题库
题库.
按主题和难度筛选，然后逐题练习。每道题都有详细解析。
没有符合这些筛选条件的题目。
-
If 2x + 4 = 14, what is x?
- 3
- 4
- 5 ✓
- 6
解析
Subtract 4 from both sides: 2x = 10. Divide by 2: x = 5.
-
If x/3 − 2 = 4, what is x?
- 6
- 12
- 18 ✓
- 24
解析
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
解析
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
解析
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
解析
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
解析
Divide both sides by 4: x = 5.
-
If 7 − 3x = −11, what is x?
- 3
- 4
- 5
- 6 ✓
解析
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
解析
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
解析
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
解析
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
解析
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 ✓
解析
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
解析
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
解析
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
解析
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
解析
The slope a = Δy/Δx = 10/4 = 5/2.
-
If x + y = 8 and x − y = 2, what is x?
- 3
- 4
- 5 ✓
- 6
解析
Add the equations: 2x = 10 → x = 5.
-
Given 2x + y = 7 and x − y = 2, what is x + y?
- 3
- 4 ✓
- 5
- 6
解析
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
解析
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 ✓
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
(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
解析
−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 ✓
解析
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 ✓
解析
"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
解析
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
解析
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
解析
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
解析
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
解析
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)
解析
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
解析
−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
解析
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
解析
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 ✓
解析
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
解析
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
解析
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)
解析
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
解析
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)
解析
(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
解析
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
解析
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
解析
(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
解析
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
解析
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
解析
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
解析
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)
解析
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
解析
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
解析
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
解析
(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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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 ✓
解析
2⁵ = 32.
-
The function f(x) = a · 2^x passes through (0, 4) and (3, 32). What is a?
- 2
- 3
- 4 ✓
- 8
解析
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
解析
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
解析
g(2) = 2(2) + 1 = 5. f(5) = 25 − 4 = 21.
-
If f(x) = 3x − 2, what is f(4)?
- 8
- 9
- 10 ✓
- 11
解析
f(4) = 3(4) − 2 = 12 − 2 = 10.
-
If g(x) = x² − 3x + 2, what is g(−1)?
- 4
- 5
- 6 ✓
- 7
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
A = length × width = 8 × 5 = 40.
-
What is the area of a triangle with base 10 and height 6?
- 16
- 30 ✓
- 60
- 80
解析
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
解析
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
解析
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
解析
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°
解析
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°
解析
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
解析
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°
解析
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°
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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)
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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%
解析
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%
解析
78/120 × 100 = 65%.
-
After a 20% increase, a price is $84. What was the original price?
- $60
- $65
- $70 ✓
- $75
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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 ✓
解析
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
解析
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 ✓
解析
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
解析
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 ✓
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
5x > 25 → x > 5.
-
A line passes through (0, 6) and (3, 0). What is the slope of the line?
- −3
- −2 ✓
- 2
- 3
解析
Slope = (0 − 6)/(3 − 0) = −6/3 = −2.
-
Given x + 2y = 10 and x = 4, what is y?
- 2
- 3 ✓
- 4
- 5
解析
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
解析
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
解析
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
解析
−4x + 20 = 0 → 4x = 20 → x = 5.
-
If 2x + 3y = 18 and x = 3, what is y?
- 3
- 4 ✓
- 5
- 6
解析
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
解析
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
解析
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
解析
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
解析
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 ✓
解析
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 ✓
解析
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
解析
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
解析
−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
解析
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
解析
−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
解析
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
解析
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
解析
6x − 12 − 2x − 2 = 4x − 14.
-
Which is equivalent to (5x²)(−2x⁴)?
- −10x⁶ ✓
- −10x⁸
- 3x⁶
- 10x⁶
解析
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)
解析
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
解析
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
解析
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
解析
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)
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
(x − 5)(x + 2) = x² + 2x − 5x − 10 = x² − 3x − 10 = 0.
-
Simplify: (x³ · x²)/x⁴
- x ✓
- x²
- x³
- x⁴
解析
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
解析
(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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
A = πr² = π(6²) = 36π.
-
A rectangular prism has length 5, width 4, and height 3. What is its volume?
- 40
- 47
- 60 ✓
- 120
解析
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
解析
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 ✓
解析
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 ✓
解析
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
解析
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°
解析
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 ✓
解析
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
解析
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 ✓
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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°
解析
tan θ = opposite/adjacent = 20/20 = 1. θ = arctan(1) = 45°.
-
A circle has radius 9. What is its area?
- 9π
- 18π
- 81π ✓
- 27π
解析
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
解析
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)
解析
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
解析
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° ✓
解析
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)
解析
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)
解析
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
解析
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
解析
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
解析
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
解析
(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
解析
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
解析
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%
解析
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
解析
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
解析
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%
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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
解析
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%
解析
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
解析
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.