This module contains 16 multiple-choice questions covering algebra, advanced math, problem-solving & data analysis, and geometry/trigonometry, written at an elevated difficulty level. Questions are not in a fixed order — scroll to preview all questions and answer them in any order you like using the navigator below.
Factored, f(x) = (x + 4)(x − 1)². Setting each factor equal to 0 gives x = −4 (a simple root, where the graph crosses the x-axis) and x = 1 (a double root, where the graph only touches the x-axis). Even though x = 1 is a repeated root, it is still just one distinct x-value. So f(x) = 0 at exactly two distinct values of x.
The equation is x(x − 27) = 1,540, which expands to x² − 27x − 1,540 = 0. This factors as (x − 55)(x + 28) = 0, giving x = 55 or x = −28. Since x must be positive, x = 55. (Check: 55 × (55 − 27) = 55 × 28 = 1,540.)
Angle ACD and angle ACB form a linear pair, so angle ACB = 180° − 152° = 28°. Since AB = AC, triangle ABC is isosceles with base angles ABC and ACB equal, so angle ABC = 28° as well. The angles of triangle ABC sum to 180°, so angle BAC = 180° − 28° − 28° = 124°. (Check with the exterior angle theorem: exterior angle ACD = sum of remote interior angles = 28° + 124° = 152°. ✓)
From the dot plot, set A has mean = (3·2 + 6·3 + 9·4 + 12·3 + 15·2) ÷ 14 = 126 ÷ 14 = 9 cm, and range = 15 − 3 = 12 cm. Adding a constant (23 cm) to every value shifts the mean by that same constant: new mean = 9 + 23 = 32 cm. However, adding the same constant to every value does not change the spread of the data, so the range stays 12 cm.
Line j has slope −5/6. Since line k is parallel to line j, line k also has slope −5/6. Writing line k's equation in slope form: y = −(20/r)x + 48/r, so its slope is −20/r. Setting −20/r = −5/6 gives r = 24. Substituting r = 24 into line k's equation: 20x + 24y = 48. At the y-intercept, x = 0, so 24y = 48, giving y = b = 2.
"a is 3,600% of c" means a = 36c. "c is 20% of b" means c = 0.2b, so b = c / 0.2 = 5c. Substituting into a − b = wc: 36c − 5c = wc, so 31c = wc, giving w = 31.
Subtracting a constant (14) from every y-value shifts the entire line of best fit down by 14, without changing its slope. The new y-intercept is 58 − 14 = 44, so the line of best fit for set B is y = −0.9x + 44.
Density = mass ÷ volume = 741 g ÷ 39 cm³ = 19 g/cm³.
Substitute y = 5x + 12 into 4y = 9x − 18: 4(5x + 12) = 9x − 18 → 20x + 48 = 9x − 18 → 11x = −66 → x = −6. Then y = 5(−6) + 12 = −18. So x − y = −6 − (−18) = 12.
f(8) = c⁸ and f(4) = c⁴, so the equation is c⁸ = 16c⁴. Dividing both sides by c⁴ (c ≠ 0): c⁴ = 16, so c = ⁴√16 = 2 (taking the positive root, since c > 1).
Substitute y = 19 into both equations: −x − 19w = −300, so x = 300 − 19w. Also 3x − 19w = −12, so x = (−12 + 19w) / 3. Setting these equal: 300 − 19w = (−12 + 19w)/3 → 900 − 57w = −12 + 19w → 912 = 76w → w = 12. (This gives x = g = 300 − 19(12) = 72, which checks in both equations.)
"When the cup was placed on the table" means t = 0. At t = 0, the exponent −0.083(0) = 0, so (1.8)⁰ = 1. Then h(0) = 410 + (520 − 410)(1) = 410 + 110 = 520 kelvins.
We want the form where plugging in x = 1 makes the exponent equal to 0, so that f(1) simply equals the coefficient out front. That happens when the exponent is (x − 1), since (1 − 1) = 0: f(1) = 100(2.5)⁰ = 100 = k. In every other form the exponent isn't 0 at x = 1, so the coefficient doesn't directly equal k. (All four forms are algebraically equivalent — you can check that each equals 100(2.5)ˣ⁻¹ once simplified.)
Because a > 4, the base a/4 is greater than 1, so (a/4)x+c is a strictly increasing exponential function whose output is always positive, ranging continuously from 0 (as x → −∞) to +∞ (as x → +∞). Multiplying by 7 keeps it positive and increasing, so 7(a/4)x+c ranges over (0, ∞). Subtracting the positive constant b shifts this range to (−b, ∞). Since b > 0, this interval contains 0, and because the function is continuous and strictly increasing, it crosses y = 0 at exactly one point. (The condition b > 2c isn't actually needed to answer the question — it's extra information.)
Isolate xyz: xyz = 1/(8yz) − 1/(3xy). Combine the right side over a common denominator of 24xyz: 1/(8yz) = 3x/(24xyz) and 1/(3xy) = 8z/(24xyz), so xyz = (3x − 8z)/(24xyz). Multiply both sides by 24xyz: 24x²y²z² = 3x − 8z. Divide by 24x²z²: y² = (3x − 8z)/(24x²z²). Since y is positive, take the positive square root: y = √[(3x − 8z)/(24x²z²)].
| x | g(x) |
|---|---|
| −25 | 3 |
| −5 | 0 |
| 10 | 3 |
Since g(−5) = 0, f(−5)/(−5+5) would be undefined unless f(−5) = 0 — so f has a root at x = −5, meaning f(x) = m(x + 5) for some slope m. Using g(−25) = 3: f(−25)/(−25+5) = 3 → f(−25) = 3(−20) = −60. Also f(−25) = m(−25+5) = −20m, so −20m = −60, giving m = 3. So f(x) = 3(x + 5) = 3x + 15. (Check with x = 10: f(10) = 3(15) = 45, and g(10) = 45/15 = 3 ✓.) The y-intercept of y = f(x) is at x = 0: f(0) = 15, so the point is (0, 15).
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