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SAT Comprehensive Practice Exam With Each Question Including The Correct Answer And A Detailed Rationale

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Part 2: Math (Questions 41-100) 41. If a − b = 3 and a + b = 4, what is the value of (a − b)(a² − b²)? A) 12 B) 24 C) 36 D) 48 Correct Answer: C *Rationale: * *First, factor a² − b² as a difference of two squares: (a + b)(a − b). Then (a − b)(a² − b²) = (a − b)(a + b)(a − b) = (a − b)²(a + b). Substitute the given values: (3)² × 4 = 9 × 4 = 36 .* 42. x² - 9 / x - 3 = -1 What is the solution set of the equation above? A) {−4} B) {0} C) {3} D) {3, 4} Correct Answer: A *Rationale: * *Multiply both sides by (x − 3): x² − 9 = −1(x − 3). Simplify: x² − 9 = −x + 3. Rearrange: x² + x − 12 = 0. Factor: (x + 4)(x − 3) = 0, so x = −4 or x = 3. Check for extraneous solutions: x = 3 makes the denominator zero, so it's invalid. Therefore, the solution set is {−4} .* 43. The bar graph shows the number of points scored by a football team in each of four games: Game 1: 14, Game 2: 9, Game 3: 23, Game 4: 22. For these four games, how much greater is the median number of points scored than the mean number of points scored? A) 1 B) 2 C) 3 D) 4 Correct Answer: A *Rationale: * *Order the values: 9, 14, 22, 23. Median = (14 + 22)/2 = 18. Mean = (14 + 9 + 23 + 22)/4 = 68/4 = 17. The median (18) is 1 greater than the mean (17) .* 44. An equation of a circle in the xy-plane with radius 2 and center at (5, −4) is x² + y² + ax + by = c. What is the value of c? A) −37 B) −25 C) 25 D) 37 Correct Answer: A *Rationale: * *Start with standard form: (x − 5)² + (y + 4)² = 4. Expand: x² − 10x + 25 + y² + 8y + 16 = 4. Rearrange: x² + y² − 10x + 8y + 41 = 4. Subtract 41: x² + y² − 10x + 8y = −37. Therefore, c = −37 .* 45. If the minimum value of r is 16 less than 8 times the value of x, which inequality shows the possible values of r? A) r ≤ 8x − 16 B) r ≥ 8x − 16 C) r ≤ 16 − 8x D) r ≥ 16 − 8x Correct Answer: B *Rationale: * "16 less than 8 times x" translates to 8x − 16. The phrase "minimum value of r is" means that r must be greater than or equal to that expression: r ≥ 8x − 16 . 46. A soda machine charges $0.75 for a bottle of water and $1.25 for a bottle of soda. If in one day it sells 40 items and collects $40, how many bottles of soda did it sell? A) 15 B) 20 C) 25 D) 30 Correct Answer: B *Rationale: * *Let s = soda, w = water. Two equations: s + w = 40 and 1.25s + 0.75w = 40. Multiply the first equation by 0.75: 0.75s + 0.75w = 30. Subtract from the second equation: (1.25s − 0.75s) + (0.75w − 0.75w) = 40 − 30 → 0.5s = 10 → s = 20 .* 47. Which of the following systems of linear equations has no solution? A) x = −1, y = −1 B) y = 5 − 2x, y = −3 C) y = −3x, y = 5 − 3x D) y = 1 − 4x, y = 1 + 4x Correct Answer: C *Rationale: * *A system has no solution when the lines are parallel (same slope, different y-intercepts). In Choice C, both equations have slope −3, but y-intercepts are 0 and 5, making them parallel with no intersection .* 48. Last week, a nonprofit organization recorded 1,980 items used for meal packs. The equation 4x + 10y = 1,980 represents this situation, where x is the number of snack packs and y is the number of dinner packs. Which of the following is the best interpretation of 10 in this context? A) There were 10 total items used in snack packs. B) There were 10 items used per snack pack. C) There were 10 total items used in dinner packs. D) There were 10 items used per dinner pack. Correct Answer: D *Rationale:

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