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Lọc theo chủ đề và độ khó, sau đó làm từng câu một. Giải thích chi tiết cho mọi câu trả lời.
Không có câu hỏi phù hợp với bộ lọc này.
-
If 2x + 4 = 14, what is x?
- 3
- 4
- 5 ✓
- 6
Giải thích
Subtract 4 from both sides: 2x = 10. Divide by 2: x = 5.
-
If x/3 − 2 = 4, what is x?
- 6
- 12
- 18 ✓
- 24
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
Divide both sides by 4: x = 5.
-
If 7 − 3x = −11, what is x?
- 3
- 4
- 5
- 6 ✓
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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 ✓
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
The slope a = Δy/Δx = 10/4 = 5/2.
-
If x + y = 8 and x − y = 2, what is x?
- 3
- 4
- 5 ✓
- 6
Giải thích
Add the equations: 2x = 10 → x = 5.
-
Given 2x + y = 7 and x − y = 2, what is x + y?
- 3
- 4 ✓
- 5
- 6
Giải thích
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
Giải thích
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 ✓
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
(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
Giải thích
−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 ✓
Giải thích
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 ✓
Giải thích
"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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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)
Giải thích
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
Giải thích
−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
Giải thích
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
Giải thích
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 ✓
Giải thích
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
Giải thích
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
Giải thích
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)
Giải thích
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
Giải thích
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)
Giải thích
(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
Giải thích
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
Giải thích
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
Giải thích
(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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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)
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
(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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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 ✓
Giải thích
2⁵ = 32.
-
The function f(x) = a · 2^x passes through (0, 4) and (3, 32). What is a?
- 2
- 3
- 4 ✓
- 8
Giải thích
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
Giải thích
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
Giải thích
g(2) = 2(2) + 1 = 5. f(5) = 25 − 4 = 21.
-
If f(x) = 3x − 2, what is f(4)?
- 8
- 9
- 10 ✓
- 11
Giải thích
f(4) = 3(4) − 2 = 12 − 2 = 10.
-
If g(x) = x² − 3x + 2, what is g(−1)?
- 4
- 5
- 6 ✓
- 7
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
A = length × width = 8 × 5 = 40.
-
What is the area of a triangle with base 10 and height 6?
- 16
- 30 ✓
- 60
- 80
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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°
Giải thích
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°
Giải thích
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
Giải thích
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°
Giải thích
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°
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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)
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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%
Giải thích
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%
Giải thích
78/120 × 100 = 65%.
-
After a 20% increase, a price is $84. What was the original price?
- $60
- $65
- $70 ✓
- $75
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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 ✓
Giải thích
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
Giải thích
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 ✓
Giải thích
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
Giải thích
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 ✓
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
5x > 25 → x > 5.
-
A line passes through (0, 6) and (3, 0). What is the slope of the line?
- −3
- −2 ✓
- 2
- 3
Giải thích
Slope = (0 − 6)/(3 − 0) = −6/3 = −2.
-
Given x + 2y = 10 and x = 4, what is y?
- 2
- 3 ✓
- 4
- 5
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
−4x + 20 = 0 → 4x = 20 → x = 5.
-
If 2x + 3y = 18 and x = 3, what is y?
- 3
- 4 ✓
- 5
- 6
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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 ✓
Giải thích
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 ✓
Giải thích
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
Giải thích
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
Giải thích
−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
Giải thích
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
Giải thích
−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
Giải thích
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
Giải thích
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
Giải thích
6x − 12 − 2x − 2 = 4x − 14.
-
Which is equivalent to (5x²)(−2x⁴)?
- −10x⁶ ✓
- −10x⁸
- 3x⁶
- 10x⁶
Giải thích
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)
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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)
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
(x − 5)(x + 2) = x² + 2x − 5x − 10 = x² − 3x − 10 = 0.
-
Simplify: (x³ · x²)/x⁴
- x ✓
- x²
- x³
- x⁴
Giải thích
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
Giải thích
(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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
A = πr² = π(6²) = 36π.
-
A rectangular prism has length 5, width 4, and height 3. What is its volume?
- 40
- 47
- 60 ✓
- 120
Giải thích
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
Giải thích
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 ✓
Giải thích
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 ✓
Giải thích
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
Giải thích
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°
Giải thích
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 ✓
Giải thích
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
Giải thích
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 ✓
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
cos θ = adjacent/hypotenuse = 5/13.
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In a right triangle, sin θ = 7/25. What is cos θ?
- 7/24
- 24/25 ✓
- 25/7
- 7/25
Giải thích
opp = 7, hyp = 25. adj = √(625 − 49) = √576 = 24. cos θ = 24/25.
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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
Giải thích
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
Giải thích
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°
Giải thích
tan θ = opposite/adjacent = 20/20 = 1. θ = arctan(1) = 45°.
-
A circle has radius 9. What is its area?
- 9π
- 18π
- 81π ✓
- 27π
Giải thích
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
Giải thích
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)
Giải thích
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
Giải thích
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° ✓
Giải thích
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)
Giải thích
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)
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
(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
Giải thích
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
Giải thích
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%
Giải thích
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
Giải thích
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
Giải thích
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%
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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
Giải thích
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%
Giải thích
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
Giải thích
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.