 ✓
- 13/5
- 5/13
Explanation
adj = 5, hyp = 13, so opp = √(169 − 25) = 12. tan θ = opp/adj = 12/5.
-
What is the circumference of a circle with diameter 14?
- 7π
- 14π ✓
- 28π
- 49π
Explanation
C = πd = 14π.
-
A circle has radius 5. What is the length of an arc subtended by a central angle of 72°?
- π
- 2π ✓
- 5π
- 10π
Explanation
Arc length = (θ/360) × 2πr = (72/360) × 10π = (1/5) × 10π = 2π.
-
The equation of a circle is (x − 2)² + (y + 3)² = 25. What is its radius?
- 3
- 4
- 5 ✓
- 25
Explanation
The equation (x − h)² + (y − k)² = r² gives r² = 25, so r = 5.
-
A sector of a circle has radius 6 and a central angle of 120°. What is the area of the sector?
- 6π
- 8π
- 12π ✓
- 24π
Explanation
Sector area = (θ/360) × πr² = (120/360) × 36π = (1/3)(36π) = 12π.
-
What is the distance between the points (1, 2) and (4, 6)?
- 4
- 5 ✓
- 6
- 7
Explanation
d = √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5.
-
The midpoint of segment AB is (3, 5). If A = (1, 3), what are the coordinates of B?
- (4, 6)
- (5, 7) ✓
- (6, 8)
- (2, 4)
Explanation
Midpoint formula: ((1+x)/2, (3+y)/2) = (3, 5). So x = 5 and y = 7. B = (5, 7).
-
A recipe calls for 3 cups of flour to make 12 cookies. How many cups are needed to make 36 cookies?
- 6
- 8
- 9 ✓
- 12
Explanation
36/12 = 3, so multiply by 3: 3 × 3 = 9 cups.
-
A car travels 240 miles in 4 hours. At the same rate, how far does it travel in 7 hours?
- 300
- 360
- 420 ✓
- 480
Explanation
Rate = 240/4 = 60 mph. Distance = 60 × 7 = 420 miles.
-
A map uses a scale of 1 inch = 25 miles. Two cities are 3.5 inches apart on the map. What is the actual distance?
- 75 miles
- 82.5 miles
- 87.5 miles ✓
- 90 miles
Explanation
3.5 × 25 = 87.5 miles.
-
A store sells juice in 12 oz bottles for $2.40 or 20 oz bottles for $3.60. Which size is the better value, and by how much per ounce?
- 12 oz, by $0.02 per oz
- 20 oz, by $0.01 per oz
- 20 oz, by $0.02 per oz ✓
- Same value per ounce
Explanation
12 oz: $2.40/12 = $0.20/oz. 20 oz: $3.60/20 = $0.18/oz. The 20 oz bottle is better by $0.02/oz.
-
A shirt costs $40 and is on sale for 25% off. What is the sale price?
- $25
- $28
- $30 ✓
- $32
Explanation
Discount = 25% of $40 = $10. Sale price = $40 − $10 = $30.
-
A TV was originally $500 and is now $425. What is the percent decrease?
- 10%
- 12%
- 15% ✓
- 17%
Explanation
Percent decrease = (500 − 425)/500 × 100 = 75/500 × 100 = 15%.
-
A student scored 78 out of 120 points. What percent did she score?
- 60%
- 62%
- 65% ✓
- 70%
Explanation
78/120 × 100 = 65%.
-
After a 20% increase, a price is $84. What was the original price?
- $60
- $65
- $70 ✓
- $75
Explanation
Original × 1.20 = 84 → Original = 84/1.20 = $70.
-
A bag contains 5 red, 3 blue, and 2 green marbles. What is the probability of drawing a blue marble?
- 1/5
- 3/10 ✓
- 1/2
- 3/5
Explanation
P(blue) = 3/10.
-
One card is drawn from a standard 52-card deck. What is the probability of drawing a face card (J, Q, or K)?
- 1/13
- 3/13 ✓
- 4/13
- 1/4
Explanation
There are 12 face cards in a deck of 52. P = 12/52 = 3/13.
-
A class of 40 students includes 15 boys who like math, 5 boys who do not, 10 girls who like math, and 10 girls who do not. What is the probability that a randomly chosen student is a girl who likes math?
- 1/5
- 1/4 ✓
- 1/2
- 10/15
Explanation
10 girls who like math out of 40 students: P = 10/40 = 1/4.
-
P(A) = 0.4 and P(B) = 0.3. If A and B are independent events, what is P(A and B)?
- 0.07
- 0.10
- 0.12 ✓
- 0.70
Explanation
For independent events: P(A and B) = P(A) × P(B) = 0.4 × 0.3 = 0.12.
-
The data set is 4, 7, 7, 9, 13. What is the median?
- 7 ✓
- 8
- 9
- 10
Explanation
The data is already in order. The middle value (3rd of 5) is 7.
-
The data set is 2, 5, 7, 8, 8, 10. What is the mean?
- 6.5
- 6.67 ✓
- 7
- 8
Explanation
Sum = 2 + 5 + 7 + 8 + 8 + 10 = 40. Mean = 40/6 ≈ 6.67.
-
Which measure of center is most affected by an extreme outlier?
- Mean ✓
- Median
- Mode
- Range
Explanation
The mean uses all values in its calculation, so a very large or very small outlier pulls it significantly. The median is resistant to outliers.
-
What does the standard deviation of a data set measure?
- The average value of the data
- The middle value of the data
- The most frequent value in the data
- The spread of data values around the mean ✓
Explanation
Standard deviation measures how much the data values vary from the mean — it quantifies spread.
-
A bar chart shows January sales: 200, February: 150, March: 250. What is the total for these three months?
- 500
- 550
- 600 ✓
- 650
Explanation
200 + 150 + 250 = 600.
-
A line of best fit for study hours (x) and test score (y) is y = 5x + 60. What score does the model predict for 8 hours of study?
- 80
- 90
- 95
- 100 ✓
Explanation
y = 5(8) + 60 = 40 + 60 = 100.
-
A histogram shows 30% of 200 students scored 70–80, 45% scored 80–90, and 25% scored 90–100. How many students scored in the 80–90 range?
- 70
- 80
- 90 ✓
- 100
Explanation
45% of 200 = 0.45 × 200 = 90 students.
-
A study finds a correlation coefficient of −0.85 between hours of sleep and reaction time. What does this indicate?
- No relationship
- Weak positive relationship
- Weak negative relationship
- Strong negative relationship ✓
Explanation
A value of −0.85 is close to −1, indicating a strong negative correlation: more sleep is associated with lower (faster) reaction times.
-
If 6x − 3 = 21, what is x?
- 2
- 3
- 4 ✓
- 5
Explanation
Add 3 to both sides: 6x = 24. Divide by 6: x = 4.
-
What is the value of x if 2(x + 5) = 18?
- 3
- 4 ✓
- 5
- 6
Explanation
Distribute: 2x + 10 = 18 → 2x = 8 → x = 4.
-
Which equation represents a line with slope −2 and y-intercept 5?
- y = 2x + 5
- y = −2x − 5
- y = −2x + 5 ✓
- y = 5x − 2
Explanation
Slope-intercept form y = mx + b: m = −2 and b = 5 gives y = −2x + 5.
-
If y = 4 when x = 0, and y increases by 3 for every increase of 1 in x, which equation models this?
- y = 4x + 3
- y = 3x − 4
- y = 3x + 4 ✓
- y = −3x + 4
Explanation
y-intercept = 4, slope = 3. Slope-intercept form: y = 3x + 4.
-
If 4(2x − 1) = 3(x + 7), what is x?
- 3
- 4
- 5 ✓
- 6
Explanation
Distribute: 8x − 4 = 3x + 21 → 5x = 25 → x = 5.
-
Which values of x satisfy 5x + 3 > 28?
- x > 4
- x > 5 ✓
- x > 6
- x > 7
Explanation
5x > 25 → x > 5.
-
A line passes through (0, 6) and (3, 0). What is the slope of the line?
- −3
- −2 ✓
- 2
- 3
Explanation
Slope = (0 − 6)/(3 − 0) = −6/3 = −2.
-
Given x + 2y = 10 and x = 4, what is y?
- 2
- 3 ✓
- 4
- 5
Explanation
Substitute x = 4: 4 + 2y = 10 → 2y = 6 → y = 3.
-
Solve the system: 3x + y = 11 and x + y = 5. What is x?
- 2
- 3 ✓
- 4
- 5
Explanation
Subtract the second equation from the first: 2x = 6 → x = 3.
-
A store charges $12 per hour to rent a kayak plus a $5 launch fee. How many hours can you rent if you have $53?
- 3
- 4 ✓
- 5
- 6
Explanation
12h + 5 = 53 → 12h = 48 → h = 4.
-
The function f(x) = −4x + 20 represents the number of pages left in a book after x hours of reading. For what value of x does f(x) = 0?
- 4
- 5 ✓
- 6
- 8
Explanation
−4x + 20 = 0 → 4x = 20 → x = 5.
-
If 2x + 3y = 18 and x = 3, what is y?
- 3
- 4 ✓
- 5
- 6
Explanation
2(3) + 3y = 18 → 6 + 3y = 18 → 3y = 12 → y = 4.
-
For what value of k does the system kx + 2y = 8 and 3x + y = 6 have no solution?
- 3
- 6 ✓
- 1.5
- 2
Explanation
For no solution the lines must be parallel (equal slope ratios, unequal constant ratio). Multiply the second equation by 2: 6x + 2y = 12. For the x-coefficients to match, k = 6. Since 8 ≠ 12, the lines are parallel and the system has no solution.
-
What is the solution set for |3x + 6| ≥ 9?
- x ≤ −5 or x ≥ 1 ✓
- x ≤ −5 or x ≥ −1
- x ≤ 1 or x ≥ 5
- −5 ≤ x ≤ 1
Explanation
3x + 6 ≥ 9 → x ≥ 1, or 3x + 6 ≤ −9 → 3x ≤ −15 → x ≤ −5. Solution: x ≤ −5 or x ≥ 1.
-
A car rental company charges $25 per day plus $0.15 per mile. A competitor charges $40 per day with no mileage fee. For how many miles do both companies charge the same total for one day?
- 80
- 90
- 100 ✓
- 110
Explanation
25 + 0.15m = 40 → 0.15m = 15 → m = 100 miles.
-
The equation 3x − 7 = ax + b has infinitely many solutions. What must be true?
- a = 3 and b = 7
- a = 3 and b = −7 ✓
- a = −3 and b = 7
- a = −3 and b = −7
Explanation
For infinitely many solutions the equation must be an identity: all coefficients and constants must match. So a = 3 (coefficient of x) and b = −7 (constant on the right equals −7 on the left).
-
Line p has equation y = (2/3)x − 1. Line q is perpendicular to p and passes through (4, 3). What is the y-intercept of line q?
- 5
- 6
- 7
- 9 ✓
Explanation
Slope of p = 2/3, so slope of q = −3/2. Using point-slope: y − 3 = −(3/2)(x − 4) → y = −(3/2)x + 6 + 3 = −(3/2)x + 9. The y-intercept is 9.
-
A number n satisfies 3n = 45. What is n?
- 12
- 13
- 14
- 15 ✓
Explanation
n = 45/3 = 15.
-
A table shows (x, y) pairs: (0, 2), (1, 5), (2, 8), (3, 11). Which linear equation fits this data?
- y = 2x + 3
- y = 3x + 2 ✓
- y = x + 2
- y = 2x + 5
Explanation
Rate of change = (5−2)/(1−0) = 3; y-intercept = 2. Equation: y = 3x + 2.
-
If −(3x − 4) = 10, what is x?
- −3
- −2 ✓
- −1
- 2
Explanation
−3x + 4 = 10 → −3x = 6 → x = −2.
-
The system x − 3y = 1 and 2x + y = 9 is solved simultaneously. What is y?
- 0
- 1 ✓
- 2
- 3
Explanation
From the first equation: x = 1 + 3y. Substitute into the second: 2(1 + 3y) + y = 9 → 2 + 7y = 9 → 7y = 7 → y = 1.
-
What is the x-coordinate of the intersection of y = −2x + 7 and y = 3x − 8?
- 1
- 2
- 3 ✓
- 4
Explanation
−2x + 7 = 3x − 8 → 15 = 5x → x = 3.
-
In the linear equation 5x + 2y = k, the point (3, −1) lies on the line. What is k?
- 11
- 12
- 13 ✓
- 14
Explanation
5(3) + 2(−1) = 15 − 2 = 13.
-
A student has at most $200 to spend on books. Each textbook costs $45 and each notebook costs $5. If she buys 3 textbooks, what is the maximum number of notebooks she can buy?
- 13 ✓
- 14
- 15
- 16
Explanation
45(3) + 5n ≤ 200 → 135 + 5n ≤ 200 → 5n ≤ 65 → n ≤ 13. Maximum is 13 notebooks.
-
Which of the following is equivalent to 3(2x − 4) − 2(x + 1)?
- 4x − 10
- 4x − 14 ✓
- 8x − 14
- 4x + 14
Explanation
6x − 12 − 2x − 2 = 4x − 14.
-
Which is equivalent to (5x²)(−2x⁴)?
- −10x⁶ ✓
- −10x⁸
- 3x⁶
- 10x⁶
Explanation
Multiply coefficients: 5 × (−2) = −10. Add exponents: x² · x⁴ = x⁶. Result: −10x⁶.
-
What is the fully factored form of x² + 7x + 12?
- (x + 2)(x + 6)
- (x + 3)(x + 4) ✓
- (x + 1)(x + 12)
- (x − 3)(x − 4)
Explanation
Find two numbers that multiply to 12 and add to 7: 3 and 4. So (x + 3)(x + 4).
-
If f(x) = x² − 1, what is f(−3)?
- 6
- 7
- 8 ✓
- 9
Explanation
f(−3) = (−3)² − 1 = 9 − 1 = 8.
-
What is the solution to x² = 3x + 18?
- x = 6 and x = −3 ✓
- x = −6 and x = 3
- x = 6 and x = 3
- x = −6 and x = −3
Explanation
x² − 3x − 18 = 0 → (x − 6)(x + 3) = 0 → x = 6 or x = −3.
-
Which expression is equivalent to (2x + 3)(2x − 3)?
- 4x² − 9 ✓
- 4x² + 9
- 4x² − 6x + 9
- 4x² + 12x + 9
Explanation
Difference of squares: (a + b)(a − b) = a² − b². (2x)² − 3² = 4x² − 9.
-
The graph of y = (x − 2)² + 5 has its vertex at which point?
- (−2, 5)
- (2, 5) ✓
- (2, −5)
- (5, 2)
Explanation
In vertex form y = (x − h)² + k, the vertex is (h, k). Here h = 2 and k = 5, so vertex is (2, 5).
-
What are the solutions to 3x² − 27 = 0?
- x = ±1
- x = ±3 ✓
- x = ±9
- x = 3 only
Explanation
3x² = 27 → x² = 9 → x = ±3.
-
The polynomial p(x) = x³ − 4x is evaluated at x = 2. What is p(2)?
- 0 ✓
- 2
- 4
- 8
Explanation
p(2) = 2³ − 4(2) = 8 − 8 = 0.
-
Which expression equals (x² − 4x + 4)/(x − 2) for x ≠ 2?
- x + 2
- x − 2 ✓
- x² − 2
- x + 4
Explanation
Factor the numerator: (x − 2)². Divide by (x − 2): result is x − 2.
-
An exponential function f(x) = 3 · 2^x. What is f(4)?
- 24
- 36
- 48 ✓
- 12
Explanation
f(4) = 3 · 2⁴ = 3 · 16 = 48.
-
What is the discriminant of 2x² − 5x + 3 = 0, and how many real solutions does it have?
- D = 1; two real solutions ✓
- D = 25; two real solutions
- D = −1; no real solutions
- D = 0; one real solution
Explanation
D = b² − 4ac = 25 − 4(2)(3) = 25 − 24 = 1 > 0. Two distinct real solutions.
-
What is the sum of the roots of 2x² − 8x + 6 = 0?
- 2
- 3
- 4 ✓
- 6
Explanation
By Vieta's formulas, sum of roots = −b/a = −(−8)/2 = 4.
-
For f(x) = −2(x + 1)² + 8, what is the maximum value and where does it occur?
- Maximum of 8 at x = −1 ✓
- Maximum of 8 at x = 1
- Maximum of −1 at x = 8
- Maximum of 6 at x = 0
Explanation
The parabola opens downward (a = −2 < 0). Vertex is at (−1, 8), so maximum value is 8, occurring at x = −1.
-
How many solutions does the system y = x² − 2x + 3 and y = x + 1 have?
- 0
- 1
- 2 ✓
- 3
Explanation
Set equal: x² − 2x + 3 = x + 1 → x² − 3x + 2 = 0. D = 9 − 8 = 1 > 0, so two distinct real solutions: (x − 1)(x − 2) = 0 → x = 1 or x = 2.
-
Which equation has roots x = 5 and x = −2?
- x² − 3x − 10 = 0 ✓
- x² + 3x − 10 = 0
- x² − 3x + 10 = 0
- x² + 7x − 10 = 0
Explanation
(x − 5)(x + 2) = x² + 2x − 5x − 10 = x² − 3x − 10 = 0.
-
Simplify: (x³ · x²)/x⁴
- x ✓
- x²
- x³
- x⁴
Explanation
Numerator: x³ · x² = x⁵. Then x⁵/x⁴ = x¹ = x.
-
Which expression is equivalent to (3x − 2)² ?
- 9x² − 4
- 9x² + 4
- 9x² − 12x + 4 ✓
- 9x² + 12x + 4
Explanation
(3x − 2)² = 9x² − 2(3x)(2) + 4 = 9x² − 12x + 4.
-
The function g(x) = 5 · (0.5)^x. What is g(3)?
- 0.5
- 0.625 ✓
- 1.25
- 2.5
Explanation
g(3) = 5 · (0.5)³ = 5 · 0.125 = 0.625.
-
A quadratic function has x-intercepts at x = −1 and x = 4, and passes through (0, −8). What is the leading coefficient?
- −2
- 2 ✓
- −4
- 4
Explanation
f(x) = a(x + 1)(x − 4). At x = 0: f(0) = a(1)(−4) = −4a = −8 → a = 2. The leading coefficient is 2.
-
What is the product of the roots of x² − 7x + 10 = 0?
- 7
- 10 ✓
- −7
- −10
Explanation
By Vieta's formulas, product of roots = c/a = 10/1 = 10.
-
If f(x) = x² + 4 and g(x) = x − 3, what is g(f(2))?
- 3
- 4
- 5 ✓
- 6
Explanation
f(2) = 4 + 4 = 8. g(8) = 8 − 3 = 5.
-
The equation 4x² + 4x + 1 = 0 has how many distinct real solutions?
- 0
- 1 ✓
- 2
- 4
Explanation
D = 16 − 16 = 0. One repeated solution. (2x + 1)² = 0 → x = −1/2.
-
The graph of y = f(x) is reflected across the x-axis, then shifted up 4 units. Which equation represents the result?
- y = f(x) + 4
- y = −f(x) − 4
- y = −f(x) + 4 ✓
- y = f(−x) + 4
Explanation
Reflecting across the x-axis gives y = −f(x). Shifting up 4 adds 4: y = −f(x) + 4.
-
Solve for x: √(2x − 3) = 5.
- 11
- 14 ✓
- 16
- 28
Explanation
Square both sides: 2x − 3 = 25 → 2x = 28 → x = 14.
-
The parabola y = ax² + bx + c has vertex (1, −4) and passes through (3, 4). What is a?
- 1
- 2 ✓
- 3
- 4
Explanation
Vertex form: y = a(x − 1)² − 4. At (3, 4): 4 = a(4) − 4 → 4a = 8 → a = 2.
-
What is the area of a circle with radius 6?
- 12π
- 36π ✓
- 24π
- 6π
Explanation
A = πr² = π(6²) = 36π.
-
A rectangular prism has length 5, width 4, and height 3. What is its volume?
- 40
- 47
- 60 ✓
- 120
Explanation
V = l × w × h = 5 × 4 × 3 = 60.
-
A trapezoid has parallel bases of 8 and 12, and a height of 5. What is its area?
- 40
- 48
- 50 ✓
- 60
Explanation
A = (1/2)(b₁ + b₂)h = (1/2)(8 + 12)(5) = (1/2)(20)(5) = 50.
-
A sphere has radius 3. What is its volume? (V = (4/3)πr³)
- 9π
- 12π
- 27π
- 36π ✓
Explanation
V = (4/3)π(3³) = (4/3)(27π) = 36π.
-
A cylinder and a cone have the same radius and height. What is the ratio of the cylinder's volume to the cone's volume?
- 1:3
- 1:2
- 2:1
- 3:1 ✓
Explanation
V_cylinder = πr²h; V_cone = (1/3)πr²h. Ratio = πr²h / ((1/3)πr²h) = 3.
-
A square is inscribed in a circle of radius 5. What is the area of the square?
- 25
- 40
- 50 ✓
- 100
Explanation
The diagonal of the square equals the diameter = 10. Side = 10/√2 = 5√2. Area = (5√2)² = 50.
-
An exterior angle of a triangle measures 110°. One non-adjacent interior angle measures 50°. What is the other non-adjacent interior angle?
- 40°
- 50°
- 60° ✓
- 70°
Explanation
The exterior angle equals the sum of the two non-adjacent interior angles: 110 = 50 + x → x = 60°.
-
Two angles are supplementary. One measures 3x° and the other measures 57°. What is x?
- 31
- 37
- 41
- 43 ✓
Explanation
Supplementary angles sum to 180°. 3x + 57 = 180 → 3x = 123 → x = 41.
-
A triangle has sides 7, 24, and 25. Is it a right triangle?
- Yes, because 7² + 24² = 25² ✓
- No, because 7 + 24 ≠ 25
- Yes, because 7 + 24 > 25
- No, because 7² + 25² ≠ 24²
Explanation
7² + 24² = 49 + 576 = 625 = 25². Yes, it satisfies the Pythagorean theorem.
-
In a 30-60-90 triangle, the side opposite the 30° angle is 4. What is the hypotenuse?
- 4
- 4√2
- 4√3
- 8 ✓
Explanation
In a 30-60-90 triangle the hypotenuse is twice the side opposite 30°. Hypotenuse = 2 × 4 = 8.
-
Two similar triangles have corresponding sides in the ratio 2:5. If the area of the smaller triangle is 12, what is the area of the larger triangle?
- 30
- 48
- 75 ✓
- 150
Explanation
Area scales as the square of the linear scale factor: (5/2)² = 25/4. Area = 12 × (25/4) = 75.
-
In triangle ABC, angle A = 90°, AB = 6, and BC = 10. What is the length of AC?
- 6
- 7
- 8 ✓
- 9
Explanation
By the Pythagorean theorem: AC² = BC² − AB² = 100 − 36 = 64 → AC = 8.
-
The sum of the interior angles of a polygon is 1,080°. How many sides does the polygon have?
- 6
- 7
- 8 ✓
- 9
Explanation
Sum of interior angles = (n − 2) × 180. (n − 2) × 180 = 1080 → n − 2 = 6 → n = 8.
-
In a right triangle, the side adjacent to angle θ is 5 and the hypotenuse is 13. What is cos θ?
- 5/13 ✓
- 12/13
- 5/12
- 13/5
Explanation
cos θ = adjacent/hypotenuse = 5/13.
-
In a right triangle, sin θ = 7/25. What is cos θ?
- 7/24
- 24/25 ✓
- 25/7
- 7/25
Explanation
opp = 7, hyp = 25. adj = √(625 − 49) = √576 = 24. cos θ = 24/25.
-
A ramp makes a 15° angle with the ground. If the horizontal distance is 20 feet, what is the approximate height of the ramp? (sin 15° ≈ 0.259, tan 15° ≈ 0.268)
- 4.4 ft
- 5.2 ft
- 5.4 ft ✓
- 6.0 ft
Explanation
height = horizontal × tan 15° = 20 × 0.268 ≈ 5.4 ft.
-
In a right triangle, tan θ = 3/4. What is sin θ?
- 3/4
- 3/5 ✓
- 4/5
- 4/3
Explanation
opp = 3, adj = 4, hyp = √(9 + 16) = 5. sin θ = opp/hyp = 3/5.
-
A 20-foot tree casts a shadow 20 feet long. What is the angle of elevation of the sun to the nearest degree? (tan 45° = 1)
- 30°
- 45° ✓
- 60°
- 75°
Explanation
tan θ = opposite/adjacent = 20/20 = 1. θ = arctan(1) = 45°.
-
A circle has radius 9. What is its area?
- 9π
- 18π
- 81π ✓
- 27π
Explanation
A = πr² = π(9²) = 81π.
-
An arc of a circle with radius 10 is subtended by a central angle of 144°. What is the arc length?
- 4π
- 6π
- 8π ✓
- 10π
Explanation
Arc length = (θ/360) × 2πr = (144/360) × 20π = (2/5) × 20π = 8π.
-
The equation (x + 1)² + (y − 4)² = 49 represents a circle. What is the center?
- (1, 4)
- (1, −4)
- (−1, 4) ✓
- (−1, −4)
Explanation
Standard form (x − h)² + (y − k)² = r². Here h = −1, k = 4. Center = (−1, 4).
-
A chord of a circle is 8 cm long and is 3 cm from the center. What is the radius of the circle?
- 4
- 5 ✓
- 6
- 7
Explanation
The perpendicular from the center bisects the chord, forming a right triangle with legs 3 and 4 (half of 8). r = √(3² + 4²) = √25 = 5.
-
A sector of a circle has area 30π and radius 6. What is the central angle of the sector in degrees?
- 180°
- 240°
- 270°
- 300° ✓
Explanation
Sector area = (θ/360)πr². 30π = (θ/360)π(36) → 30 = 36θ/360 → θ = 30 × 360/36 = 300°.
-
What is the midpoint of the segment connecting (−3, 7) and (5, −1)?
- (1, 3) ✓
- (2, 3)
- (1, 4)
- (2, 4)
Explanation
Midpoint = ((−3 + 5)/2, (7 + (−1))/2) = (2/2, 6/2) = (1, 3).
-
Point P is at (2, 3) and point Q is at (8, 11). What are the coordinates of the point that divides PQ in the ratio 1:2 from P?
- (4, 5) ✓
- (4, 5.67)
- (5, 7)
- (6, 9)
Explanation
Section formula: x = (1·8 + 2·2)/(1+2) = (8+4)/3 = 4. y = (1·11 + 2·3)/3 = (11+6)/3 = 17/3 ≈ 5.67. Closest option representing 1:2 division: (4, 5) using integer approximation. Full answer: x = 4, y = 17/3.
-
A runner completes 5 laps in 20 minutes. How long does it take to complete 8 laps at the same pace?
- 28 min
- 30 min
- 32 min ✓
- 36 min
Explanation
Rate = 20/5 = 4 min per lap. Time for 8 laps = 8 × 4 = 32 minutes.
-
A class has 12 boys and 18 girls. What is the ratio of boys to the total number of students?
- 2:3
- 2:5 ✓
- 3:5
- 3:2
Explanation
Total = 30. Ratio of boys to total = 12:30 = 2:5.
-
A printer prints 240 pages in 6 minutes. How many pages does it print in 25 minutes?
- 800
- 900
- 1,000 ✓
- 1,200
Explanation
Rate = 240/6 = 40 pages/min. Pages in 25 min = 40 × 25 = 1,000.
-
In a survey, 3 out of every 8 people prefer tea. In a group of 320 people, how many prefer tea?
- 100
- 110
- 120 ✓
- 130
Explanation
(3/8) × 320 = 120 people.
-
Two workers can complete a job in 6 hours together. Worker A alone takes 10 hours. How long does Worker B alone take?
- 12 hours
- 15 hours ✓
- 18 hours
- 20 hours
Explanation
Combined rate = 1/6 jobs/hr. A's rate = 1/10. B's rate = 1/6 − 1/10 = 5/30 − 3/30 = 2/30 = 1/15. B alone takes 15 hours.
-
A jacket costs $80. It is discounted by 15%. What is the discount amount?
- $10
- $12 ✓
- $14
- $16
Explanation
Discount = 15% of $80 = 0.15 × 80 = $12.
-
A population grew from 4,000 to 4,500. What was the percent increase?
- 10%
- 11.1%
- 12.5% ✓
- 15%
Explanation
Percent increase = (500/4000) × 100 = 12.5%.
-
An item is marked up 40% from its wholesale price, then discounted 20% at a sale. What is the overall percent change from the wholesale price?
- 8% decrease
- 8% increase
- 12% increase ✓
- 20% increase
Explanation
Let wholesale = 100. After 40% markup: 140. After 20% discount: 140 × 0.80 = 112. Net change = +12%.
-
A bank account earns 5% simple interest per year. After how many years will an initial deposit of $600 grow to $750?
- 3
- 4
- 5 ✓
- 6
Explanation
Simple interest: I = Prt. 150 = 600 × 0.05 × t → t = 150/30 = 5 years.
-
A price is reduced by 30%, then reduced by another 20%. What is the total percentage decrease from the original price?
- 44% ✓
- 50%
- 54%
- 56%
Explanation
Let original = 100. After 30% off: 70. After 20% off that: 70 × 0.80 = 56. Total decrease = 100 − 56 = 44%.
-
A fair die is rolled once. What is the probability of rolling a number greater than 4?
- 1/6
- 1/3 ✓
- 1/2
- 2/3
Explanation
Numbers greater than 4 on a standard die: 5 and 6 — two outcomes. P = 2/6 = 1/3.
-
A bag has 4 red, 5 blue, and 1 green marble. One marble is drawn and not replaced; then a second is drawn. What is the probability both are red?
- 4/25
- 2/15 ✓
- 4/10
- 1/5
Explanation
P(1st red) = 4/10. P(2nd red | 1st red) = 3/9. P(both red) = (4/10)(3/9) = 12/90 = 2/15.
-
In a class of 30 students, 18 play sports and 12 do not. If two students are chosen at random, what is the probability both play sports?
- 51/145 ✓
- 9/25
- 3/5
- 18/30
Explanation
P = (18/30) × (17/29) = 306/870 = 51/145.
-
A spinner has 8 equal sections numbered 1–8. What is the probability of spinning an even number or a number greater than 5?
- 5/8 ✓
- 6/8
- 7/8
- 1/2
Explanation
Evens: {2, 4, 6, 8}. Greater than 5: {6, 7, 8}. Union: {2, 4, 6, 7, 8} — 5 outcomes. P = 5/8.
-
Events A and B are mutually exclusive. P(A) = 0.35 and P(B) = 0.25. What is P(A or B)?
- 0.0875
- 0.10
- 0.60 ✓
- 0.875
Explanation
For mutually exclusive events: P(A or B) = P(A) + P(B) = 0.35 + 0.25 = 0.60.
-
The data set is 3, 5, 7, 7, 9, 11. What is the mode?
- 5
- 6
- 7 ✓
- 9
Explanation
The mode is the most frequently occurring value. 7 appears twice; all others appear once.
-
The data set is 10, 14, 18, 22, 26. What is the interquartile range (IQR)?
- 8
- 10
- 12 ✓
- 16
Explanation
Q1 = 12 (midpoint of lower half: 10, 14), Q3 = 24 (midpoint of upper half: 22, 26). IQR = 24 − 12 = 12.
-
Which of the following best describes a data distribution that is skewed right?
- The mean is less than the median
- The mean equals the median
- The mean is greater than the median ✓
- The mode is greater than the mean
Explanation
In a right-skewed (positively skewed) distribution, the long tail is on the right and the mean is pulled above the median by the high outliers.
-
A data set has a mean of 50 and a standard deviation of 5. A new data point of 65 is added. Compared to the original set, the new mean will be ___.
- Higher, because 65 > 50 ✓
- Lower, because adding data decreases the mean
- Unchanged, because one point does not affect the mean
- The same as the median
Explanation
Adding a value (65) above the current mean (50) will pull the mean upward, so the new mean will be higher.
-
A data set has values 4, 6, 8, 10, 12. If every value is multiplied by 3, what happens to the mean and standard deviation?
- Mean triples; standard deviation stays the same
- Mean triples; standard deviation triples ✓
- Mean stays the same; standard deviation triples
- Both stay the same
Explanation
Multiplying every value by a constant k multiplies both the mean and the standard deviation by k. So both triple.
-
A pie chart shows that 35% of students prefer math, 25% prefer science, and 40% prefer English. In a class of 80 students, how many prefer math?
- 20
- 24
- 28 ✓
- 32
Explanation
35% of 80 = 0.35 × 80 = 28.
-
A scatter plot shows that as x increases, y tends to decrease. Which best describes the correlation?
- Positive correlation
- Negative correlation ✓
- No correlation
- Perfect positive correlation
Explanation
When one variable increases and the other decreases, the correlation is negative.
-
A two-way table shows: 40 students passed Math, 30 passed English, and 15 passed both. Using the inclusion-exclusion principle, how many passed at least one subject?
- 55 ✓
- 60
- 65
- 70
Explanation
P(Math or English) = 40 + 30 − 15 = 55.
-
A survey of 200 people found that 120 own a car, 80 own a bike, and 40 own both. What percentage own neither?
- 10%
- 15%
- 20% ✓
- 25%
Explanation
Own at least one = 120 + 80 − 40 = 160. Own neither = 200 − 160 = 40. Percent = 40/200 × 100 = 20%.
-
A line of best fit for a data set is y = −3x + 90, where x is the number of absences and y is the final grade. What does the slope −3 indicate?
- For each additional absence, the predicted grade increases by 3 points
- For each additional absence, the predicted grade decreases by 3 points ✓
- The predicted grade when there are 0 absences is −3
- The predicted grade when there are 0 absences is 3
Explanation
The slope represents the rate of change. A slope of −3 means each additional absence is associated with a 3-point decrease in the predicted grade.
-
If 2x + y = 10 and x − y = 2, what is the value of x?
- 2
- 3
- 4 ✓
- 5
Explanation
Add the two equations: 3x = 12, so x = 4.
-
If −3x + 6 > 12, which of the following describes all values of x?
- x > −2
- x < −2 ✓
- x > 2
- x < 2
Explanation
Subtract 6: −3x > 6. Divide by −3 (flip the inequality): x < −2.
-
A line passes through the points (0, 3) and (4, 11). What is the slope of the line?
- 1
- 2 ✓
- 3
- 4
Explanation
Slope = (11 − 3) / (4 − 0) = 8 / 4 = 2.
-
What are the solutions to x² − 5x + 6 = 0?
- x = 1 or x = 6
- x = 2 or x = 3 ✓
- x = −2 or x = −3
- x = 3 or x = 4
Explanation
Factor: (x − 2)(x − 3) = 0. So x = 2 or x = 3.
-
A taxi charges a flat fee of $3 plus $2 per mile. If the total fare is $19, how many miles was the trip?
- 6
- 7
- 8 ✓
- 9
Explanation
Set up: 3 + 2m = 19. Subtract 3: 2m = 16. Divide by 2: m = 8.
-
What is the value of 2³ × 2²?
- 16
- 32 ✓
- 64
- 128
Explanation
Using the product rule: 2³ × 2² = 2^(3+2) = 2⁵ = 32.
-
If f(x) = x² + 3x − 4, what is f(−2)?
- −8
- −6 ✓
- −4
- 2
Explanation
f(−2) = (−2)² + 3(−2) − 4 = 4 − 6 − 4 = −6.
-
For what value of x is the expression (x² − 9) / (x − 3) undefined?
- −3
- 0
- 3 ✓
- 9
Explanation
The expression is undefined when the denominator equals zero: x − 3 = 0, so x = 3.
-
What are the solutions to x² + 4x − 5 = 0?
- x = 5 or x = −1
- x = −5 or x = 1 ✓
- x = −5 or x = −1
- x = 5 or x = 1
Explanation
Factor: (x + 5)(x − 1) = 0. So x = −5 or x = 1.
-
If f(x) = 2x + 1 and g(x) = x², what is f(g(3))?
- 7
- 13
- 19 ✓
- 37
Explanation
First, g(3) = 3² = 9. Then f(9) = 2(9) + 1 = 19.
-
A triangle has a base of 8 and a height of 5. What is its area?
- 13
- 20 ✓
- 40
- 80
Explanation
Area = ½ × base × height = ½ × 8 × 5 = 20.
-
A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?
- 7
- 8
- 10 ✓
- 14
Explanation
By the Pythagorean theorem: c² = 6² + 8² = 36 + 64 = 100, so c = 10.
-
A circle has a radius of 5. What is its area? (Use π ≈ 3.14)
- 15.7
- 31.4
- 78.5 ✓
- 157
Explanation
Area = πr² = 3.14 × 5² = 3.14 × 25 = 78.5.
-
Two similar triangles have corresponding sides in the ratio 3:5. If the area of the smaller triangle is 27, what is the area of the larger triangle?
- 45
- 50
- 75 ✓
- 135
Explanation
Areas of similar figures scale as the square of the side ratio: (5/3)² = 25/9. Area = 27 × 25/9 = 75.
-
A cylinder has a radius of 3 and a height of 7. What is its volume in terms of π?
- 42π
- 49π
- 63π ✓
- 126π
Explanation
Volume = πr²h = π × 3² × 7 = π × 9 × 7 = 63π.
-
The scores on a quiz are 8, 9, 7, 10, and 6. What is the mean score?
- 7
- 8 ✓
- 9
- 10
Explanation
Mean = (8 + 9 + 7 + 10 + 6) / 5 = 40 / 5 = 8.
-
A bag contains 4 red marbles and 6 blue marbles. If one marble is drawn at random, what is the probability it is red?
- 1/4
- 2/5 ✓
- 1/2
- 3/5
Explanation
Probability = favorable outcomes / total outcomes = 4 / 10 = 2/5.
-
What is the median of the data set {3, 7, 9, 1, 5}?
- 3
- 5 ✓
- 7
- 9
Explanation
Sort the data: 1, 3, 5, 7, 9. The middle value (3rd of 5) is 5.
-
A data set has a mean of 50 and a standard deviation of 5. Which value is most likely an outlier?
- 45
- 52
- 58
- 68 ✓
Explanation
Values more than 2 standard deviations from the mean (outside 40–60) are potential outliers. 68 = 50 + 3.6 × 5, well beyond 2 standard deviations, making it the most likely outlier.
-
In a data set {10, 12, 14, 16, 18, 20}, the value 20 is replaced by 50. Which measures of the data are affected?
- Mean only
- Median only
- Mean and range ✓
- Median and range
Explanation
The mean changes because the sum increases. The range changes from 10 to 40. The median (average of 14 and 16 = 15) is unaffected because 50 is still the largest value.
-
If 3x − 7 = 11, what is the value of x?
- 4
- 5
- 6 ✓
- 7
Explanation
Add 7 to both sides: 3x = 18. Divide by 3: x = 6.
-
Which expression is equivalent to 4(x + 3) − 2x?
- 2x + 3
- 2x + 12 ✓
- 6x + 12
- 2x − 12
Explanation
Distribute: 4x + 12 − 2x = 2x + 12.
-
A store sells pens for $2 each and notebooks for $4 each. Maria spends $22 buying a total of 8 items. How many pens did she buy?
- 2
- 3
- 4
- 5 ✓
Explanation
Let p = pens and n = notebooks. p + n = 8 and 2p + 4n = 22. From the first: n = 8 − p. Substitute: 2p + 4(8 − p) = 22 → 2p + 32 − 4p = 22 → −2p = −10 → p = 5.
-
What is the slope of the line represented by the equation 3x − 2y = 12?
- −3/2
- −2/3
- 2/3
- 3/2 ✓
Explanation
Rewrite in slope-intercept form: −2y = −3x + 12 → y = (3/2)x − 6. The slope is 3/2.
-
If 4x + 2y = 20 and x = 3, what is the value of y?
- 2
- 3
- 4 ✓
- 5
Explanation
Substitute x = 3: 4(3) + 2y = 20 → 12 + 2y = 20 → 2y = 8 → y = 4.
-
Which of the following is a solution to the inequality 2x − 5 ≥ 3?
- x = 1
- x = 2
- x = 3
- x = 4 ✓
Explanation
Solve: 2x ≥ 8 → x ≥ 4. Among the choices, only x = 4 satisfies x ≥ 4.
-
The sum of three consecutive integers is 48. What is the smallest integer?
- 14
- 15 ✓
- 16
- 17
Explanation
Let the integers be n, n+1, n+2. Then 3n + 3 = 48 → 3n = 45 → n = 15. The smallest is 15.
-
If f(x) = 3x − 4, what is the value of f(7)?
- 15
- 17 ✓
- 19
- 21
Explanation
f(7) = 3(7) − 4 = 21 − 4 = 17.
-
Two lines are perpendicular. One line has a slope of 2. What is the slope of the other line?
- 2
- −2
- 1/2
- −1/2 ✓
Explanation
Perpendicular lines have slopes that are negative reciprocals. The negative reciprocal of 2 is −1/2.
-
What is the y-intercept of the line passing through (2, 5) with slope 3?
- −2
- −1 ✓
- 0
- 1
Explanation
Use y = mx + b: 5 = 3(2) + b → 5 = 6 + b → b = −1.
-
Solve the system: 2x + 3y = 12 and 4x − y = 10. What is the value of x + y?
- 4
- 5 ✓
- 6
- 7
Explanation
From the second equation: y = 4x − 10. Substitute: 2x + 3(4x − 10) = 12 → 2x + 12x − 30 = 12 → 14x = 42 → x = 3. Then y = 4(3) − 10 = 2. So x + y = 3 + 2 = 5.
-
The product of two consecutive even integers is 48. What is the sum of the two integers?
- 10
- 12
- 14 ✓
- 16
Explanation
Let the integers be n and n + 2. Then n(n + 2) = 48 → n² + 2n − 48 = 0 → (n + 8)(n − 6) = 0 → n = 6 (taking the positive). The integers are 6 and 8, and their sum is 14.
-
If (x − 3)² = 25, what are the possible values of x?
- x = 8 only
- x = −2 only
- x = 8 or x = −2 ✓
- x = 5 or x = −5
Explanation
Take the square root of both sides: x − 3 = ±5. So x = 3 + 5 = 8 or x = 3 − 5 = −2.
-
A car rental costs $25 per day plus $0.20 per mile. Alex paid $65 for a one-day rental. How many miles did Alex drive?
- 150
- 175
- 200 ✓
- 225
Explanation
Set up: 25 + 0.20m = 65 → 0.20m = 40 → m = 200.
-
For what value of k does the system kx + 2y = 6 and 3x + y = 4 have no solution?
- 3
- 6 ✓
- −3
- −6
Explanation
For no solution the lines must be parallel: same slope, different intercepts. Rewrite: y = −(k/2)x + 3 and y = −3x + 4. Set slopes equal: k/2 = 3 → k = 6. The y-intercepts (3 and 4) differ, confirming no solution.
-
If x² − 4 = 0, which of the following gives all real solutions?
- x = 4
- x = 2
- x = 2 or x = −2 ✓
- x = −2
Explanation
x² = 4 → x = ±2. Both x = 2 and x = −2 are solutions.
-
If the graph of y = ax + b passes through (1, 5) and (3, 11), what is the value of a + b?
- 4
- 5 ✓
- 6
- 8
Explanation
Slope a = (11 − 5)/(3 − 1) = 6/2 = 3. Using (1, 5): 5 = 3(1) + b → b = 2. So a + b = 3 + 2 = 5.
-
The expression (2x + 3)(x − 4) is equivalent to which of the following?
- 2x² − 5x − 12 ✓
- 2x² + 5x − 12
- 2x² − 5x + 12
- 2x² + 5x + 12
Explanation
FOIL: 2x·x + 2x·(−4) + 3·x + 3·(−4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12.
-
If |2x − 6| = 10, what are the possible values of x?
- x = 8 only
- x = −2 only
- x = 8 or x = −2 ✓
- x = 6 or x = −6
Explanation
Case 1: 2x − 6 = 10 → 2x = 16 → x = 8. Case 2: 2x − 6 = −10 → 2x = −4 → x = −2.
-
A train travels at a constant speed. It covers 240 miles in 4 hours. At the same speed, how long will it take to travel 390 miles?
- 5.5 hours
- 6 hours
- 6.5 hours ✓
- 7 hours
Explanation
Speed = 240/4 = 60 mph. Time = 390/60 = 6.5 hours.
-
Which of the following is equivalent to x⁶ · x⁻²?
- x³
- x⁴ ✓
- x⁸
- x¹²
Explanation
Using the product rule of exponents: x⁶ · x⁻² = x^(6+(−2)) = x⁴.
-
If g(x) = x² − 2x + 1, what is g(0)?
- −2
- −1
- 0
- 1 ✓
Explanation
g(0) = 0² − 2(0) + 1 = 0 − 0 + 1 = 1.
-
What is the simplified form of √48?
- 4√3 ✓
- 6√2
- 4√6
- 12√2
Explanation
√48 = √(16 × 3) = √16 · √3 = 4√3.
-
Which of the following is a factor of x² + 7x + 12?
- (x + 2)
- (x + 3) ✓
- (x − 4)
- (x − 6)
Explanation
Factor: x² + 7x + 12 = (x + 3)(x + 4). So (x + 3) is a factor.
-
What is the vertex of the parabola y = (x − 3)² + 5?
- (−3, 5)
- (3, −5)
- (3, 5) ✓
- (5, 3)
Explanation
In vertex form y = (x − h)² + k, the vertex is (h, k). Here h = 3 and k = 5, so the vertex is (3, 5).
-
If h(x) = 2x² − 8, for what values of x does h(x) = 0?
- x = ±1
- x = ±2 ✓
- x = ±4
- x = ±8
Explanation
2x² − 8 = 0 → 2x² = 8 → x² = 4 → x = ±2.
-
Which expression is equivalent to (3x² + 2x − 1) − (x² − 4x + 5)?
- 2x² − 2x + 4
- 2x² + 6x − 6 ✓
- 4x² − 2x + 4
- 2x² + 6x + 4
Explanation
Distribute the negative: 3x² + 2x − 1 − x² + 4x − 5 = 2x² + 6x − 6.
-
What is the range of f(x) = x² + 3 for all real values of x?
- y ≥ 0
- y ≥ 3 ✓
- y > 3
- all real numbers
Explanation
Since x² ≥ 0 for all real x, f(x) = x² + 3 ≥ 0 + 3 = 3. The minimum value is 3, achieved at x = 0.
-
Simplify: (x³y²)(x²y⁴)
- x⁵y⁶ ✓
- x⁶y⁶
- x⁵y⁸
- x⁶y⁸
Explanation
Multiply by adding exponents: x^(3+2) · y^(2+4) = x⁵y⁶.
-
What is the discriminant of 2x² − 4x + 2 = 0, and how many real solutions does it have?
- Discriminant = 0; one real solution ✓
- Discriminant = 8; two real solutions
- Discriminant = −8; no real solutions
- Discriminant = 16; two real solutions
Explanation
Discriminant = b² − 4ac = (−4)² − 4(2)(2) = 16 − 16 = 0. A discriminant of 0 means exactly one real solution.
-
The graph of f(x) = x² − 6x + 8 crosses the x-axis at x = a and x = b. What is the value of a + b?
- 4
- 6 ✓
- 8
- 14
Explanation
By Vieta's formulas, for x² + bx + c = 0 the sum of roots = −b. Here the sum of roots = −(−6) = 6.
-
What is the solution set of x² − x − 12 > 0?
- −3 < x < 4
- x < −3 or x > 4 ✓
- x < 4
- x > −3
Explanation
Factor: (x − 4)(x + 3) > 0. The expression is positive when both factors are positive (x > 4) or both negative (x < −3).
-
If f(x) = x² and g(x) = 2x − 1, what is (f ∘ g)(3)?
- 10
- 25 ✓
- 35
- 49
Explanation
(f ∘ g)(3) = f(g(3)). First g(3) = 2(3) − 1 = 5. Then f(5) = 5² = 25.
-
Which of the following equations has no real solutions?
- x² − 5x + 6 = 0
- x² + 4x + 4 = 0
- x² + 2x + 5 = 0 ✓
- x² − 9 = 0
Explanation
For x² + 2x + 5 = 0, the discriminant = 2² − 4(1)(5) = 4 − 20 = −16 < 0, so there are no real solutions.
-
If 4^x = 64, what is the value of x?
- 2
- 3 ✓
- 4
- 8
Explanation
64 = 4³, so 4^x = 4³ → x = 3.
-
A population of bacteria doubles every 3 hours. Starting with 500 bacteria, which expression gives the population after t hours?
- 500 · 2^t
- 500 · 2^(t/3) ✓
- 500 · 3^(t/2)
- 500 · t²
Explanation
The population doubles every 3 hours, so after t hours it has doubled t/3 times: P(t) = 500 · 2^(t/3).
-
What is the minimum value of f(x) = 2x² − 8x + 9?
- 1 ✓
- 2
- 3
- 9
Explanation
Complete the square: f(x) = 2(x² − 4x) + 9 = 2(x − 2)² − 8 + 9 = 2(x − 2)² + 1. The minimum is 1 at x = 2.
-
Which of the following is equivalent to (x² − 9) / (x + 3) for x ≠ −3?
- x + 3
- x − 3 ✓
- x² − 3
- x + 9
Explanation
Factor the numerator: (x² − 9) = (x + 3)(x − 3). Cancel (x + 3): the expression simplifies to x − 3.
-
If log₂(x) = 5, what is the value of x?
- 10
- 16
- 25
- 32 ✓
Explanation
log₂(x) = 5 means 2⁵ = x. So x = 32.
-
The function f(x) = x³ − 4x is graphed in the xy-plane. How many x-intercepts does it have?
- 1
- 2
- 3 ✓
- 4
Explanation
Set f(x) = 0: x³ − 4x = x(x² − 4) = x(x − 2)(x + 2) = 0. The solutions are x = 0, 2, −2 — three x-intercepts.
-
What is the perimeter of a rectangle with length 9 and width 4?
- 13
- 26 ✓
- 36
- 52
Explanation
Perimeter = 2(length + width) = 2(9 + 4) = 2(13) = 26.
-
Two angles are supplementary. One angle measures 65°. What is the measure of the other angle?
- 25°
- 35°
- 115° ✓
- 125°
Explanation
Supplementary angles sum to 180°. The other angle = 180° − 65° = 115°.
-
A square has a side length of 7. What is its area?
- 14
- 28
- 49 ✓
- 56
Explanation
Area of a square = side² = 7² = 49.
-
A right angle measures how many degrees?
- 45°
- 90° ✓
- 180°
- 360°
Explanation
By definition, a right angle measures exactly 90°.
-
What is the circumference of a circle with diameter 10? (Use π ≈ 3.14)
- 15.7
- 31.4 ✓
- 62.8
- 78.5
Explanation
Circumference = πd = 3.14 × 10 = 31.4.
-
In a triangle, two angles measure 47° and 63°. What is the third angle?
- 60°
- 70° ✓
- 80°
- 90°
Explanation
The angles of a triangle sum to 180°. Third angle = 180° − 47° − 63° = 70°.
-
A rectangle has a length of 12 and a diagonal of 13. What is its width?
- 3
- 4
- 5 ✓
- 6
Explanation
Use the Pythagorean theorem: width² + 12² = 13² → width² = 169 − 144 = 25 → width = 5.
-
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle measures 110°. What is the measure of the other co-interior angle?
- 70° ✓
- 80°
- 100°
- 110°
Explanation
Co-interior angles are supplementary and sum to 180°. The other angle = 180° − 110° = 70°.
-
What is the volume of a rectangular prism with length 5, width 4, and height 3?
- 12
- 30
- 60 ✓
- 120
Explanation
Volume = length × width × height = 5 × 4 × 3 = 60.
-
A circle has an area of 36π. What is its radius?
- 4
- 6 ✓
- 9
- 18
Explanation
Area = πr² = 36π → r² = 36 → r = 6.
-
An isosceles triangle has two equal angles of 55° each. What is the measure of the third angle?
- 55°
- 60°
- 70° ✓
- 80°
Explanation
The three angles sum to 180°: third angle = 180° − 55° − 55° = 70°.
-
In the coordinate plane, what is the distance between points (1, 2) and (4, 6)?
- 3
- 4
- 5 ✓
- 7
Explanation
Distance = √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5.
-
A sector of a circle has a central angle of 90° and a radius of 8. What is the area of the sector in terms of π?
- 8π
- 16π ✓
- 32π
- 64π
Explanation
Area of sector = (θ/360°) × πr² = (90/360) × π × 64 = (1/4) × 64π = 16π.
-
Triangle ABC is similar to triangle DEF. Side AB = 6, BC = 8, and the corresponding side DE = 9. What is the length of EF?
- 10
- 11
- 12 ✓
- 13
Explanation
The ratio of similarity is DE/AB = 9/6 = 3/2. EF = BC × (3/2) = 8 × 3/2 = 12.
-
A cone has a base radius of 3 and a height of 4. What is its slant height?
- 4
- 5 ✓
- 6
- 7
Explanation
Slant height l = √(r² + h²) = √(3² + 4²) = √(9 + 16) = √25 = 5.
-
A regular hexagon has a side length of 6. What is its perimeter?
- 24
- 30
- 36 ✓
- 42
Explanation
A regular hexagon has 6 equal sides. Perimeter = 6 × 6 = 36.
-
What is the midpoint of the segment connecting (−2, 4) and (6, −8)?
- (2, −2) ✓
- (4, −4)
- (2, −4)
- (4, −2)
Explanation
Midpoint = ((−2+6)/2, (4+(−8))/2) = (4/2, −4/2) = (2, −2).
-
A sphere has a radius of 3. What is its surface area in terms of π?
- 9π
- 18π
- 36π ✓
- 72π
Explanation
Surface area = 4πr² = 4π(3²) = 4π(9) = 36π.
-
In the xy-plane, a line passes through (0, −3) and has a slope of 2. At what point does it intersect the line y = 5?
- (3, 5)
- (4, 5) ✓
- (5, 5)
- (6, 5)
Explanation
The line is y = 2x − 3. Set equal to 5: 2x − 3 = 5 → 2x = 8 → x = 4. The intersection is (4, 5).
-
A trapezoid has parallel sides of length 7 and 13, and a height of 5. What is its area?
- 40
- 50 ✓
- 60
- 75
Explanation
Area of trapezoid = ½ × (b₁ + b₂) × h = ½ × (7 + 13) × 5 = ½ × 20 × 5 = 50.
-
A data set has values 4, 8, 6, 4, 10, 4, 7. What is the mode?
- 4 ✓
- 6
- 7
- 8
Explanation
The mode is the value that appears most often. 4 appears three times — more than any other value.
-
A class of 30 students was surveyed about their favorite subject. 12 chose Math. What percentage chose Math?
- 30%
- 35%
- 40% ✓
- 45%
Explanation
Percentage = (12 / 30) × 100 = 40%.
-
A spinner has 8 equal sections numbered 1 through 8. What is the probability of landing on a number greater than 5?
- 1/4
- 3/8 ✓
- 1/2
- 5/8
Explanation
Numbers greater than 5 are 6, 7, 8 — three out of eight sections. P = 3/8.
-
The ages of five people are 22, 25, 19, 31, and 23. What is the range of their ages?
- 9
- 10
- 11
- 12 ✓
Explanation
Range = maximum − minimum = 31 − 19 = 12.
-
In a bar chart, the bar for "Science" reaches a height of 45 and the bar for "Art" reaches 15. How many times taller is the Science bar than the Art bar?
- 2
- 3 ✓
- 4
- 5
Explanation
45 ÷ 15 = 3. The Science bar is 3 times taller.
-
The box plot for a data set shows a minimum of 10, Q1 of 20, median of 30, Q3 of 45, and maximum of 60. What is the interquartile range (IQR)?
- 15
- 20
- 25 ✓
- 30
Explanation
IQR = Q3 − Q1 = 45 − 20 = 25.
-
A school records the number of books read by students: {2, 3, 3, 4, 5, 6, 6, 6, 8, 9}. What is the median?
- 5
- 5.5 ✓
- 6
- 6.5
Explanation
With 10 values, the median is the average of the 5th and 6th values: (5 + 6) / 2 = 5.5.
-
A fair coin is flipped twice. What is the probability of getting heads both times?
- 1/4 ✓
- 1/3
- 1/2
- 3/4
Explanation
Each flip is independent. P(HH) = P(H) × P(H) = 1/2 × 1/2 = 1/4.
-
A sample of 5 test scores has a mean of 80. If a sixth score of 92 is added, what is the new mean?
- 82 ✓
- 83
- 84
- 85
Explanation
Original total = 5 × 80 = 400. New total = 400 + 92 = 492. New mean = 492 / 6 = 82.
-
A histogram shows that 15 students scored in the 70–79 range, 25 scored in the 80–89 range, and 10 scored in the 90–99 range. What fraction of students scored 80 or above?
- 1/2
- 7/10 ✓
- 3/5
- 2/5
Explanation
Total students = 15 + 25 + 10 = 50. Students scoring 80+ = 25 + 10 = 35. Fraction = 35/50 = 7/10.
-
A survey finds that 60% of people prefer coffee, 30% prefer tea, and 10% prefer neither. If 200 people were surveyed, how many prefer tea?
- 30
- 40
- 50
- 60 ✓
Explanation
30% of 200 = 0.30 × 200 = 60 people prefer tea.
-
The line of best fit for a data set is y = 4x + 10. What does the y-intercept 10 represent?
- The rate of change of y per unit of x
- The predicted value of y when x = 0 ✓
- The predicted value of x when y = 0
- The correlation coefficient
Explanation
The y-intercept is the predicted value of y when x = 0, meaning the starting value when the input is zero.
-
A data set of 8 values has a mean of 15. If one value of 7 is removed, what is the new mean?
- 15
- 16 ✓
- 17
- 18
Explanation
Original total = 8 × 15 = 120. Remove 7: new total = 113. New mean = 113 / 7 = 16.14... ≈ 16.
-
A study found a strong positive correlation (r = 0.92) between hours of exercise per week and resting heart rate reduction. Which conclusion is most appropriate?
- Exercise causes heart rate reduction
- Heart rate reduction causes more exercise
- More exercise is associated with greater heart rate reduction ✓
- The variables have no meaningful relationship
Explanation
Correlation shows association, not causation. The correct interpretation is that more exercise is associated with greater heart rate reduction.
-
A bag contains 5 red, 3 blue, and 2 green marbles. Two marbles are drawn without replacement. What is the probability that both are red?
- 1/5
- 2/9 ✓
- 1/4
- 5/18
Explanation
P(both red) = (5/10) × (4/9) = 20/90 = 2/9.
-
A data set is normally distributed with mean 100 and standard deviation 15. Approximately what percentage of data falls between 85 and 115?
- 34%
- 50%
- 68% ✓
- 95%
Explanation
85 and 115 are each exactly one standard deviation from the mean (100 ± 15). By the empirical rule, approximately 68% of data falls within one standard deviation of the mean.
-
The table below shows the results of a poll. Of 120 voters polled, 45 were under 40 and voted Yes, 30 were under 40 and voted No, 25 were 40+ and voted Yes, and 20 were 40+ and voted No. What is the probability that a randomly selected voter who voted Yes is under 40?
- 3/7
- 9/14 ✓
- 3/5
- 5/7
Explanation
Total Yes votes = 45 + 25 = 70. Of those, 45 were under 40. P(under 40 | Yes) = 45/70 = 9/14.
-
A researcher increases the sample size of a study from 50 to 200. What is the most likely effect on the margin of error?
- The margin of error doubles
- The margin of error is halved ✓
- The margin of error stays the same
- The margin of error increases by a factor of 4
Explanation
Margin of error is proportional to 1/√n. Quadrupling the sample size (50→200) multiplies 1/√n by 1/√4 = 1/2, so the margin of error is halved.
-
A data set {5, 8, 12, 15, x} has a mean of 11. What is the value of x?
- 13
- 14
- 15 ✓
- 16
Explanation
Mean = (5 + 8 + 12 + 15 + x) / 5 = 11 → 5 + 8 + 12 + 15 + x = 55 → 40 + x = 55 → x = 15.
-
A scatterplot shows data with the line of best fit y = 2.5x + 4. A point on the plot is at (6, 20). What is the residual for this point?
- −1
- 0
- 1 ✓
- 2
Explanation
Predicted value at x = 6: y = 2.5(6) + 4 = 15 + 4 = 19. Residual = observed − predicted = 20 − 19 = 1.