SAT Exam 6 (Full Math Module 2)

1. Correct answer: D. 55
   Why D is correct: Set the equations equal:
   x² − 10x + k = 6x − 9
   x² − 16x + (k + 9) = 0
   For the system to have exactly one solution, the discriminant must be 0:
   (−16)² − 4(k + 9) = 0
   256 − 4k − 36 = 0
   220 − 4k = 0
   k = 55
   Why the others are incorrect: They do not make the discriminant equal to 0, so the system would have either two real solutions or no real solutions.
2. Correct answer: C. 24
   Why C is correct: The center is (1, −3), and the line is y = 2, so the vertical distance from the center to the line is 5. With radius 13, half the chord is:
   √(13² − 5²) = √(169 − 25) = √144 = 12
   So the full chord length is 24.
   Why the others are incorrect: 10 and 12 are too short, and 26 would be the diameter only if the line passed through the center.
3. Correct answer: B. 1154
   Why B is correct: Start with:
   x + 1/x = 6
   Square both sides:
   x² + 2 + 1/x² = 36
   x² + 1/x² = 34
   Now square again:
   (x² + 1/x²)² = x⁴ + 2 + 1/x⁴
   34² = x⁴ + 2 + 1/x⁴
   1156 = x⁴ + 2 + 1/x⁴
   x⁴ + 1/x⁴ = 1154
   Why the others are incorrect: They come from arithmetic mistakes or forgetting the middle term.
4. Correct answer: B. 34
   Why B is correct: Since sin A = 8/17, the triangle follows the 8-15-17 ratio. Let the sides be 8k, 15k, and 17k. The area is:
   1/2(8k)(15k) = 240
   60k² = 240
   k² = 4
   k = 2
   So the hypotenuse is 17k = 34.
   Why the others are incorrect: They do not match the scaled 8-15-17 triangle.
5. Correct answer: A. m = 96(0.88)ᵗ
   Why A is correct: A 12% decrease means the sample keeps 88% of its mass each hour, so the multiplier is 0.88. Since the initial mass is 96 grams, the model is m = 96(0.88)ᵗ.
   Why the others are incorrect: B models growth, C uses the wrong initial amount and decay factor, and D is linear instead of exponential.
6. Correct answer: B. 14
   Why B is correct: Factor the numerator:
   x² + 11x + 30 = (x + 5)(x + 6)
   So for x ≠ −6,
   f(x) = (x + 5)(x + 6)/(x + 6) = x + 5
   Then f(9) = 14.
   Why the others are incorrect: They do not come from the simplified expression x + 5.
7. Correct answer: C. 112
   Why C is correct: tan = 7/24 gives a 7-24-25 triangle. Let the legs be 7k and 24k. Then:
   1/2(7k)(24k) = 336
   84k² = 336
   k² = 4
   k = 2
   So the sides are 14, 48, and 50, and the perimeter is 112.
   Why the others are incorrect: They do not match the scaled 7-24-25 triangle.
8. Correct answer: C. 10
   Why C is correct: Set the line and parabola equal:
   x² − 4x + 1 = −2x + 5
   x² − 2x − 4 = 0
   The solutions are x = 1 ± √5. The two intersection points are symmetric, and the distance between them is 10.
   Why the others are incorrect: They come from incomplete or incorrect distance calculations.
9. Correct answer: B. 5/4
   Why B is correct: Rewrite both sides with base 3:
   9⁽ˣ ⁺ ¹⁾ = 3⁽²ˣ ⁺ ²⁾
   27⁽²ˣ ⁻ ¹⁾ = 3⁽⁶ˣ ⁻ ³⁾
   Set exponents equal:
   2x + 2 = 6x − 3
   5 = 4x
   x = 5/4
   Why the others are incorrect: They come from solving the linear equation incorrectly.
10. Correct answer: C. 24
    Why C is correct: The side lengths are x, x + 2, and x + 4, with x + 4 as the hypotenuse. Use the Pythagorean theorem:
    x² + (x + 2)² = (x + 4)²
    x² + x² + 4x + 4 = x² + 8x + 16
    x² − 4x − 12 = 0
    (x − 6)(x + 2) = 0
    So x = 6, and the sides are 6, 8, and 10. The perimeter is 24.
    Why the others are incorrect: They do not match the resulting right triangle.
11. Correct answer: B. 7/2
    Why B is correct: The least value of |2x − 5| + |x + 1| occurs where the expression is minimized. At x = 5/2, the first absolute value becomes 0:
    |2(5/2) − 5| + |5/2 + 1| = 0 + 7/2 = 7/2
    Why the others are incorrect: They are larger than the minimum possible value.
12. Correct answer: C. 2
    Why C is correct: Let f(x) = mx + b. You are given f(3) = 11 and f(f(3)) = f(11) = 35. So the line passes through (3, 11) and (11, 35). The slope is:
    (35 − 11)/(11 − 3) = 24/8 = 3
    So f(x) = 3x + b. Using f(3) = 11:
    11 = 9 + b
    b = 2
    Therefore f(0) = 2.
    Why the others are incorrect: They do not match the linear function defined by the two conditions.
13. Correct answer: B. 56
    Why B is correct: The side lengths are 7, x, and x + 1, with x + 1 as the hypotenuse. Apply the Pythagorean theorem:
    7² + x² = (x + 1)²
    49 + x² = x² + 2x + 1
    48 = 2x
    x = 24
    So the sides are 7, 24, and 25, and the perimeter is 56.
    Why the others are incorrect: They do not equal the perimeter of the triangle.
14. Correct answer: C. 2
    Why C is correct: The graph of ||x − 1| − 2| is W-shaped. The equation ||x − 1| − 2| = k has exactly 3 distinct real solutions when the horizontal line y = k passes through the center point and both outer branches. That happens when k = 2.
    Why the others are incorrect: They do not create exactly 3 distinct intersections.
15. Correct answer: C. 7/2
    Why C is correct: If f(x) = ax + b, then
    f(f(x)) = a(ax + b) + b = a²x + b(a + 1)
    Given f(f(x)) = 9x − 10 and a > 0, we get a² = 9, so a = 3. Then:
    b(3 + 1) = −10
    4b = −10
    b = −5/2
    Now find f(2):
    f(2) = 3(2) − 5/2 = 12/2 − 5/2 = 7/2
    Why the others are incorrect: They do not come from the correct values of a and b.
16. Correct answer: B. 16
    Why B is correct: Set the equations equal:
    x² + (k − 3)x + 4 = 5x + 1
    x² + (k − 8)x + 3 = 0
    For exactly one solution, the discriminant must be 0:
    (k − 8)² − 12 = 0
    So k = 8 ± 2√3. The sum of these two values is 16.
    Why the others are incorrect: They do not equal the sum of the valid k-values.
17. Correct answer: B. 18/7
    Why B is correct: From x + 1/x = 3,
    x² + 1/x² = 3² − 2 = 7
    Also,
    x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x)
    = 27 − 9 = 18
    So
    (x³ + 1/x³)/(x² + 1/x²) = 18/7
    Why the others are incorrect: They come from incorrect identities or arithmetic mistakes.
18. Correct answer: C. 2
    Why C is correct: The equation ||x − 1| − 2| = k has exactly 3 distinct real solutions only when the horizontal line y = k hits the graph at exactly 3 points. That happens at k = 2.
    Why the others are incorrect: They produce fewer or more than 3 real solutions.
19. Correct answer: B. 1
    Why B is correct: Let t = 3ˣ. Then 3⁽⁻ˣ⁾ = 1/t, so
    t + 1/t = 10/3
    Multiply by t:
    t² − (10/3)t + 1 = 0
    Multiply by 3:
    3t² − 10t + 3 = 0
    Factor:
    (3t − 1)(t − 3) = 0
    So t = 1/3 or 3, meaning x = −1 or 1. Therefore x² = 1.
    Why the others are incorrect: They do not match the squared value of the valid solutions.
20. Correct answer: C. 50
    Why C is correct: tan A = 3/4 gives a 3-4-5 triangle. Let the legs be 3m and 4m, so the hypotenuse is 5m. The altitude from the right angle to the hypotenuse is
    (3m)(4m)/(5m) = 12m/5
    Set that equal to 24:
    12m/5 = 24
    12m = 120
    m = 10
    So the hypotenuse is 5m = 50.
    Why the others are incorrect: They do not match the scaled 3-4-5 triangle.
21. Correct answer: C. 12/5
    Why C is correct: A tangent line to x² + y² = 25 must be exactly 5 units from the origin. Rewrite the line as
    kx − y + 13 = 0
    The distance from the origin to this line is
    |13|/√(k² + 1) = 5
    So
    169 = 25(k² + 1)
    169 = 25k² + 25
    144 = 25k²
    k² = 144/25
    Since k > 0, k = 12/5.
    Why the others are incorrect: They do not satisfy the tangent-distance condition.
22. Correct answer: B. x + √x + 1
    Why B is correct: Let t = √x. Then x³ᐟ² = t³, so the expression becomes
    (t³ − 1)/(t − 1)
    Factor the numerator:
    t³ − 1 = (t − 1)(t² + t + 1)
    So the expression simplifies to
    t² + t + 1
    Substitute back t = √x:
    x + √x + 1
    Why the others are incorrect: They are incomplete or incorrect simplifications of the difference of cubes.

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