maddyhelps

Math 30-1 · Worksheets

Diploma-style challenge

Ten questions of mixed difficulty, covering Counting, Exponents and logs, Sinusoidal models, Transformations, Trig identities, Trigonometry. Print it, or work through it on screen — the answer key starts on its own page.

All worksheets

Diploma-style challenge

Math 30-1 · maddyhelps.com

Name
Date
Score
/ 10

Circle the best answer for each question. Show your work in the space provided.

  1. What is the non-permissible value of sin x ÷ (1 − cos x) for 0 ≤ x < 2π?

    1. a) x = π/2 and x = 3π/2
    2. b) x = 0
    3. c) x = π
    4. d) there is none

  2. A committee of 3 is chosen from 8 people, and one of the three is then named chair. How many outcomes are possible?

    1. a) 168
    2. b) 336
    3. c) 512
    4. d) 56

  3. How many different 5-card hands can be dealt from a standard 52-card deck?

    1. a) 270 725
    2. b) 22 100
    3. c) 311 875 200
    4. d) 2 598 960

  4. A rider's height is h(t) = −10 cos(πt/20) + 12, in metres after t seconds. When does the rider first reach the top?

    1. a) 40 seconds
    2. b) 10 seconds
    3. c) 12 seconds
    4. d) 20 seconds

  5. $1000 is invested at 6% interest, compounded annually. To the nearest tenth, how many years until it doubles?

    1. a) 33.3
    2. b) 11.9
    3. c) 16.7
    4. d) 12.5

  6. Evaluate log₅ 1.

    1. a) 5
    2. b) 0
    3. c) 1
    4. d) undefined

  7. What is the exact value of cos(4π/3)?

    1. a) −1/2
    2. b) −√3/2
    3. c) √3/2
    4. d) 1/2

  8. Using the letters A, B, C, D, E exactly once each, how many arrangements have no two vowels next to each other?

    1. a) 48
    2. b) 36
    3. c) 120
    4. d) 72

  9. The graph of y = f(x) is translated 3 units to the right. What is the new equation?

    1. a) y = f(x − 3)
    2. b) y = f(x) + 3
    3. c) y = f(x) − 3
    4. d) y = f(x + 3)

  10. How could the domain of f(x) = x² be restricted so that its inverse is also a function?

    1. a) y ≥ 0
    2. b) no restriction is needed
    3. c) x ≥ 1
    4. d) x ≥ 0

Answer key · Diploma-style challenge

Math 30-1 · maddyhelps.com

  1. b) x = 0 — The denominator is zero when cos x = 1, which happens at x = 0 in this domain. In general, x ≠ 2πn.
  2. a) 168 — Choose the group, then pick the chair: ₈C₃ × 3 = 56 × 3 = 168. You can also pick the chair first: 8 × ₇C₂ = 8 × 21 = 168.
  3. d) 2 598 960 — A hand is a group, not an order: ₅₂C₅ = 2 598 960. The larger number 311 875 200 is ₅₂P₅, which counts the same hand in every dealing order.
  4. d) 20 seconds — The top is where −10 cos(πt/20) is largest, so cos(πt/20) = −1, which first happens when πt/20 = π. That gives t = 20, halfway through a 40-second rotation.
  5. b) 11.9 — Set 2000 = 1000(1.06)ⁿ, so 2 = 1.06ⁿ and n = log 2 ÷ log 1.06 ≈ 11.9. 16.7 comes from 100 ÷ 6, which ignores compounding.
  6. b) 0 — Any base to the power 0 is 1, so log of 1 is always 0. It is log₅ 5 that equals 1.
  7. a) −1/2 — 4π/3 is in Quadrant III with reference angle π/3. cos(π/3) = 1/2, and cosine is negative in Quadrant III, so −1/2.
  8. d) 72 — Place the 3 consonants first: 3! = 6. That makes 4 gaps; choose 2 for the vowels and order them: ₄C₂ × 2! = 12. Then 6 × 12 = 72.
  9. a) y = f(x − 3) — Horizontal shifts act on x and go the opposite way to the sign: x − 3 moves the graph right. Adding 3 outside the function would move it up.
  10. d) x ≥ 0 — The inverse of the whole parabola fails the vertical line test. Keeping only one half, such as x ≥ 0, makes each output come from one input. y ≥ 0 restricts the range, not the domain.