SAT Exam 5 (Full Math Module 2)

1. Correct answer: B. 120/13
   Why: In a right triangle, the altitude to the hypotenuse is h = ab/c. The legs are 10 and 24, and the hypotenuse is √(10² + 24²) = 26. So BD = (10)(24)/26 = 120/13.
2. Correct answer: C. 13
   Why: Set the equations equal: x² – 6x + k = 2x – 3, so x² – 8x + (k + 3) = 0. For exactly one solution, the discriminant must be 0. That gives 64 – 4(k + 3) = 0, so k = 13.
3. Correct answer: C. y = 6x + 24
   Why: Multiplying every y-value by 4 multiplies both the slope and the y-intercept by 4. Starting from y = 1.5x + 6 gives y = 6x + 24.
4. Correct answer: C. 6
   Why: The center is (2, -1), the radius is 5, and the line x = 6 is 4 units from the center horizontally. Half the chord is √(5² – 4²) = 3, so the full chord is 6.
5. Correct answer: B. 25
   Why: Rewrite x + m/x = 10 as x² – 10x + m = 0. For exactly one positive solution, the discriminant must be 0. So 100 – 4m = 0, which gives m = 25.
6. Correct answer: C. 480
   Why: Since sin P = 5/13, the triangle follows the 5-12-13 ratio. If QR = 20, then 5k = 20, so k = 4. The other leg is 12k = 48. Area = 1/2(20)(48) = 480.
7. Correct answer: C. 198
   Why: Use the identity (x + 1/x)³ = x³ + 1/x³ + 3(x + 1/x). Substituting x + 1/x = 6 gives 216 = x³ + 1/x³ + 18, so x³ + 1/x³ = 198.
8. Correct answer: 10
   Why: f(x) = (x² – 9)/(x – 3) = (x – 3)(x + 3)/(x – 3) = x + 3 for x ≠ 3. So f(7) = 10.
   Note: The original answer choices were broken.
9. Correct answer: 8
   Why: For infinitely many solutions, the equations must be the same line. Multiply x + 2y = 4 by 4 to get 4x + 8y = 16, so k = 8.
   Note: The original keyed choice was broken.
10. Correct answer: -1
    Why: The slope through (-2, 5) and (4, -1) is (-1 – 5) / (4 – (-2)) = -6/6 = -1.
    Note: The original answer choices were broken.
11. Correct answer: C. 45π
    Why: Volume of a cylinder is V = πr²h. With r = 3 and h = 5, V = π(9)(5) = 45π.
12. Correct answer: 73
    Why: x² + y² = (x + y)² – 2xy = 11² – 2(24) = 121 – 48 = 73.
    Note: The original answer choices were broken.
13. Correct answer: -10
    Why: A quadratic with x-intercepts -1 and 5 and leading coefficient 2 is y = 2(x + 1)(x – 5). At x = 0, y = 2(1)(-5) = -10.
    Note: The original answer choices were broken.
14. Correct answer: 162.24
    Why: The original 9-12-15 triangle has area 54. The new scale factor is 26/15, so the new area is 54(26/15)² = 162.24.
    Note: The original answer choices were broken.
15. Correct answer: A. 194
    Why: If x + 1/x = 4, then x² + 1/x² = 14. Squaring again gives x⁴ + 1/x⁴ = 14² – 2 = 194.
16. Correct answer: B. 1
    Why: Set the equations equal: x² + (k – 4)x + 9 = 5x – 7, so x² + (k – 9)x + 16 = 0. For exactly one solution, the discriminant must be 0: (k – 9)² = 64. So k = 17 or 1, and since k < 10, k = 1.
17. Correct answer: C. 24
    Why: The center is (3, -4), the radius is 13, and the line y = 1 is 5 units above the center. Half the chord is √(13² – 5²) = 12, so the full chord is 24.
18. Correct answer: D. -24
    Why: Rewrite the first equation into slope-intercept form to get y = 3/8 x – 3/28. Rewrite the second as y = 3/(14r) – 9/r x. For no solution, the slopes must be equal: -9/r = 3/8. Solving gives r = -24.
19. Correct answer: B. 26√2
    Why: Since sin A = 5/13, the triangle is in a 5-12-13 ratio. Let the sides be 5k, 12k, and 13k. Area = 1/2(5k)(12k) = 30k² = 240, so k² = 8 and k = 2√2. The hypotenuse is 13k = 26√2.
20. Correct answer: C. 36
    Why: Use the identity (x – 1/x)³ = x³ – 1/x³ – 3(x – 1/x). Substituting x – 1/x = 3 gives 27 = x³ – 1/x³ – 9, so x³ – 1/x³ = 36.
21. Correct answer: D. 40
    Why: tan = 8/15 means the legs are in an 8-15 ratio, which is the standard 8-15-17 right triangle. The perimeter is 8 + 15 + 17 = 40.
22. Correct answer: B. 4/5
    Why: Use the altitude relation CD² = AD × DB. So 12² = 9(DB), which gives DB = 16. Then AB = 9 + 16 = 25. Next, BC² = DB × AB = 16 × 25 = 400, so BC = 20. Therefore sin A = BC/AB = 20/25 = 4/5.

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